Kepler problem

Abstract: The relation between the solutions of the time‐independent Schrodinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half‐width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well‐defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropicgermanium.

Abstract: I. The harmonic oscillator.- 1. Hamilton's equations and Sl symmetry.- 2. S1 energy momentum mapping.- 3. U(2) momentum mapping.- 4. The Hopf fibration.- 5. Invariant theory and reduction.- 6. Exercises.- II. Geodesics on S3.- 1. The geodesic and Delaunay vector fields.- 2. The SO(4) momentum mapping.- 3. The Kepler problem.- 3.1 The Kepler vector field.- 3.2 The so(4) momentum map.- 3.3 Kepler's equation.- 3.4 Regularization of the Kepler vector field.- 4. Exercises.- III The Euler top.- 1. Facts about SO(3).- 1.1 The standard model.- 1.2 The exponential map.- 1.3 The solid ball model.- 1.4 The sphere bundle model.- 2. Left invariant geodesics.- 2.1 Euler-Arnol'd equations on SO(3).- 2.2 Euler-Arnol'd equations on T1S2 x R3.- 3. Symmetry and reduction.- 3.1 Construction of the reduced phase space.- 3.2 Geometry of the reduction map.- 3.3 Euler's equations.- 4. Qualitative behavior of the reduced system.- 5. Analysis of the energy momentum map.- 6. Integration of the Euler-Arnol'd equations.- 7. The rotation number.- 7.1 An analytic formula.- 7.2 Poinsot's construction.- 8. A twisting phenomenon.- 9. Exercises.- IV. The spherical pendulum.- 1. Liouville integrability.- 2. Reduction of the Sl symmetry.- 3. The energy momentum mapping.- 4. Rotation number and first return time.- 5. Monodromy.- 6. Exercises.- V. The Lagrange top.- 1. The basic model.- 2. Liouville integrability.- 3. Reduction of the right Sl action.- 3.1 Reduction to the Euler-Poisson equations.- 3.2 The magnetic spherical pendulum.- 4. Reduction of the left S1 action.- 5. The Poisson structure.- 6. The Euler-Poisson equations.- 6.1 The Poisson structure.- 6.2 The energy momentum mapping.- 6.3 Motion of the tip of the figure axis.- 7. The energy momemtum mapping.- 7.1 Topology of ???1(h,a,b) and H?1(h).- 7.2 The discriminant locus.- 7.3 The period lattice and monodromy.- 8. The Hamiltonian Hopf bifurcation.- 8.1 The linear case.- 8.2 The nonlinear case.- 9. Exercises.- Appendix A. Fundamental concepts.- 1. Symplectic linear algebra.- 2. Symplectic manifolds.- 3. Hamilton's equations.- 4. Poisson algebras and manifolds.- 5. Exercises.- Appendix B. Systems with symmetry.- 1. Smooth group actions.- 2. Orbit spaces.- 2.1 Orbit space of a proper action.- 2.2 Orbit space of a free action.- 2.3 Orbit space of a locally free action.- 3. Momentum mappings.- 3.1 General properties.- 3.2 Normal form.- 4. Reduction: the regular case.- 5. Reduction: the singular case.- 6. Exercises.- Appendix C. Ehresmann connections.- 1. Basic properties.- 2. The Ehresmann theorems.- 3. Exercises.- Appendix D. Action angle coordinates.- 1. Local action angle coordinates.- 2. Monodromy.- 3. Exercises.- Appendix E. Basic Morse theory.- 1. Preliminaries.- 2. The Morse lemma.- 3. The Morse isotopy lemma.- 4. Exercises.- Notes.- References.- Acknowledgements.

Abstract: Superintegrable Hamiltonians in three degrees of freedom possess more than three functionally independent globally defined and single-valued integrals of motion. Some familiar examples, such as the Kepler problem and the harmonic oscillator, have been known since the time of Laplace. Here, a classification theorem is given for superintegrable potentials with invariants that are quadratic polynomials in the canonical momenta. Such systems must possess separable solutions to the Hamilton-Jacobi equation in more than one coordinate system. There are 11 coordinate systems for which the Hamilton-Jacobi equation separates in ${\mathit{openR}}^{3}$. One coordinate system may be arbitrarily rotated or translated with respect to the other, yielding 66 distinct cases. In each case, the differential equations for separability in the two coordinates are integrated to give a complete list of all superintegrable potentials with four or five quadratic integrals. The tables---which may be consulted independently of the main body of the paper---list the distinct superintegrable potentials, the separating coordinates, and the isolating integrals of the motion. If there exist five isolating integrals, then all finite classical trajectories are closed; if only four, then the trajectories are restricted to two-dimensional surface. An extraordinary consequence of the work is the discovery of perturbations to both the Kepler problem and the harmonic oscillator that do not destroy the fragile degeneracy. The perturbed systems still have five isolating integrals of the motion.

Abstract: A Primer on Line Integral Methods A general framework Geometric integrators Hamiltonian problems Symplectic methods s-stage trapezoidal methods Runge-Kutta line integral methods Examples of Hamiltonian Problems Nonlinear pendulum Cassini ovals Henon-Heiles problem N-body problem Kepler problem Circular restricted three-body problem Fermi-Pasta-Ulam problem Molecular dynamics Analysis of Hamiltonian Boundary Value Methods (HBVMs) Derivation and analysis of the methods Runge-Kutta formulation Properties of HBVMs Least square approximation and Fourier expansion Related approaches Implementing the Methods and Numerical Illustrations Fixed-point iterations Newton-like iterations Recovering round-off and iteration errors Numerical illustrations Hamiltonian Partial Differential Equations The semilinear wave equation Periodic boundary conditions Nonperiodic boundary conditions Numerical tests The nonlinear Schrodinger equation Extensions Conserving multiple invariants General conservative problems EQUIP methods Hamiltonian boundary value problems Appendix: Auxiliary Material Bibliography Index