Daniel J. Cross

The physical origin of torque and of the rotational second law

We derive the rotational form of Newtons second law τ=Iα from the translational form F→=ma→ 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 traced to its violation of the strong form of Newtons third law. This leads us to consider a slightly modified non-rigid model that respects the third law, produces the correct rotational second law, and makes explicit the importance of the product of the tangential force with the radial distance: the torque.

When the charge on a planar conductor is a function of its curvature

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 curvature function is non-injective, the conductor must transform under one of the planar symmetry groups. In particular, for the charge density on a closed conductor with smooth boundary to be a function of its curvature, the conductor must possess dihedral symmetry with a mirror line running through each curvature extremum. Several examples are presented along with explicit charge-curvature functions. Both increasing and decreasing functions were found.

A Schwinger disentangling theorem

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.

Equivariant differential embeddings

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 diffeomorphic to the original. However, if the original dynamical system is symmetric, it is natural to search for reconstructions that preserve this symmetry. These generally do not exist. We demonstrate that a differential reconstruction of any nonlinear dynamical system preserves at most a twofold symmetry.

The relativistic gamma factor from Newtonian mechanics and Einstein's equivalence of mass and energy

We obtain the expressions for the energy and momentum of a relativistic particle by incorporating the equivalence of mass and energy into Newtonian mechanics.