Daniel J Cross

Daniel J Cross
Geneva College · Department of Physics

PhD, Physics

About

21
Publications
2,217
Reads
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98
Citations
Additional affiliations
August 2015 - present
Geneva College
Position
  • Professor (Associate)
August 2012 - July 2015
Haverford College
Position
  • Visiting Assistant Professor

Publications

Publications (21)
Article
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...
Article
We obtain the expressions for the energy and momentum of a relativistic particle by incorporating the equivalence of mass and energy into Newtonian mechanics.
Article
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...
Article
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...
Article
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...
Article
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...
Article
Full-text available
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...
Article
Full-text available
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...
Article
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...
Article
Full-text available
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.
Article
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...
Article
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...
Article
Full-text available
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...
Article
Full-text available
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...
Article
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...
Article
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...
Article
Full-text available
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...
Article
Full-text available
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.
Article
Full-text available
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...
Article
Full-text available
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...

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