Completing the Li\'enard-Wiechert potentials: The origin of the delta function fields for a charged particle in hyperbolic motion

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-Wiechert potentials misses some terms. We present new derivations of the missing field and potential terms by considering only pure hyperbolic motion, and we amend the offending step in the standard Li\'enard-Wiechert construction.Comment: 4 pages, no figure

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

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 conductingsurface 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

Equivariant Dierential 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 anynonlinear dynamical system preserves at most a twofold symmetry

Differential Embedding of the Lorenz Attractor

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 rotation symmetry, are not ideal. While the differential embedding technique happens to yield an embedding of the Lorenz attractor in three dimensions, it does not yield an embedding of the entire flow. An embedding of the flow requires at least four dimensions. The four dimensional embedding produces a flow restricted to a twisted three dimensional manifold in R4. This inversion symmetric three-manifold cannot be projected into any three dimensional Euclidean subspace without singularities

The Physical Origin of Torque and of the Rotational Second Law

We derive the rotational form of Newton\u27s 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 traced to its violation of the strong form of Newton\u27s 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. (C) 2015 American Association of Physics Teachers

Representation Theory for Strange Attractors

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 the systematic ways inequivalent embeddings become equivalent as the embedding dimension increases until there is finally only one “universal” embedding in a suitable dimension