I have undergraduate degrees in mathematics and physics from Cedarville University and a PhD in physics from Drexel University. I taught physics for several years at Bryn Mawr College and Haverford College before coming to Geneva.
Research Interests
I have a general interest in the intersection of mathematics and physics, and I find immense joy in the beauty of creation that mathematical investigation into the natural world reveals. As Richard Feynman observed, “To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature… If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.”
Some specific interests include finding order in chaos, classical and relativistic electricity and magentism, symmetry, topology, and massively parallel computation.
It is a right, yes a duty, to search in cautious manner for the numbers, sizes, and weights, the norms for everything [God] has created. For He himself has let man take part in the knowledge of these things… For these secrets are not of the kind whose research should be forbidden; rather they are set before our eyes like a mirror so that by examining them we observe to some extent the goodness and wisdom of the Creator." Johannes Kepler
Current Projects
I am interested in how we can better understand the complexities of low dimensional chaotic behavior. While there exists a robust theory for highly dissipative chaotic system in three dimensions, the case of weakly- and non-dissipative systems has received far less attention. Here are two particular systems that interest me. First is the double pendulum: where you hang one pendulum from a second one. While periodic (nicely repeating) motions are possible, chaotic motion is typical. Nevertheless, understanding the periodic behaviors would shed light on the chaotic motions.
Figure 1 - Time lapse following the chaotic motion of the bottom pendulum.
The second system is the magnetic field lines near current carrying wires. While closed filed lines are possible, a typical field line will meander around the wires endlessly (chaotically). Understanding the closed (periodic) field lines would shed light on the chaotic ones.
Figure 2 - A trefoil knot carrying an electric current generates magnetic field lines that can be quasi-periodic or chaotic.
Figure 3 - Cartoon illustration of a (knotted) closed magnetic field line generated by a figure 8 knot.
Recent Student Projects
The motion of a gyroscopic pendulum.
Is the speed of light the same in all directions?
What we can learn about the mass of a photon by measuring Jupiter’s magnetic field.
Every isochronous potential is shear-equivalent to a harmonic potential, Am. J. Phys., 86, 198 (2018).
The relativistic gamma factor from Newtonian mechanics and Einstein's equivalence of mass and energy, Am. J. Phys., 84, 384 (2016).
Completing the Liénard-Wiechert Potentials: The Origin of the Delta Function Fields for a Charged Particle in Hyperbolic Motion, Am. J. Phys., 83, 349 (2015).
The Physical Origin of Torque and of the Rotational Second Law, Am. J. Phys., 83, 121 (2015).
When the Charge on a Planar Conductor is a Function of its Curvature, J. Math. Phys., 55, 123504 (2014).
Dressed Return Maps Distinguish Chaotic Mechanisms, Phys. Rev. E 87, 012919 (2013) with R. Gilmore.
Complete Set of Representations for Dissipative Chaotic Three-Dimensional Dynamical Systems, Phys. Rev. E 82, 056211 (2010) with R. Gilmore.
A Schwinger Disentangling Theorem, J. Math. Phys., 51, 103515 (2010) with R. Gilmore.
Equivariant Differential Embeddings, J. Math. Phys., 51, 092706 (2010) with R. Gilmore.
Differential Embedding of the Lorenz Attractor, Phys. Rev. E 81, 066220 (2010) with R. Gilmore.
A Biological Algorithm for Data Reconstruction, Phys. Rev. E 81, 036217 (2010) with R. Michaluk, and R. Gilmore.
Representation Theory for Strange Attractors, Phys. Rev. E 80, 056207 (2009) with R. Gilmore.
Unpublished Papers
Resolution of the Mansuripur Paradox} D. J. Cross, arXiv:1205.5451 (2012).
Comment on `CPT symmetry and antimatter gravity in general relativity,' arXiv:1108.5117 (2011).
Linking Integral Projection, arXiv:0907.3446 (2010).
On the Flux Rule (2009).
On the Relation between Real and Complex Jacobian Determinants (2008).
Comments on the Cooperstock-Tieu Galaxy Model, arXiv:astro-ph/0701019 (2005).
Anisotropy of Inertia from the CMB Anisotropy (2004).
