
Daniel J CrossGeneva College · Department of Physics
Daniel J Cross
PhD, Physics
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21
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Introduction
Additional affiliations
August 2015 - present
August 2012 - July 2015
Publications
Publications (21)
Harmonic oscillations are isochronous, but so are oscillations in any potential that is shear-equivalent to a harmonic potential. We demonstrate the converse: that every possible isochronous potential is shear-equivalent to a harmonic potential. To do this we show that every potential is shear-equivalent to a unique symmetric potential, and that ev...
We obtain the expressions for the energy and momentum of a relativistic particle by incorporating the equivalence of mass and energy into Newtonian mechanics.
We derive the rotational form of Newton's second law tau = I alpha from the translational form (F) over right arrow = m (a) over right arrow by performing a force analysis of a simple body consisting of two discrete masses. Curiously, a truly rigid body model leads to an incorrect statement of the rotational second law. The failure of this model is...
While there is no general relationship between the electric charge density on a conducting surface and its curvature, the two quantities can be functionally related in special circumstances. This paper presents a complete classification of two-dimensional conductors for which charge density is a function of boundary curvature. Whenever the curvatur...
Calculating the electromagnetic field of a uniformly accelerated charged
particle is a surprisingly subtle problem that has been long discussed in the
literature. While the correct field has been obtained many times and through
various means, it remains somewhat unclear why the (supposedly general) field
expression derived from the Li\'enard-Wieche...
Chaotic data generated by a three-dimensional dynamical system can be embedded into R^{3} in a number of inequivalent ways. However, when lifted into R^{5} they all become equivalent, indicating that they all belong to a single universality class sharing a common chaos-generating mechanism. We present a complete invariant determining this universal...
The interaction of a magnetic dipole with a point charge leads to an apparent
paradox when analyzed using the 3-vector formulation of the Lorentz force.
Specifically, the dipole is subject to a torque in some frames and not in
others. We show that when analyzed according to the covariant 4-vector
formulation the paradox disappears. The torque that...
The observed accelerated cosmic expansion is problematic in that it seems to
require an otherwise unobserved dark energy for its origin. A possible
alternative explanation has been recently given, which attempts to account for
this expansion in terms of a hypothesized matter-anti-matter repulsion. This
repulsion or anti-gravity is derived by applyi...
Embeddings are diffeomorphisms between some dynamical phase space and a reconstructed image. Different embeddings may or may not be equivalent under isotopy. We regard embeddings as representations of the dynamical phase space. We determine the topological labels required to distinguish inequivalent representations of three-dimensional dissipative...
Baker-Campbell-Hausdorff formulas are exceedingly useful for disentangling operators so that they may be more easily evaluated on particular states. We present such a disentangling theorem for general bilinear and linear combinations of multiple boson creation and annihilation operators. This work generalizes a classical result of Schwinger.
Takens [Dynamical Systems and Turbulence, Lecture Notes in Mathematics, edited by D. A. Rand and L. S. Young (Springer-Verlag, New York, 1981), Vol. 898, pp. 366-381] has shown that a dynamical system may be reconstructed from scalar data taken along some trajectory of the system. A reconstruction is considered successful if it produces a system di...
Ideally an embedding of an N -dimensional dynamical system is N -dimensional. Ideally, an embedding of a dynamical system with symmetry is symmetric. Ideally, the symmetry of the embedding is the same as the symmetry of the original system. This ideal often cannot be achieved. Differential embeddings of the Lorenz system, which possesses a twofold...
An algorithm inspired by Genome sequencing is proposed which "reconstructs" a single long trajectory of a dynamical system from many short trajectories. This procedure is useful in situations when many data sets are available but each is insufficiently long to apply a meaningful analysis directly. The algorithm is applied to the Rössler and Lorenz...
Takens [Dynamical Systems and Turbulence, Lecture Notes in Mathematics, edited by D. A. Rand and L. S. Young (Springer-Verlag, New York, 1981), Vol. 898, pp. 366-381] has shown that a dynamical system may be reconstructed from scalar data taken along some trajectory of the system. A reconstruction is considered successful if it produces a system di...
In physics, experiments form the bridge connecting theory to reality. This bridge is often quite narrow: one typically only records one of the myriad variables responsible for generating a complicated dynamical behavior. Nevertheless, every variable can usually be “reconstructed” from that single observation. Such a reconstruction provides an embed...
Embeddings are diffeomorphisms between some unseen physical attractor and a reconstructed image. Different embeddings may or may not be equivalent under isotopy. We regard embeddings as representations of the attractor, review the labels required to distinguish inequivalent representations for an important class of dynamical systems, and discuss th...
The linking integral is an invariant of the link-type of two manifolds immersed in a Euclidean space. It is shown that the ordinary Gauss integral in three dimensions may be simplified to a winding number integral in two dimensions. This result is then generalized to show that in certain circumstances the linking integral between arbitrary manifold...
The recently proposed Cooperstock-Tieu galaxy model claims to explain the flat rotation curves without dark matter. The purpose of this note is to show that this model is internally inconsistent and thus cannot be considered a valid solution. Moreover, by making the solution consistent the ability to explain the flat rotation curves is lost.
The classical electromagnetic field is described by a pair of 3-vector fields, E and B, (or a single anti-symmetric second rank tensor) which are functions of space and time. The field therefore has a total of 2 � 3 = 6 degrees of freedom at every point. It is more convenient to take the Fourier transform of the fields and consider them as function...
Mach's principle states that the local inertial properies of matter are determined by the global matter distribution in the universe. In 1958 Cocconi and Salpeter suggested that due to the quadrupolar assymetry of matter in the local galaxy about the earth, inertia on earth would be slightly anisotropic, leading to unequal level splittings of nucle...