Thomas P. Mitchell

Dynamical symmetries of the Schrödinger equation

The conditions are derived under which a symmetry operator quadratic in the momenta exists for a spin‐zero structureless point charge interacting with an externally applied uniform magnetic field in the presence of a potential field. The conditions apply to the possible forms of the potential. The explicit form of the symmetry operator in general is constructed and some particular examples are examined.

Stress analysis of a thick-walled pressure vessel nozzle junction

Abstract A thick-walled pressure vessel with an attached nozzle under internal pressure is analyzed by using the finite element method. The specified boundary displacement method has been applied to measure the stress concentration due to the geometrical discontinuity. The problem areas involved with the analysis are identified. The results are compared to the work of previous investigators and are found to provide a significant improvement.

An algorithm for finding the natural frequencies of a randomly supported string

Abstract An algorithm is developed for the determination of the natural frequencies of a finite stretched string which is supported, subject to a critical probability, by linear springs at uniform intervals between its fixed ends. The algorithm is derived by the use of transfer matrices. Examples of its implementation are presented.

Vector constants and their algebras for classical Hamiltonians

The explicit form of a vector constant of the motion for an arbitrary relativistic spherically symmetric time independent classical Hamiltonian is obtained by showing that its construction is achieved on solving a linear second order ordinary differential equation. The solution of this equation is presented and the vector, in conjunction with the angular momentum to which it is normal, is used to generate the algebras of the Euclidean group E(3), the orthogonal rotation group O(4) and the special unitary group SU(3). The mass is assumed to be structureless and to move in an externally prescribed scalar potential field.

The relativistic motion of a radiating point charge in a constant electromagnetic field

The attenuating influence of the radiation recoil momentum on the motion in constant uniform arbitrarily directed electric and magnetic fields is exhibited in detail by solving the relativistic Lorentz-Dirac equation to first order in the parameter 2e2/3mc3. The geometry of the trajectory is also elucidated.

The librational dynamics of deformable bodies

The volume average of the strain tensor in a body moving in an inverse-square force field is evaluated. The calculation is carried out assuming the satellite to be an isotropic elastic body whose center of mass moves in a planar orbit. An approximate expression, in terms of its volume and elastic properties, is presented for the strain energy in the satellite. Using this expression the equation of planar librational motion is written explicity. This equation is discussed for both circular and elliptic orbits and is modified to include the effects of energy dissipation in the body. It is shown that the concept of Adiabatic Invariants allows one to analyze the influence of slow changes in the material volume and elasticity.

Lie algebras associated with motion in axisymmetric electromagnetic fields

The existence and analytical form of a vector constant of the classical relativistic planar motion of a point charge in an arbitrary time‐independent axisymmetric electromagnetic field are established. The components of the vector are utilized, in conjunction with the angular momentum, to construct realizations of the Lie algebras of the Euclidean group E(2), of the special unitary group SU(2), and of the Ladder operators of the harmonic oscillator. The charge is assumed to move in an externally prescribed field. The formulation is gauge invariant.

RELATIVISTIC DYNAMICS OF A POINT CHARGE IN A MAGNETIC-MONOPOLE FIELD.

This paper derives and interprets the constants of the charges motion. The physical meaning of these constants and their use in discussing the over‐all motion of the charge are presented.

Perturbation of the vectorial elements of non-Keplerian orbits

A vector method of treating perturbations of orbits in arbitrary spherically symmetric fields of force is presented. This formulation makes it possible to commence the vector perturbational analysis of motion in an arbitrary non-symmetric field from an intermediate orbit which incorporates all of the spherically symmetric part of the field rather than from a simple Keplerian orbit. Only the nonsymmetric part of the field need then be considered the source of the perturbation.

Vector constants of the motion in spherically symmetric fields of force

The determination of the explicit form of vector constants of the motion for a point mass moving in an arbitrary spherically symmetric time-independent potential is reduced to the solution of an ordinary second-order linear differential equation. The vectors to be determined are assumed to be orthogonal to the angular momentum. The differential equation is solved for some particular fields of force and the corresponding vectors are constructed.