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ORCiD: 0000-0003-1871-7803
February 10, 2025
Within Extended Classical Mechanics (ECM), photon dynamics describes dark energy by positing that photons, due to their unique properties within the framework, can exhibit a "negative apparent mass," causing them to effectively repel each other and contribute to the observed accelerating expansion of the universe, which is the primary characteristic of dark energy; this negative mass arises from the complex interaction of photon momentum and energy within the ECM equations, leading to an "effective acceleration" that counteracts gravitational pull.
Photon Dynamics and Dark Energy in the Framework of Extended Classical Mechanics (ECM)
In the framework of Extended Classical Mechanics (ECM), photon dynamics and dark energy are intricately linked through the concepts of effective mass (Mᵉᶠᶠ) and apparent mass (Mᵃᵖᵖ). This framework provides a novel perspective on how gravitational interactions can induce mass in initially massless particles, such as photons, and how these interactions relate to the observed phenomena of dark energy.
Photon Dynamics and Effective Mass
Effective Mass and Apparent Mass:
In ECM, the effective mass (Mᵉᶠᶠ) of a photon is a dynamic property that combines the rest mass (Mᴍ) and the apparent mass (Mᵃᵖᵖ). For photons, which have zero rest mass, their apparent mass dictates their energy-momentum exchanges and response to forces. This leads to the reformulated force equation:
Fₚₕₒₜₒₙ =−Mᵃᵖᵖ aᵉᶠᶠ
The apparent mass (Mᵃᵖᵖ) can be negative, which is crucial for understanding antigravitational effects and dark energy.
Gravitational Redshift and Photon Energy:
The total energy of a photon is analysed as the sum of its inherent energy (E) and gravitational interaction energy (Eg). As photons escape a gravitational field, they retain their inherent energy while gradually expending their gravitational energy. This leads to gravitational redshift, where the photon's frequency shifts due to the gravitational potential.
Dark Energy and Negative Effective Mass
Dark Energy as a Gravitational Interaction:
In ECM, dark energy is not treated as a conventional field or particle but as a gravitationally interactive background that influences mass distributions at intergalactic scales. It acts on cosmic scales by modifying the gravitational potential, leading to the observed cosmic acceleration.
Negative Effective Mass and Antigravitational Effects:
The negative effective mass (Mᵉᶠᶠ<0) is a key feature of ECM, particularly in the context of dark energy. This negative mass can lead to antigravitational effects, where objects experience repulsion rather than attraction. This phenomenon echoes the behaviour of dark energy, which accelerates the universe's expansion by generating antigravitational effects.
Gravitational Mass and Dark Energy:
The gravitational mass (Mg) in ECM is given by:
Mɢ = Mᴍ + (-Mᵃᵖᵖ)
At intergalactic scales, the interaction of dark matter with dark energy results in an effective mass contribution (Mᴅᴇ), which is represented by:
Mɢ = Mᴍ + Mᴅᴇ
This additional inferred mass component (Mᴅᴇ) is an emergent gravitational effect, not a fundamental mass term.
Implications for Photon Dynamics and Dark Energy
Unified Framework:
ECM provides a unified framework that bridges classical mechanics, quantum principles, and cosmological implications. By incorporating the concept of apparent mass, ECM offers a cohesive mechanism to reconcile classical, quantum, and cosmological phenomena.
Cosmic Acceleration:
The negative effective mass associated with dark energy explains the observed cosmic acceleration. This antigravitational effect is crucial for understanding the expansion of the universe and the role of dark energy in shaping cosmic dynamics.
Gravitational Collapse at the Planck Scale:
At the Planck scale, gravitational interactions can induce mass in massless particles, leading to gravitational collapse. This transition from massless to massive states is a direct consequence of ECM's mass induction principle, where increasing energy (via frequency) leads to mass acquisition.
Conclusion
The framework of Extended Classical Mechanics (ECM) offers a detailed and nuanced understanding of photon dynamics and dark energy. By incorporating the concepts of effective mass and apparent mass, ECM provides a unified perspective on gravitational interactions across quantum and cosmological scales. This approach not only aligns with fundamental principles but also offers potential explanations for cosmic-scale phenomena involving dark matter, dark energy, and exotic gravitational effects.
The assertion that "antigravity is an unlikely phenomenon" is inconsistent with established observations. The behaviour of photons provides direct evidence to the contrary. According to its energy, a photon possesses effective mass and is observed to escape gravitational wells, demonstrating a counteracting effect against gravity. Extended Classical Mechanics (ECM), a framework built upon classical mechanics principles, provides a clear formulation of this phenomenon. ECM reveals that photons exert an antigravitational force on massive bodies, accelerating at twice the speed of light within gravitational influence. As a photon leaves a gravitational well, it expends energy but retains its inherent energy, continuing to travel at the speed of light in free space. This indicates that antigravitational effects are an intrinsic feature of certain mass-energy interactions, contradicting the claim that antigravity is unlikely.
The notion that "dark energy is not real" is only partially correct. While dark energy is not a physical object with rest mass, its effects are observable. ECM equations establish dark energy as a form of potential energy with a dynamic nature, existing only as a consequence of gravitational and motion dynamics of massive bodies. Rather than being an independent entity, dark energy emerges from the interplay of gravity and motion at cosmic scales, reinforcing its role in large-scale universal dynamics.
Similarly, the claim that "negative mass lacks a physical description and remains unproven" overlooks key insights provided by ECM. Rather than considering negative mass as a standalone entity, ECM introduces the concept of negative apparent mass, which arises from motion and gravitational interactions. This phenomenon does not imply an intrinsic negative mass but rather an emergent property influenced by both baryonic matter and dark matter. ECM principles illustrate how apparent mass contributes to gravitational effects, expanding the understanding of mass-energy interactions beyond conventional classical mechanics.
These refinements in ECM extend classical mechanics while maintaining consistency with empirical observations, providing a structured approach to understanding gravitational repulsion, dark energy, and the role of apparent mass in astrophysical phenomena.
We all know that energy density entanglements undoubtedly exist since they are measured almost daily.
We also know that all the giant scientists Einstein, Schrödinger, Minkowski, Hilbert, Hamilton... . etc. didn't understand the physics of entanglement.
We assume that the reason is that the entanglement is related to the correct definition of time, which is more complex and hidden than the entanglement itself.
In the following, some true but simplified quotes about entanglement:
1- Most of the time, everything gets tangled up with everything.
2-The true geometry of nature implies that
The problem of time and entanglement is a conceptual conflict between quantum mechanics and general relativity.
3-Quantum mechanics considers the flow of time as universal and absolute, while general relativity considers the flow of time as malleable and relative.
4-why time seems to flow in only one direction, (Arrow of time)
5- In classical mechanics and quantum mechanics, time has a separate status in the sense that it is treated as a classical background parameter, external to the system itself.
6-In Copenhagen interpretation of quantum mechanics: all measurements of observables are made at certain instants of time and probabilities are assigned only to these measurements. Additionally, the Hilbert space used in quantum theory relies on a complete set of observables that commute at a specific time.
7-What is the entanglement theory of time?
Work started by Don Page and William Wootters suggests that the universe appears to evolve for observers on the inside because of energy entanglement between an evolving system and a clock system, both within the universe. In this way the overall system can remain timeless while parts experience time via entanglement.
8- It is possible to go to quantum mechanics from statistical mechanics by PROJECTION OF CLASSICAL STATISTICAL MECHANICS ONTO PRESPACE.
This means in that 4D statistical classical physical space is equivalent to 3D quantum space and so on.
9- etc .. .etc.
*Looking forward to responses and comments from our respectable readers and contributors, please focus on a few ideas for a thorough break.
February 07, 2025
The foundation of Extended Classical Mechanics (ECM) is constructed upon classical mechanics principles, as formulated by Newton, Lagrange, and Hamilton, yet it aims to transcend the limitations encountered at quantum scales, relativistic speeds, and in complex systems.
A central innovation within ECM is the introduction of the concepts of apparent mass (Mᵃᵖᵖ) and effective mass (Mᵉᶠᶠ). These constructs extend the traditional framework to incorporate the effects of dark matter and dark energy, offering a more comprehensive understanding of gravitational dynamics.
The concept of apparent mass (Mᵃᵖᵖ) is established in classical mechanics, specifically through the fundamental relationship between force, mass, and acceleration (F = ma). However, it also integrates observational evidence from phenomena like dark energy, bridging classical principles with contemporary cosmological insights.
Extended classical mechanics offers a unified perspective on photon dynamics. It synthesizes classical principles with modern observations, emphasizing the conservation of photon energy (E) and the symmetry of gravitational interactions (Eg). This approach posits that photons maintain their intrinsic energy (E) while interacting with gravitational fields, dynamically exchanging gravitational interactional energy (Eg) during their trajectories.
In summary, ECM weaves together classical mechanics with modern astrophysical phenomena through the constructs of Mᵃᵖᵖ and Mᵉᶠᶠ. This cohesive model not only respects the heritage of classical mechanics but also embraces the complexities revealed by modern science, offering new avenues for exploring the cosmos.
The concept of negative apparent mass in extended classical mechanics is a groundbreaking innovation. It marks a turning point in classical mechanics, introducing negative mass and expanding its capabilities beyond traditional frameworks. This extension enhances classical mechanics, making it more powerful than relativistic mechanics.
Furthermore, velocity-induced relativistic Lorentz's transformations are flawed because they neglect classical acceleration between the rest and moving frames. They also overlook material stiffness in calculations, relying solely on the speed of light as the defining dynamic factor. For these reasons, extended classical mechanics stands as a far superior framework compared to the flawed foundations of relativistic mechanics.
Effective Mass and Acceleration Implications of Negative Apparent Mass in Extended Classical Mechanics (ECM):
Newton's Second Law and Acceleration:
In classical mechanics, Newton's second law is typically expressed as:
F = ma
This shows that force (F) is directly proportional to acceleration (a) and mass (m).
As force F increases, acceleration a increases proportionally. However, the relationship a ∝ 1/m means that if mass m increases, acceleration a will decrease, assuming force is constant.
In this framework, if acceleration increases while force increases, it suggests that mass must decrease to maintain the inverse relationship between acceleration and mass.
Apparent Mass and Effective Mass in ECM:
In Extended Classical Mechanics (ECM), this relationship is reflected in the equation:
F = (Mᴍ − Mᵃᵖᵖ) aᵉᶠᶠ
The term (Mᴍ − Mᵃᵖᵖ) implies that the effective mass is the difference between matter mass and apparent mass, which is a dynamic concept.
Apparent mass reduction:
If the apparent mass Mᵃᵖᵖ decreases (or becomes negative), this results in an increase in effective mass, which in turn causes an increase in acceleration a when the force F remains constant.
Thus, in ECM, a reduction in apparent mass leads to a corresponding increase in acceleration, aligning with the inverse relationship a ∝ 1/m, where m is the effective mass. This supports the idea that acceleration can increase without an actual increase in matter mass Mᴍ but rather a reduction in apparent mass Mᵃᵖᵖ.
Supporting Observational Findings:
The expression Mᵉᶠᶠ = Mᴍ + Mᴅᴇ, where Mᴅᴇ is negative, aligns with this reasoning. If the apparent mass Mᵃᵖᵖ (which could be represented Mᴅᴇ in this framework) is negative, the effective mass becomes:
Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ)
This negative apparent mass Mᵃᵖᵖ or, effective mass of dark energy (Mᴅᴇ), reduces the total effective mass, causing an increase in acceleration when force is applied, consistent with the relationship a ∝1/m.
Conclusion:
In this framework, the concept of effective mass Mᵉᶠᶠ is key to understanding how acceleration behaves when apparent mass changes. When apparent mass decreases (or becomes negative), the effective mass also decreases, leading to an increase in acceleration. This theory not only aligns with the classical force-acceleration-mass relationship but also supports observational findings, particularly the role of negative apparent mass in cosmological models or exotic gravitational effects.
It is predicted but was never experimentally confirmed because technical limitations but also physical observation limitation of HUP (Heisenberg Uncertainty Principle) that an electron at rest has an intrinsically caused jittery motion similar as an analogy, to the Brownian vibration motion observed on atoms. Although the Brownian motion of atoms we must differentiate here, in a fluid for example, is not caused intrinsically inside the atom, but rather due to extrinsic causes like different atoms bumping on each other all the time.
Nevertheless, the Zitterbewengung (German term for jittery vibrational motion) of the electron is predicted to be twice the electron's Compton frequency fc :
fc ~ 127 EHz (i.e. one ExaHertz [EHz] unit is 1018 Hz)
Thus,
fZ = 2fc~254 EHz
which is an enormous high frequency vibration.
However, if the cause of the electron's Zitterbewengung at rest, is to be attributed to the unknown possible intrinsic mechanics of the electron then this also possible mean, and also predicted by de Broglie theory about every subatomic particle, that a kind of intrinsic harmonic energy oscillation exists for the electron that is the classical cause for the electron Zitterbewengung.
My question here is and discussion topic, what do you think is the physical reason (please do not use just equations to explain or abstract effective terminology) this frequency to be twice the Compton frequency?
I found this question always to be a big mystery which however if answered could be the key for deciphering the electron.
Emmanouil
p.s. In my "Intrinsic Mechanics" for example model of the electron, the electron undergoes in superposition two overlapping different harmonic energy oscillations at a 90º angle. One latitudinal and one longitudinal harmonic oscillation of its energy string :
see animation, https://www.horntorus.com/particle-model/e_spin-up.html
This combinatoric motion is of two harmonic oscillations is in my theory and model, is the reason why the zitter vibration of the electron is twice its Compton frequency.
Relater paper of my model:
Quantum mechanics is 9ne of the most succesfull theories (if not the most, empirical) yet at its roots ontological vaqueness thrives.
In QM, physicists are not sticking to the standards they set in classical mechanics: clear ontological status.
Here we had better at least be clear whether we are talking about mental states, or physical ones out there in the world, and whether the theory implies that there uncountably many other versions of our everyday reality out there, or not.
Few however believe that remaining ambivalent about this kind of thing is just not acceptable.
"Note that this is not a choice imposed on us by physics. It is a choice that we can make depending on what kind of thing we think science should be".
The obove is the view of some such as Paul Msibwood, Ph. D Philosophy of Physics, that this approach is the only defensible option.
He thinks that exploring the basic ontology of the theory and its implications is something rich and valuable to the progress of science, and more generally, a real addition to human knowledge.
Do you agree?
The question is if the Relativistic Kinetic Energy formula:
KE_rel = (gamma -1) mc**2
Has it been verified using thermal calorimetry?
SINCE I WROTE THIS QUESTION, I FOUND OUT ABOUT THE BERTOZZI EXPERIMENT ON RELATIVISTIC ELECTRONS. THAT SETTLES THE ISSUE OF RELATIVISTIC KINETIC ENERGY. THAT IS THE CORRECT ANSWER.
MY THEORY PREDICTS WEAKENED FORCES AT HIGH ABSOLUTE VELOCITIES. THAT IS CONSISTENT WITH OBSERVATIONS AND ANATOLI BUGORSKI ACCIDENT.
READ ABOUT THE HYPERGEOMETRICAL UNIVERSE THEORY HERE:

2) How is the formation of the universe?
The universe, at its most fundamental level, appears to operate according to the principles of quantum mechanics, where uncertainty and indeterminacy play key roles in shaping its evolution. In classical computational theory, Turing’s Halting Problem demonstrates that it is impossible to predict whether a system will reach a final state or run indefinitely. This raises profound questions about the nature of the universe: could it, too, one day halt, reaching a state where no further evolution is possible? However, the inherent unpredictability of quantum mechanics—through phenomena like superposition, quantum fluctuations, and entanglement—may offer a safeguard against such a scenario. This paper explores the intersection of quantum mechanics and the Halting Problem, suggesting that quantum uncertainty prevents the universe from settling into a static, final state. By continuously introducing randomness and variation into the fabric of reality, quantum processes ensure the universe remains in perpetual motion, avoiding a halting condition. We will examine the scientific and philosophical implications of this theory and its potential to reshape our understanding of cosmology.
Stam Nicolis added a reply:
The evolution of the universe, from the inflationary epoch onwards, is described by classical, not quantum, gravity.
Stam Nicolis added a reply:
Turing's halting problem doesn't have anything to do with the subject of cosmology, or any subject, where the equations that describe the evolution of the system under study are known.
In particular the answer to the question of the evolution of the universe is known: It's described by the de Sitter solution to Einstein's equations, that is its expansion is accelerating, although with a very slow rate. The question, whose answer isn't, yet, known is what happened before the inflationary epoch. It is for this question that a new theory is needed, that can match to classical description of spacetime and the quantum description of matter that emerged from it.
Stam Nicolis added a reply:
That quantum mechanics provides a probabilistic description isn't particular to it. Classical mechanics, also provides a probabilistic description, since classical systems are, typically, chaotic and integrable systems are the exception, not the rule. The only difference between a quantum system and its classical limit is the space of states.
This is a simple proof the guitar is Hamiltonian. Then by deconstruction so is string vibration because the string is the smallest open set on guitar.
The time-independent Hamiltonian has the form H(p, q) = c and dH/dt = 0.
All I need is to define p and q.
So p will be the center of harmonic motion, and q will be a potential energy gradient that reads off the differential between any two points.
Consider the set of notes for the guitar tuning known as standard: E A D G B E.
The tuning naturally separates into two vectors in this way: Indexing the tuning notes by counting up from the low E the pitch values are equivalent to p: 0 5 10 15 19 24.
Now taking the intervals between the notes we have a second vector q: 0 5 5 5 5 4.
It is important to notice that tuning vectors p and q are equal, opposite, and inverse, which is expected since the orbit and center have this relation in the Hamiltonian.
For instance, p is the summation of q and q is the differential of p. The points in p and the intervals in q together make a unit interval in R.
Most important, p = 1/q means the tuning is the identity of the guitar. If you know the tuning, you know everything (all movement). You can learn guitar without learning anything but the tuning.
The proof the vectors are Hamiltonian is this, p is the center of motion in R6, and q is the gradient of the potential field surface in R5 where every vibrational state is presented by a single point.
The coordinates of notes on guitar chord charts given by the gradient function
form a union as a smooth atlas.
Therefore, it must be true the guitar is Hamiltonian. How else could the symplectic manifold be smooth?
Physicists and mathematicians have no choice but to accept that one degree of freedom is better than two. The fact that they cannot see it implies an illness of the public mind that cannot think straight about classical mechanics.
A minion is a low-level official protecting a bureaucracy form challengers.
A Kuhnian minion (after Thomas Kuhn's Structure of Scientific Revolutions) is a low-power scientist who dismisses any challenge to existing paradigm.
A paradigm is a truth structure that partitions scientific statement as true to the paradigm or false.
Recently, I posted a question on Physics Stack Exchange that serves as a summary of the elastic string paradigm. My question was: “Is it possible there can be a non-Fourier model of string vibration? Is there an exact solution?”
To explain, I asked if they knew the Hamiltonian equation for the string vibration. They did not agree it must exist. I pointed out there are problems with the elastic model of vibration with its two degrees of freedom and unsolvable equations of motion can only be approximated by numerical methods. I said elasticity makes superposition the 4th Newtonian law. How can a string vibrate in an infinite number of modes without violating energy conservation?
Here are some comments I got in response:
“What does string is not Fourier mean? – Qmechanic
“ ‘String modes cannot superimpose!’ Yet, empirically, they do.” – John Doty
“ A string has an infinite number of degrees of freedom, since it can be modeled as a continuous medium. If you manage to force only the first harmonic, the dynamics of the system only involve the first harmonic and it’s a standing wave: this solution does depend on time, being (time dependence in the amplitude of the sine). No 4th Newton’s law. I didn’t get the question about Hamilton equation.
“What do you mean with ‘archaic model’? Can I ask you what’s your background that makes you do this sentence? Physics, Math, Engineering? You postulate nothing here. You have continuum mechanics here. You have PDEs under the assumption of continuum only. You have exact solutions in simple problems, you have numerical methods approximating and solving exact equations. And trust me: this is how the branch of physics used in many engineering fields, from mechanical, to civil, to aerospace engineering.” – basics
I want to show the rigid versus elastic dichotomy goes back to the calculus wars. Quoting here from Euler and Modern Science, published by the Mathematical Association of America:
"We now turn to the most famous disagreement between Euler and d’Alembert … over the particular problem of the theory of elasticity concerning a string whose transverse vibrations are expressed through second-order partial differential equations of a hyperbolic type later called the wave equation. The problem had long been of interest to mathematicians. The first approach worthy of note was proposed by B. Taylor, … A decisive step forward was made by d’Alembert in … the differential equation for the vibrations, its general solution in the form of two “arbitrary functions” arrived at by means original with d’Alembert, and a method of determining these functions from any prescribed initial and boundary conditions.”
[Editorial Note: The boundary conditions were taken to be the string endpoints. The use of the word hyperbolic is, I believe, a clear reference to Taylor’s string. A string with constant curvature can only have one mathematic form, which is the cycloid, which is defined by the hyperbolic cosh x function. The cosh x function is the only class of solutions that are allowed if the string cannot elongate. The Taylor/Euler-d’Alembert dispute whether the string is trigonometric or hyperbolic.
Continuing the quote from Euler and Modern Science:
"The most crucial issue dividing d’Alembert and Euler in connection with the vibrating string problem was the compass of the class of functions admissible as solutions of the wave equation, and the boundary problems of mathematical physics generally, D’Alembert regarded it as essential that the admissible initial conditions obey stringent restrictions or, more explicitly, that the functions giving the initial shape and speed of the string should over the whole length of the string be representable by a single analytical expression … and furthermore be twice continuously differentiable (in our terminology). He considered the method invalid otherwise.
"However, Euler was of a different opinion … maintaining that for the purposes of physics it is essential to relax these restrictions: the class of admissible functions or, equivalently, curves should include any curve that one might imagine traced out by a “free motion of the hand”…Although in such cases the analytic method is inapplicable, Euler proposed a geometric construction for obtain the shape of the string at any instant. …
Bernoulli proposed finding a solution by the method of superimposition of simple trigonometric functions, i.e. using trigonometric series, or, as we would now say, Fourier series. Although Daniel Bernoulli’s idea was extremely fruitful—in other hands--, he proved unable to develop it further.
Another example is Euler's manifold of the musical key and pitch values as a torus. To be fair, Euler did not assert the torus but only drew a network show the Key and Pitch can move independently. This was before Mobius's classification theorem.
My point is it should be clear the musical key and pitch do not have different centers of harmonic motion. But in my experience, the minions will not allow Euler to be challenged by someone like me. Never mind Euler's theory of music was crackpot!
In trigonometry we know that frequency and amplitude are independent because they have independent variables.
Then frequency and amplitude do not have the same equation of motion.
But according to Newtonian determinism, all of the motion of a system is determined an equation that depends only on the initial state of the string, being the totality of points on string and their velocities. The initial velocity is zero.
In a closed system, all of the movement must include both frequency and amplitude. That is, frequency and amplitude have the same equation of motion.
On the elastic string, the false assumption the string wave is trigonometric by itself implies amplitude and frequency have independent equations. Indeed, in the literature when mathematicians and physicists want the standing wave to stand down, they just add another arbitrary real-valued function. The frequency and amplitude are parameterized by sine wave and exponential functions, and each has its own time variable. Frequency and amplitude do not map on to the same interval of time.
But under one degree of freedom the standing wave never stands down because it is a surface defined by the potential energy. The surface being precisely those lines of motion along which energy is conserved.
So please tell why are two equations better than one? Why are two degrees of freedom better than one? Some even say the string has infinite degrees of freedom as if the string is not subject holonomic constraint.
You guy’s think the frequency is a velocity, but it's not. Frequency is a potential. Constant velocity and constant potential are both measure by a time unit.
Apparently, physicists and mathematicians think the velocity of the string is constant right up to the point in time when the string stops moving. Because the frequency is constant. That is, you think dv/dt = df/dt = 0. Then you write a partial differential equation that has the form of a sine wave. But your equation in the form u(x. t) is parameterized by time but contain coefficients that are not determined by the initial condition of the string. And it is not continuous on the lower limit.
That is to say the trigonometric string cannot map onto the string at rest. The trigonomtric string has no natural vector field.
Furthermore, the assumption of a continuous trig function implies that you are not required to have a lower semi-continuous boundary, without which it is not possible to formulate the law of string motion in terms of a minimum principle. (See Critical Point Theory by Mawhin and Willem)
There is a stumbling block here because it may seem that it is obvious that amplitude is dependent on time, since it occupies an interval of time. In fact, it is independent of time because decay always consumes the same amount of time regardless of amplitude magnitude.
the rate of amplitude decay da/dt2 = 0 is constant just like the frequency. They have the same Hamiltonian minimizing functions.
The equation da/dt2 = 0 is possible mathematically if the external derivative of amplitude decay is a tautochrone formed by the cycloidal involution of the cycloidal string manifold.
On a tautochrone, a rolling ball always arrives at the bottom of the curve at the same time regardless of how high the ball in dropped from.
This shows that frequency and amplitude are subject to the same holonomic restraint imposed by energy conservation.
When you give up your false assumption frequency is a velocity and change to frequency is a potential, you should see energy conservation is equivalent to volume preservation according to the principle of Liouville integration.
In attached diagrams I show the string manifold and amplitude decay manifold are both minimal surfaces of revolution and they have the same submanifold in Liouville integration except that amplitude is the involution of the cycloid at constant volume. Both manifolds uniform rectilinear motion. The frequency and amplitude run on the same time interval and clearly are not independent.
The trigonometric law of frequency/amplitude independence is not a natural Newtonian law, it is just an illusion that results from the assumption that frequency itself is sinusoidal.
But potential energy is a real number. You guys are just assuming frequency is real (so continuity seems to demand a trigonometric form).
Finally, if the moving string keeps moving until external force stops it, what force stops the string? Clearly not gravity, friction, or viscosity.
The answer is that the motion of the string is quasi-periodic meaning that perturbation involves only the loss of kinetic energy. Potential and kinetic energy do not alternate like a pendulum. When the string is deformed, the potential increases, but quickly the excess goes to kinetic energy and never returns to potential energy. Amplitude decay is simply the loss of kinetic energy doing work against the inertial mass of the string. Since it must be true that potential and kinetic energy have the same Hamiltonian equation, they cannot be independent.
Fig 1 The string manifold and amplitude decay manifold have the same submanifold
Fig 2 Amplitude Decay Manifold
Fig 3 Path of a Cycloidal Pendulum
Fig 4 Amplitude decay is the cycloidal involution of the Cycloidal Manifold.
Fig 5 Volume-preserving Liouville Integration
Fig 6 Constructing a cycloid geometrically using a horocycle give the string a constant radius of curvature.





+1
1. Dark energy has been a subject of considerable debate since its discovery due to its association with the accelerated expansion of the universe. Traditionally perceived as an unknown force or substance, dark energy is better understood as a by-product of the universe’s dynamic processes, particularly the transformation of potential energy into kinetic energy during and after the Big Bang. This work explores the interconnected roles of gravitational forces, kinetic energy, and apparent negative mass, highlighting that dark energy results from the complex interplay between these elements rather than being an independent substance.
2. Initial State of the Universe and Energy Transformation
Immediately after the Big Bang, the universe's total energy consisted of potential and kinetic components:
Eᴛₒₜ,ᴜₙᵢᵥ = PEᴜₙᵢᵥ + KEᴜₙᵢᵥ
In the earliest moments, the universe was dominated by potential energy, which rapidly approached zero as kinetic energy surged from zero to infinity:
PEᴜₙᵢᵥ: ∞ → 0, KEᴜₙᵢᵥ: 0 → ∞
This energetic shift was driven by gravitational dynamics, where the rapid conversion of potential energy into kinetic energy fuelled the universe’s expansion.
3. Emergence of Dark Energy: A Dynamic Outcome
Dark energy did not pre-exist the universe but emerged from the dynamic interactions between mass, gravity, and kinetic energy. As the universe’s initial potential mass accelerated due to gravitational forces, an apparent negative mass effect arose, which we interpret as dark energy:
Fᴜₙᵢᵥ = (Mᴘᴇ,ᴜₙᵢᵥ - Mᵃᵖᵖ,ᴜₙᵢᵥ)·aᵉᶠᶠ,ᴜₙᵢᵥ
Here, the apparent mass (Mᵃᵖᵖ,ᴜₙᵢᵥ) represents the dynamic influence of dark energy, emerging from the acceleration of potential mass under universal forces.
4. Inverse Relationship Between Potential and Kinetic Energy
The universe’s potential energy is inversely related to its kinetic energy, illustrating the natural balance that dictates cosmic evolution:
PEᴜₙᵢᵥ ∝ 1/KEᴜₙᵢᵥ
This relationship underscores the continuous transformation and reactivation of dark energy as the kinetic energy of the universe’s matter evolves.
5. Dark Energy's Dormancy and Reactivation
Dark energy enters a dormant state when kinetic energy and potential energy achieve equivalence. However, as the universe’s matter mass persists in motion, dark energy reactivates, leading to the accelerated expansion observed today. This cyclical behaviour underscores the transient nature of dark energy:
When PEᴜₙᵢᵥ = KEᴜₙᵢᵥ , Mᵃᵖᵖ = 0
As the universe continues to expand, dark energy becomes dominant once again, reflecting the evolving interplay of mass-energy dynamics.
6. Conclusion
Dark energy is not a fundamental substance but a manifestation of the universe’s dynamic processes. The accelerated expansion is driven by the continuous transformation of kinetic and potential energies, highlighting that dark energy is a consequence of the cosmic gravitational and kinetic interplay. This understanding shifts the perspective from viewing dark energy as an isolated force to recognizing it as an emergent property of the universe’s mass-energy transformations.
Can somebody point to publications describing essential logic gates (e.g. AND and NOT) implemented in the most basic model of classical mechanics - bistable systems consisting of point masses (point charges) in external potential fields, manipulated by other external potential fields.
E.g., a single bit can be implemented in this classical mechanics framework as a bistable system consisting of a single charged particle trapped in one of two nearby potential wells of the field interacting with the charge of the particle. The time-dependent external field can be applied to move the particle between these two wells, thus implementing flipping the bit between 0 and 1. Perhaps, other time-dependent fields can be suggested, acting on a single and multiple bits, thus implementing the logic gates? See below.
The question is motivated by the physics of quantum computing (QC), where QC practitioners use external fields to manipulate qubits, thus implementing quantum gates, claiming that many/most/all quantum gates can be implemented by properly modulated external fields.
Thank you!
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Here is some information on mechanical logic gates and some thoughts on potential approaches to the implementation of such logic gates in simple classical mechanics model.
Mechanical logic gates exist - see, for instance:
However, these mechanical logic gates are more complex than just point charges in external fields - they involve solid bodies (of certain dimensions and shapes), interacting with each other.
For the bit implemented as a single charged particle trapped in one of two adjacent potential wells, as described above, perhaps the following approaches can be used to implement the NOT and AND logic gates.
To implement the NOT logic gate, consider the field that would drag the particle from the well 0 to the well 1, combined with the simultaneously applied field, dragging the particle from the well 1 to the well 0, along the non-intersecting paths. Regardless of the original position of the particle (well 0 or well 1), such time-varying field will flip the value of the bit from 0 to 1 or from 1 to 0. This time-varying field implements the NOT gate.
The AND gate can be constructed using the fact that the field from the charged particle trapped in the well compensates, in part, the field of this well. So, if the "depth" of the well is properly chosen, and the second particle is dragged through the well with another particle already trapped there, this second particle will not be trapped in this well, which can be used to implement the AND gate.
Discussion:
This discussion explores the inherent limitations of the Lorentz transformation, a cornerstone of special relativity, particularly concerning its treatment of acceleration. While the Lorentz transformation adeptly describes relativistic effects such as time dilation, length contraction, and mass increase, it falls short in directly accommodating acceleration. This discrepancy becomes pronounced when the velocity-dependent Lorentz transformation fails to reconcile velocities between rest and inertial frames without the presence of acceleration, thus highlighting a significant gap in its applicability.
The discussion delves into the historical context of the Lorentz transformation, acknowledging its development by Mr. Lorentz and its status as a final form in science. However, it also underscores the expectation for accurate physics within its framework, especially considering the pre-existence of the concept of acceleration predating Mr. Lorentz. This expectation includes honouring Isaac Newton's second law, which governs the dynamics of accelerated motion in classical mechanics.
While the scientific community initially accepted the Lorentz transformation without questioning its treatment of acceleration, there is now a growing recognition of the importance of integrating principles from classical mechanics, such as Newton's second law, to address these limitations. The discussion emphasizes the need for a more comprehensive theoretical framework that harmonizes the principles of classical mechanics and relativity, thereby offering a more unified and accurate depiction of physical phenomena.
The Impact of Acceleration on Kinetic Energy in the Relativistic Lorentz Factor in Motion?
The Lorentz factor (γ) becomes relevant when the object attains its desired velocity and is in motion relative to the observer. Initially, when both reference frames are at rest, the object's energetic state reflects its lack of motion, resulting in zero kinetic energy (KE). As the frames separate, the moving object undergoes acceleration until it reaches its desired velocity. At this stage, the object's energetic state reflects its motion, and it possesses kinetic energy (KE) due to its acceleration. This acceleration is not accounted for in the Lorentz factor (γ). Once the object reaches its desired velocity, its energetic state reflects its motion, and it possesses kinetic energy (KE) due to its velocity. The Lorentz factor (γ) and kinetic energy (KE) play significant roles in relativistic motion. However, the acceleration component is not considered in the Lorentz factor (γ).
This discussion delves into the intricate relationship between acceleration, inertial reference frames, and Relativistic Lorentz transformation. It scrutinizes how the necessity of different velocities for separated reference frames underscores the pivotal role of acceleration in achieving this transition. By integrating classical mechanics concepts like Newton's second law and Hooke's Law with relativistic physics theories, the discussion enriches our comprehension of motion in diverse reference frames.
The initial motion and separation of inertial reference frames are crucial for their physics, but once they separate, they must have different velocities, with the first frame's velocity (v₀) and the second frame's velocity (v₁) needing acceleration to achieve v₁ > v₀. This acceleration is essential in both classical mechanics and Relativistic Lorentz transformation. The Lorentz factor (γ) is a velocity-dependent factor that involves velocity-induced forces, affecting the behaviour of objects in motion. It is based on the equation E = KE + PE, where KE is treated as 'effective mass'. Piezoelectric materials can convert mechanical energy from vibrations, shocks, or stress into electrical energy, typically an alternating current (AC). This process involves force-mass conversion, where the force applied to the piezo actuator results in a deformation or displacement. The displacement ΔLɴ of the actuator is inversely proportional to the stiffness, highlighting the interplay between force, stiffness, and displacement in force-mass conversion.
This discussion aims to delve deeper into the interconnectedness of classical mechanics, wave mechanics, and relativistic physics, alongside an exploration of piezoelectricity, guided by the fundamental concepts of velocity, speed, and gravitational dynamics. Drawing inspiration from recent research, we seek to unravel the intricate relationships between external forces, atomic structures, and wave phenomena, shedding light on potential advancements in materials science, physics, and engineering.
Join us as we navigate this interdisciplinary landscape and contemplate its implications for future scientific endeavours.
I can measure positions in classical mechanics because my measurements do not disturb the state of the system.
Why measurements cannot be used without perturbation of the system for atomic or subatomic interactions, for example using smaller scale interactions like neutrinos?

In classical mechanics, an important principle is the principle of relativity: the physical laws are invariant with respect to the transformation from one inertial frame into another. Maxwell's equations seem to violate this principle, because they contain a distinguished speed -- the speed of light c. It was this apparent conflict between mechanics and electrodynamics that led Albert Einstein in 1905 to his special theory of relativity. By a careful analysis of the concept of time, he realized that Maxwell's equations do indeed obey the relativity principle, although the transformation law becomes more complicated (Lorentz instead of Galileo transformations).
Einstein was very aware of the problem of the speed of light inconcitensy with Newton's laws (and Maxwell's equations). We was also aware of the observer effects (relative velocity effects) and he married the two, being the fist to explain the constancy of speed of light or reconcile classical mechanics with that fact.
This has been the only (albeit successful) attempt. Are you aware of any others >
In classical mechanics, kinetic energy is KE = ½mv², where m is mass and v is velocity. So mass multiplied by the square of the speed is an energy. The concept of energy plays a fundamental role in understanding the behaviour of objects in motion. One of the key forms of energy is kinetic energy, which is intimately linked to an object's mass and velocity. Additionally, in the realm of relativity, Einstein's famous equation E = mc² introduces a profound understanding of energy in terms of mass and the speed of light. This discussion aims to delve into the classical expression for kinetic energy KE = ½mv² and its connection to relativistic energy (mc²).
We assume that the accepted definition of a quantum particle is one subject to Schrödinger dynamics as opposed to Newtonian dynamics.
This implies some limitation on the size or volume of the quantum particle V.
In other words there exists a critical volume Vc where if V<< Vc the particle obeys quantum dynamics and for V >> Vc the particle is subject to Newtonian classical mechanics.
The question arises: is there an accepted estimate of the critical size Vc?
Since Einstein proposed the fundamental shiftbin thinking, that time and space are interwoven, people began to doubt time's fundamental it. Althoughtbitbis still ladenly agreed to be so.
Einstein made the proposal to deal with incompatibilities between properties of light and principles of classical mechanics, the science of describing motion its causes and what its possible (given certain kinematics and so-called dynamics parameters in a system) . This then led to the most hroad, novel and consistent confirmstions of any theory in science. Yet, doubt remains.
Current classical mechanics, Newtonian, are force causality centered.
However, linear ordering of events (LOoE) based causality and teleological are also options. Here I present the basics of LOoE and a glimpse of the challenges to odopting it to low speed mechanics.
Linear ordering based
**Speed of light-based i.e light cone causality is fundamental
** Newtonian force-based causality is fundamental in SR, GR but needs to be extented in all physics
Challenges
**problem of adopt ing low speed Requirements of classic mechanics to light speed causality
** force causality must be eliminated and still low speed mechanics work
To develop a full outline of the challenges, what do you think one should add ?
Modern quantum mechanics is based on the Shrödinger equation (SE) with the Bohr-Copenhagen interpretation.
As far as I understand, SE creates its own independent world outside of classical mechanics or any other world of measurement and/or interaction.
The box itself, with a cat and a bomb inside, constitutes the separation or boundary between the QM world and the other worlds of measurement and interaction.
In other words, the box creates a pure quantum mechanical statistical system (world) with its own QM states where their linear superposition is possible.
And yet, there are many solutions for SEs where the essential box is poorly taken into account, how?
Most masters focus on general review of qm, classical mechanics, assesing students skills in classical yet heneric and self-value calculative and interpreting capabilities.
The English MSc's on the other hand, provide an introduction to the physical principles and mathematical techniques of current research in:
general relativity
quantum gravity
quantum f. Theory
quantum information
cosmology and the early universe
There is also a particular focus on topics reflecting research strengths.
Graduates are more well equiped to contribute to research and make impressive ph. D dissertations.
Of course instructors that teach masters are working in classical and quantum gravity, geometry and relativity, to take the theoretical physics sub-domain, in all universities but the emphasis on current research's mathematical techniques and principles is only found in English university'masters offerings.
I think these phenomena follow the scientific facts defined in the classical mechanics, but not theories or definitions in Einstein's mechanics.
A review of analogies used in high energy physics is a 2020 Ph.D dissertation by Gunnar Kreisel, Analogies in Physics — Analysis of an Unplanned Epistemic Strategy, Gottfried Wilhelm Leibniz Universitat Hannover.
Are there any books or articles for analogies in classical mechanics?
We assume that the conservation of probability does not exist in quantum mechanics as in the case of classical mechanics.
According to Shrodinger's equation, there is a continuous generation of probability density in subatomic systems.
I uploaded an article on research gate with a wrong date of publication. The article is "Investigating Students’ Physico-mathematical Difficulties in Classical Mechanics and Designing an Instructional Model". The article was published in 2018 by American Journal of Educational Research, 2018, Vol. 6, No. 8, 1127.
When uploaded on research gate it picked the date it was uploaded April 2023
How can the date of publication be corrected?
Waiting to hear from you soon.
I am new to modeling, i am only familiar with Gaussian (quantum mechanics only)
And I want to model the adsorption mechanism of Dye Methylene blue on metal carbonate because therir are many psoible mechanisms like elecrostatic attraction force or Hydrogen bond. I don't know if it works with quantum mechanics or classical mechanics.
plz advice me
Physics is Stalled by Politics - Paper with solutions to 64 significant problems published in #3 journal but rejected by arXiv.org.
The new paper entitled, Measurement Quantization, published Jan. 25, 2023 in the Intl. J. Geom. Methods Mod. Phys.,
Article Measurement Quantization
lays out the foundations of quantum behavior using existing classical expressions, expanded to account for the discrete internal frame of the universe. The paper presents predictions of a length contraction effect unrelated to that described by Einstein and then presents measurement data to support the approach. It then derives the physical constants and the laws of nature from first principles. It unifies gravity with electromagnetism and writes both SR and GR anew from first principles, therein leading to a derivation of the equivalence principle. It presents simple classical solutions to dark energy, dark matter and a no free parameter description of early universe events.
But we should restate, this paper is classical mechanics, offering 530 equations describing phenomena across the entire measurement domain. While extensive support is offered, additional support rests on a long history of support for classical mechanics.
In regards to posts regarding the heated debate about the absurdity of new research being filtered, I agree! Even though this paper is published in the #3 mathematical physics journal in the world (by SJR ranking) and is indexed to NASA’s ADS, it was rejected by arXiv.org. I then sought the assistance of a top five ranked astrophysicist in the world. The case was reopened, reconsidered and then rejected a second time, as though classical mechanics was of questionable scientific merit. The point is, classical mechanics is worthy of arXiv.org.
We must conclude that the paper was rejected because of a higher cultural mandate, that breakthroughs that impact the existing funding model cannot appear as though they enjoy support by the community. For insight, see this post by Avi Loeb:
https://avi-loeb.medium.com/how-to-navigate-academia-6e8c4feea460
I will state, this paper presents solutions to 82.5% of all outstanding problems in cosmology among many other classical and quantum problems.
If community leaders really want to effect change, they would use well-vetted publications as example of this cultural absurdity and the need for change. And the best way to affect that change is to begin by supporting breakthroughs in existing classical mechanics on their blogs, in videos, in presentations and at conferences. Community leaders should not be posting literature as unanswered (i.e., dark matter, dark energy) where existing classical mechanics offers insight with straight-forward calculations. Otherwise, such individuals mimic the same cultural bias they are arguing against.Start with a purely classical case to define vocabulary. A charged marble (marble instead of a point particle to avoid some singularities) is exposed to an external electromagnetic (E&M) field. "External" means that the field is created by all charges and currents in the universe except the marble. The marble is small enough for the external field to be regarded as uniform within the marble's interior. The external field causes the marble to accelerate and that acceleration causes the marble to create its own E&M field. The recoil of the marble from the momentum carried by its own field is the self force. (One piece of the charged marble exerts an E&M force on another piece and, contrary to Newton's assumption of equal but opposite reactions, these forces do not cancel with each other if the emitted radiation carries away energy and momentum.) The self force can be neglected if the energy carried by the marble's field is negligible compared to the work done by the external field on the marble. Stated another way, the self force can be neglected if and only if the energy carried by the marble's field is negligible compared to the change in the marble's energy. Also, an analysis that neglects self force is one in which the total force on the marble is taken to be the force produced by external fields alone. The key points from this paragraph are the last two sentences repeated below:
(A) An analysis that neglects self force is one in which the total force on the marble is taken to be the force produced by external fields alone.
(B) The self force can be neglected if and only if the energy carried by the marble's field is negligible compared to the change in the marble's energy.
Now consider the semi-classical quantum mechanical (QM) treatment. The marble is now a particle and is treated by QM (Schrodinger's equation) but its environment is an E&M field treated as a classical field (Maxwell's equations). Schrodinger's equation is the QM analog for the equation of force on the particle and, at least in the textbooks I studied from, the E&M field is taken to be the external field. Therefore, from Item (A) above, I do not expect this analysis to predict a self force. However, my expectation is inconsistent with a conclusion from this analysis. The conclusion, regarding induced emission, is that the energy of a photon emitted by the particle is equal to all of the energy lost by the particle. We conclude from Item (B) above that the self force is profoundly significant.
My problem is that the analysis starts with assumptions (the field is entirely external in Schrodinger's equation) that should exclude a self force, and then reaches a conclusion (change in particle energy is carried by its own emitted photon) that implies a self force. Is there a way to reconcile this apparent contradiction?
I am studying Finite element method and Classical Mechanics. I have come across three important terms
- Principle of virtual work (found in Classical Mechanics)
- Principle of minimum potential energy (found in Finite element method)
- Calculus of variation (found in Mathematics while searching concept of Variational method of Finite element method)
All above terms are being used interchangeably and in bits and pieces in different book and no book did not explain properly about the relation of those three terms.
I feel that there are some relationship but not able to figure it out. Can someone explain all these three terms and how these are inter related?
No way to find out unless you do the actual proposed experiment:
Research Proposal Galton Board Double Slit Experiment
This guy here:
claims he made an interference Galton board https://en.wikipedia.org/wiki/Galton_board experiment and got an interference pattern. This would explain that quantum randomness originates from determinism and is a result of hidden local variables possible in the photon's environment in contrast to the Bell inequality EPR experiment?
Note:
As a reminder this is nothing extremely new, actually a deterministic explanation of the quantum DS single photon experiment was previously demonstrated by this experimental application of the pilot-wave theory using bouncing droplets:
The photons which are epicenters of electromagnetic distortions when translating in space distort the EM mass fiel of the environment they move in.These distortions of the mass field environment are feedback at the photon as alterations in its motion trajectory. Photons as massless particles may pass through each other without being affected but dynamic EM flux coming from the mass field of their environment they interact with can affect their trajectory.

Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics. It uses a different mathematical formalism, providing a more abstract understanding of the theory. Historically, it was an important reformulation of classical mechanics, which later contributed to the formulation of statistical mechanics and quantum mechanics.
The electron and photon are extensively interrelated, and therefore should share many common properties, equalities and laws. According to De Broglie, one such law is the Planck-Einstein relation, stating that as wavelength (i.e. radial orbit path) decreases, energy increases. An electron at a smaller orbit (today believed to be the ground state) most assuredly has a lower wavelength, and therefore should display a higher energy, not a “lower” energy level of the so called ground state.
Bringing this question up previously, two primary responses are typically offered... the "negative energy" aspect taught in high school, and the "pay no attention to classical mechanics" response. The last I checked, negative energy does not exist. Even it if did, a "zero" electron energy believed to exist at ionization is still greater than a -13.6eV at the believed minimum radius ground state, and therefore does not answer the question. As far as the second typical response, some sort of waving of the hands is offered, stating that the electron does not obey classical mechanics. Ok, I'll bite for argument sake, but then please explain why even the Schrodinger probability wave function also states that electron energy is minimum at a smaller radius ground state, and increases with a radial distance increase away from the proton. No so-called classical mechanics here.
I propose a model of Hydrogen for consideration, and welcome any peer review comments:
1) When the electron gains photonic energy, its orbiting radius is reduced and therefore its orbiting path per cycle decreases, equating to a higher cyclic frequency, equating to a higher energy per the Planck-Einstein relationship [so far so good, nothing strange here].
2) As it’s radius (r) decreases its Coulomb attraction force to the proton increases by r squared, and therefore so does its Potential Energy (PE) by 1/r (since E=F x dist). Therefore to remain in this particular lower radius increased energy level state, its orbiting velocity must then also increase to remain in this stable orbit, thereby increasing its associated Kinetic Energy (KE). [if you’re going to wave your hands here and say that classical mechanics doesn't apply to atomics, then state WHY and give an explanation, please don’t just regurgitate something you have read. Electrons have mass, and it is orbiting another mass. A very "classical" situation].
3) This increasing orbital velocity (frequency) has two limits: a) The speed of light, and b) Ionization "escape velocity" of the electron mass.
4) Once either of these velocities occur, the electron must then, a) convert completely into a photon at the speed of light, b) completely ionize (escape) from the proton, or c) convert part of its energy into a photon plus transform itself to a lower energy (higher radius) energy level [nothing too strange here unless there is a classical hand waving fetish].
In other words, attempt to purge the incorrect visualization thinking that as an electron gets further and further away from the proton, that it is closer and closer to becoming ionized. Attempt to think about WHY and HOW the electron would want to ionize, and you’ll come up with the above postulate. In fact, attempt to think about WHY only particular wavelength photons will be absorbed by the Hydrogen electron. Perhaps it is because it has the same geometric orbital size or a harmonic of that size. I have actually derived this harmonic to be the Fine Structure Constant / 2 utilizing classical mechanics, and by doing so, believe I have also discovered why and how the quantum aspect of the electron energy levels must occur.
- J.L. Brady
Radial gravitational wave study, physical interpretation of the fine-structure constant, resolution of the problem of wave-particle duality for electromagnetic radiations, and quantization of space-time :
Have a nice day :)
I am asking this question on the supposition that a classical body may be broken down in particles which are so small in size that quantum mechanics is applicable on each of these small particles. Here number of particles tends to uncountable (keeping number/volume as constant).
Now statistical mechanics is applicable if practically infinite no. of particles are present. So if practically infinite number of infinitely small sized particles are there, Quantum Statistical Mechanics may be applied to this collection. (Please correct me if I have a wrong notion).
But this collection of infinitesimally small particles make up the bulky body, which can be studied using classical mechanics.
Dear Sirs,
R Feynman in his lectures, vol 1, chapter 12, Characteristics of force wrote:
"The real content of Newton’s laws is this: that the force is supposed to have some independent properties, in addition to the law F=ma; but the specific independent properties that the force has were not completely described by Newton or by anybody else, and therefore the physical law F=ma is an incomplete law. ".
Other researchers may consider the 2nd Newton's law as a definition of force or mass. But R. Feynman did not agree with them in the above chapter.
What is your view on the 2nd Newton's law?
Dear Sirs,
Everyone knows the derivation of Lorentz transformations from electromagnetic wave front propagation. But Lorentz transformations are the basis of the general mechanics theory. It seems to me it is logically correct to derive the transformations from purely mechanical grounds. But how to do this? Mechanical (sound) waves are not of course applicable here. Or there is only purely mathematical approach? I The later is also not good in physics. Could it be derived from gravitational wave propagation? If it is so is there any controversy because General relativity is based on special relativity? I would be grateful for your suggestions.
Dear Sirs,
I would like to find out whether galilean relativity principle (which means the same
form of three Newton's laws in all inertial frames) is derived from the three Newton's laws or
any other classical mechanics statements.
Hello Dear colleagues:
it seems to me this could be an interesting thread for discussion:
I would like to center the discussion around the concept of Entropy. But I would like to address it on the explanation-description-ejemplification part of the concept.
i.e. What do you think is a good, helpul explanation for the concept of Entropy (in a technical level of course) ?
A manner (or manners) of explain it trying to settle down the concept as clear as possible. Maybe first, in a more general scenario, and next (if is required so) in a more specific one ....
Kind regards !
Many have asked, why does a theory fail? The challenge is establishing a physical connection. You have to step back and look at it this way. Anyone can take classical mechanics and introduce a variable to produce a ‘new’ approach. This is a critical problem pervasive throughout the community. This is NOT what physicists are looking for.
You might ask, isn’t that the point? No. Here’s why. Let us say that classical mechanics is a model described by A+B=C. But there is some unknown quality such that many results are askew.
Someone introduces a new idea E. It doesn’t matter what it is. But consider E(A+B)=EC. Now, maybe EC=h. So we have E(A+B)=h. Boom! We think E is this missing physics that solves amazing problems. All calculations result in perfect matches to the measured values.
Why is this a failure? We took something we already knew and multiplied it by a random value. Most physicists see through this right away. We’ve done this so many times that we can see it immediately.
MQ differs. Frames of reference are well understood. Discrete and non-discrete measure are well-understood. Planck Units are well-known. We ask only that one frame be discrete, the other not. This is a clear physical foundation without a concept E.
For clarity, LQG considers all frames discrete. Supersymmetry does not address discreteness. String Theory is in all likelihood and example of E, but so complex we cannot resolve it.
Article Physical Significance of Measure
Dear Sirs,
The 1st law in Newton`s principia are now understood as two statements: the determination of inertial frame reference (if F=0 then a=0 and if F is not equal 0 then there is some body accelleration "a"); there is in nature at least one inertial frame reference. Theoretically I can understand it a little bit. As we have such a determination of inertial frame reference then the 2 nd Newton law is not directly followed from the 1 st law, or this determination is partly independent of the 2nd law. So it looks like logically good.
But what we have in experiment? I do not know whether there is any research on experimental determination of any particular inertial system (like International Celestial Reference System) using the 1 st Newton law. So in practice we use the 2 nd law (e.g. school example - foucault pendulum plane rotation). Could you clarify on the experimental and theoretical determination of inertial frame reference. You know there are teachers that see the 1st law as the consequence of the 2nd law.
I got a question (in a Question paper) as follows:-
A three-sphere is like a two-sphere. It consists of all points equidistant from a fixed point (the origin) in four dimensional space. Consider a particle free to move on a three sphere. How many conserved quantities does this system possess?
The answer say's 6 conserved quantities are there, but how is it possible? Can anyone kindly explain.
Suppose, in fact, our 4-dimensional space is closed and represents the product of a sphere onto a torus or a sphere onto a cylinder, and the observer's coordinate lines are helical lines of a torus or cylinder. Then in such a closed world a material particle can be represented by a rotating ring on a torus or cylinder. Let the intrinsic angular velocity of rotation of the ring be the mass of the particle, and let the winds blow in our closed world, which blow our ring according to the principle of the least number of revolutions of the ring. Does not it resemble the principle of least action of classical mechanics. And since in this case the action is the number of revolutions of the ring, then at full revolutions we get a quantized action.
If you are interested in continuing, then welcome to
Research Proposal MATHEMATICAL NOTES ON THE NATURE OF THINGS
One of the consequences of relativistic physics is the rejection of the well-known concepts of space and time in science, and replacing them with the new concept of Minkowski space-time or simply space-time.
In classical mechanics, the three spatial dimensions in Cartesian coordinates are usually denoted by x, y and z. The dimensional symbol of each is L. Time is represented by t with the dimensional symbol of T.
In relativistic physics x, y and z are still intactly used for the three spatial dimensions, but time is replaced by ct. It means its dimension has changed from T to L. Therefore, this new time is yet another spatial dimension. One thus wonders where and what is time in space-time?
Probably, due to this awkwardness, ct is not commonly used by physicists as the notion for time after more than a century since its introduction and despite the fact that it applies to any object at any speed.
The root of this manipulation of time comes directly from Lorentz transformations equations. But what are the consequences of this change?
We are told that an observer in any inertial reference frame is allowed to consider its own frame to be stationary. However, the space-time concept tells us that if the same observer does not move at all in the same frame, he or she still moves at the new so-called time dimension with the speed of light! In fact, every object which is apparently moving at a constant speed through space is actually moving with the speed of light in space-time, divided partially in time and partially in spatial directions. The difference is that going at the speed of light in the time direction is disassociated with momentum energy but going at the fraction of that speed in the other three dimensions accumulates substantial momentum energy, reaching infinity when approaching the speed of light.
In discussing Quantum Mechanics (QM), I shall restrict myself here to Schroedinger's Non-Relativistic Wave Mechanics (WM), as Dirac showed (in his 1930 text) [using Hilbert State Vectors] that Heisenberg's Matrix Mechanics (MM) was simply mathematically equivalent.
WM was invented in 1925 when Schroedinger adopted de Broglie's radical proposal that a quantum particle, like an electron, could "have" both contradictory point particle properties (like momentum, P) and wave properties, like a wave-length or wave-number K) by: K = h P; where h is Planck's constant (smuggling in quantization). Next he ASSUMED that a free electron could be represented as a spherical wave described by the Wave Equation. Then, he "joined the QM Club" by restricting his math approach to an isolated hydrogen atom, with its one orbital electron moving around the single proton (each with only one electron charge,e) at a spatial separation r at time t (i.e. x;t). He then linearized out the time by assuming a harmonic form: Exp{i w t) along with Einstein's EM frequency (photon) rule: E = h w. This gave him his famous Wave Equation [using W instead of Greek letter, psi]: H W = E W where H was the classical particle Hamiltonian H =K+U with K the kinetic energy [K= p2/2m] and U the Coulomb potential energy [U = e2/r]. Replacing the quadratic momentum term gave the Laplacian in 3D spherical polar co-ordinates [r, theta, phi]. He then remembered this resembled the 19th century oscillating sphere model with its known complete (infinite series) solution for n=1 to N=infinity for W=Y(l:cos theta) exp[i m phi] introducing the integer parameters l [angular momentum] and m [rotation around the Z axis]. By assuming the math solution is separable, he was left with the linear radial equation that could be solved [with difficulty] but approximated to Bohr's 1913 2D circular [planetary] model E values.
The "TRICK" was to isolate out from the infinite sums, all terms that only included EACH of the finite n terms [measured from n=1 to 6]. This was Dirac's key to match the nth wave function W(n:x,t) with his own Hilbert ket vector: W(n:x,t) = |n, x, t>.
So, I maintain that QM has failed to map its mathematics to a SINGLE hydrogen atom [the physical assumptions used therein] but to the full [almost infinite] collection of atoms present in real experiments. This then results in multiple epistemological nonsense such as Born's probability theory, wave function collapse and the multiverse theory.
This is NOT needed IF we reject the Continuum Hypothesis [imported from Classical Mechanics] and stick to finite difference mathematics.
I am a materials science undergrad, interested to know the algorithms for numerical integration of equation of motion in computational materials science, like molecular dynamics. It is said that, time-reversal symmetry is essential for such simulations, while classic integration schemes like Trapezoidal, simpsons or weddle methods handle previous and next time step differently. So verlet algorithm is used instead.
Position verlet indeed adds previous and past timesteps and maintains time-reversal symmetry. But velocity verlet doe not. Why is time-reversal symmetry not important for velocity? is it because time reversal symmetry is meaningful only for position and its even derivatives, as in newton's law of motion?
My knowledge on Numerical analysis is only of introductory level, and i have not deeply studied Lagrangian, chaos theory, group theory or hyper-dimensional geometry yet.
Why quantum mechanic exist?
Why the elementary particle doesn't follow newton law?
Why we need a quantum mechanic for the tiny object?
Bohr said that quantum mechanics does not produce classical mechanics in a similar way as classical mechanics arises as an approximation of special relativity at velocities very slow than light speed.
He argued that classical mechanics exists independently of quantum mechanics and cannot be derived from it.
Max Jammer has said: quantum mechanics and classical dynamics are built on fundamentally different foundations!
No one can derive the newton law from the Schrodinger equation.
only the behavior of systems described by quantum mechanics reproduces in a statistical way the classical mechanics in the limit of large quantum numbers.
I don't understand why scientists don't give this point a big attention.
The real interpretation of quantum mechanics need to give an answer to this question:
why quantum mechanic exist?
I start from the concept of the motion itself and assume that the motion (in the quantum world and classical world) is a sequence of appearances and disappearances events in space and time:
The idea that affirms that the motion happened by disappearing and appearing actions give us God willing a beautiful answer about this question.
The Newton law said:
"In an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force".
But this law is not compatible with the disappearance and appearance idea!
this law is not always acceptable since the particle might easily appear (if the quantum jump is enough) in a forbidden (have a variation to a very large potential field like for example the particle in an infinite potential well) place after some quantum jumps in the direction of the movement of the particle!
so for a huge number of particles that jump in the subatomic level, the newton law may put our universe in an unstable situation!, and this might happen specifically when the length of the jump is close (or greater) to the length of the field's fluctuations (that frequently happens in subatomic level).
But in the case where the length of the quantum jump is very small (for example for large quantum numbers or in classical case) compared to the length of the field's fluctuations then the first law of Newton will be applicable because in this case, we can be sure that the particle will feel the force before that the force gets altered.
We usually deal with the motion like it was related only with the particle itself, but in my opinion, this is not true, I think we have two players in the motion:
1- The particle itself.
2- The space-time itself.
At each time, space itself allows the particle to appear in some multiple positions with certain preferences based on a new quantum action principle named "alike action principle" (that can lead us to Schrodinger equation) that ensures the existence of physical harmony within our universe, and the particle chooses randomly between these preferences.
So this is the role of space-time in the motion process, like for example preventing the particle from easily reaching to forbidden locations (guarded by fields of great forces). Therefore, in general, this new constraint in movement could be valid at multiple positions at the same time, so in general, we have multiple acceptable positions to appear at it. Thus the probability of existence came up in our descriptions of the movement in the quantum world.
With kindest regards.
The 2nd law of the thermodynamics says that the entropy only increases. From this law we derived a time-axis that has only one direction - forward. Similarly, from our life-experience we know that we only grow older (fact which seems to be a consequence of the thermodynamics 2nd law), and from that we also derived a unidirectional time-axis.
The time-axis seems to be a concept emerging from irreversible processes.
However, not every process in the world has to do with an increase of entropy. The movement of the electrons in the ground state of an atom does not have to do with an increase of entropy, s.t. the atoms, at least in their stable state, are identical today with thouse that existed billions of years ago. There is an internal dynamics in these atoms, but it doesn't need a time-axis, because this dynamics is reversible.
What I am asking is which proof do we have that the macroscopic world does not admit an influence backwards in time?
My question is motivated by the fact that in the quantum mechanics, the results of measurements produced by entangled systems tested at different times, show a clear interdependence between present and future. None of the systems produces its response independently. The response of each system depends on the type of experiment done on the other systems, and their responses, even if other systems are tested later.
NOTE: I am not asking why the time flows only in forward direction. It is we, the human beings that choose this direction because we grow only older, not younger, because our experience of life increases, etc. I am asking if we have any evidence that a future event influences a past process.
Dear Sirs,
I would like to find out more precisely whether the 2nd Newton law is valid or not in wide range of masses, accelerations, forces. Particulary I have a question whether the inertial property of body (inertial mass) is able to stop the body for small external forces or not. I have found in the Internet the fresh articles with tests of the 2nd Newton law for small accelerations (10^-10), small forces (10^-13) and SMALL masses (about 1 kg). The articles deal with the question of dark matter and MOND theory in astrophysics.
But I am interested in BIG masses. Could the test be carried out in planetary scale? Maybe for the Moon or asteroids? Or for masses like 1000 kg? Thank you very much for any references.
Dear Sirs,
1, 2, 3 laws of Newtons need closed system (net force is zero). How do we practically realize, create such closed system?
One example. Let us look at a body motion. One can say If the body velocity is constant, e.g. zero then no forces act to it. Is it true? I think no. According to the 1st Newton law the velocity constance is the CONSEQUENCE of F=0.
So are there precise ways to construct closed system? Or all physical theory is just a mean to generate a hypothesis which has more higher probability to be true then other random thought?
Are there examples of lossless, yet time irreversible systems? I would be pleased to know if there are any in Quantum Mechanics.
Could you give references on mass measurement from the 3rd law (with different forces: gravitational, elastic, etc)? E.g. old articles by Saint-Venant.
The physical system of a charged particle under the action of crossed electric and magnetic fields is a well-known problem in classical mechanics. In such an approach, the motion equations can be obtained for different regimens of electric and magnetic field intensities.
From a quantum mechanics point of view, it is common to solve it for the null electric field case (as can found in several textbooks); even, the coherent states formalism has been applied to describe the charged particle dynamics. However, have such treatment been also applied to the non-null electric field case?
I have not found any information about that case and I am interested in knowing how that problem has been discussed.
Regards.
Dear Colleagues,
1. It is known that the correspondence principle states that the behavior of systems described by quantum mechanics reproduces in a statistical way the classical mechanics in the limit of large quantum numbers, Bohr said that quantum mechanics does not produce classical mechanics in a similar way as classical mechanics arises as an approximation of special relativity at velocities very slow than light speed.
He argued that classical mechanics exists independently of quantum mechanics and cannot be derived from it.
2. For example in the case of "particle in box", if we examine the probability density for finding the particle when n growth (case of high energy) we found a sequence of peaks separated by a distance equal to half of De Broglie wavelength, so if the correspondence principle describes exactly the reality we need to accept that the motion in classical limit does not continue.
So first it is natural to assume that the motion (in classical world) is a sequence of appearances and disappearances events in space and time.
Then if we go back to the quantum world, and say okay, why the particle does not do the same for low energies too? I mean we can suppose too that it also disappears and appears as separate events in space and time, but of course, according to another law of motion(not the half of De Broglie wavelength), in fact I suggest this law in my theory:
it is called "alike action principle" it is similar to the least action principle and can lead us simply to the path integral formulation.
What do you think?
In textbooks or introductory texts on quantum mechanics, one may read that the quantum mechanical wave function changes by two fundamentally different processes. One is the deterministic and continuous evolution according to the Schrödinger equation, the other the collapse provoked by a measurement, usually discontinuous, nonlocal and disruptive. I would like to argue that this is due to overburdening the wave function in the Schrödinger picture by requiring it to describe both the state and the dynamics of a system.
The first version of modern quantum mechanics, "matrix mechanics" given in 1925 by Heisenberg, did not do this. Actually, the notion of a wave function was not present in it, even though we may nowadays apply it to Heisenberg's state concept. In the Heisenberg picture, the wave function describes only the initial state of a system. It does not change in time, only on measurement. This change is due to a change in knowledge about the system and the necessity to adapt the probability amplitude to the new knowledge and it corresponds to the collapse. But it is not a dynamical change.
Dynamics is described by the time dependence of observables, i.e. operators. It is governed by Heisenberg's equation of motion, an equation that is equivalent to the Schrödinger equation. So in the Heisenberg picture, dynamics and collapse are neatly separated. Also, the collapse of the wave function cannot be said to be due to a wave packet interacting with a detector, because the wave packet does not change in time, it never gets close to a detector, if it is far from all of them initially. It is, in fact, not a spatiotemporal wave, it is just a spatial distribution. Interaction and dynamics is between observables only. States only describe initial conditions.
Experimental observations neither concern states nor operators directly. They always refer to matrix elements, involving both. So the Schrödinger and Heisenberg pictures are equivalent, giving the same matrix elements. Nevertheless, they attribute the dynamics to different entities.
The Heisenberg picture was invented before the Schrödinger equation. Moreover, it is closer to classical mechanics in that it is straightforward to get from Hamilton's classical equations of motion to the Heisenberg equation of motion, as soon as we know how to quantize phase space functions. Once we know how to construct quantum mechanical operators, all we have to do is to replace phase space functions by operators and the Poisson brackets of classical mechanics by i \hbar times the commutator. The way to the Schrödinger equation is more convoluted. Nevertheless, essentially everybody dashed at the Schrödinger equation, once it became available in 1926, and it soon became the predominant description. Only when we are dealing with multiple-time correlation functions, we prefer the Heisenberg picture, because the consideration of multiple-time correlations functions is difficult to justify in the Schrödinger picture (where the operators whose correlations are of interest remain time independent), whereas it has a clear motivation in the Heisenberg picture.
The reason for this rush at Schrödinger's bonanza obviously was that people knew well how to work with partial differential equations but were unfamiliar with infinite-dimensional matrices. Which then led to the (doubtful) enterprise of assigning more meaning to the wave function than follows from physical considerations.
Consider wave-particle duality, for example. There is a tendency among physicists to overemphasize one of these classical limiting ways for a quantum object "to express itself". Bohmianists give the particle aspect ontological dominance. The wave function is an additional ingredient, but when we measure a quantum particle, we always measure a positional aspect (a pointer variable), and so it is a particle, and the wave is there only to guide it. Others tend to say there is only waves and their interactions with detectors are such that a particle illusion is created. There is the fraction of field theorists saying that there are no particles, only fields, but there are also some serious scientists emphasizing "field theory without fields" by pointing out that the whole physics of a field theory is present in its particle contents.
What does the Heisenberg picture suggest on the question of wave-particle duality? The dynamical entities in the Heisenberg picture are operators. Those are neither particle nor wave. The position operator does not describe an object at a particular position. It has a spectrum containing all possible positions. When it evolves in time (because it is time-dependent in the Heisenberg picture), the relative weights of the positions and, in particular, the expectation value of the position change. But the property "position", as described by the position operator, is not univalent. So the quantum object having that dynamical property cannot be a localized particle that has only one position. On the other hand, the property wave vector is proportional to the momentum operator, evolving in time, too, and having more than one momentum value in its spectrum. The quantum object having that dynamical property cannot be a wave, not a "pure" one at least, characterized by a single wave vector (or a narrow spectrum of such wave vectors). If we take the fact at face value that dynamical variables in quantum mechanics are described by operators, it immediately becomes clear, that the quantum objects are neither waves nor particles but something different -- that's why their description is by operators. Note that in the Heisenberg picture, the double slit experiment gives the same result as in the Schrödinger picture, even though no wave is moving around there (the wave function keeps its initial distribution throughout the experiment until detection of the quantum object). What is changing are the "position" and "momentum" of the quantum object and these are influenced by the presence of both slits. Because they take into account a whole set of possibilities. (The third formulation of quantum mechanics, Feynman's path integral approach turns this set into the possible paths of particles.)
What are quantum objects? Quantum objects are characterized by their properties, as are classical objects. Properties such as mass and charge are simple parameters for the elementary objects as in classical mechanics, whereas dynamic properties such as position and momentum are characterized by operators and hence different for quantum objects from corresponding properties in classical objects. Only in certain limiting cases, descri