Discrete-Time Hamiltonian Dynamics (DTH Dynamics)

Discrete-Time Hamiltonian Dynamics (DTH Dynamics)

Yosi Shibberu

Mathematics Department

Rose-Hulman Institute of Technology

Terre Haute, IN 47803

U.S.A.

Background

My work on a discrete-time theory for Hamiltonian dynamical systems (DTH dynamics) originated from my effort to obtain the exact energy and momentum conserving properties of the discrete mechanics of my thesis advisor, Professor Donald Greenspan, from the variational principle used in the discrete mechanics of Nobel Laureate T.D. Lee.

Existence and uniqueness results as well as preliminary work on the coordinate invariance of DTH dynamics is given in my 1992 Ph.D. Thesis [1]. In my 1994 paper [2], I proved that DTH dynamics is symplectic and hence a symplectic-energy-momentum integrator. I also described points in phase space which appeared to be singularities in DTH dynamics. My 1995 paper [3] explored other ways for using conservation of energy to adapt the step size of the midpoint scheme. In my 1997 paper [4], I characterized the asymptotic behavior of time in DTH dynamics. In my 2005 paper [5], I explain how to regularize the points described in my 1994 paper [2] in a manner which preserves the symplectic-energy-momentum properties of DTH dynamics. In my 2006 paper [6], I generalize the existence and uniqueness results given in my 1992 Ph.D. Thesis [1].

[1] Discrete-Time Hamiltonian Dynamics. Ph.D. Thesis, Univ. of Texas at Arlington, 1992. (PDF)

[2] Time-Discretization of Hamiltonian Dynamical Systems, Computers and Mathematics with Applications, 28(10-12):123-145, 1994. (PDF)

[3] A Variable Time-Step Midpoint Scheme for Hamiltonian Systems, RHIT Mathematics Technical Report MS TR 95-03, May 1995. (PDF)

[4] A Discrete-Time Formulation of Hamiltonian Dynamics, June 28, 1997 (unpublished). (PDF). Submitted to Physica D June 1997. Why symplectic-energy-momentum integration does not violate Ge’s Theorem is explained for the first time in this manuscript! This manuscript was freely shared with Professor Jerrold Marsden during a discussion I had with him after his lecture on discrete mechanics at the IAS/Park City Mathematics Institute’s summer session on Symplectic Geometry and Topology, on July 18, 1997, Park City, Utah, U.S.A. The manuscript predates C. Kane, J.E. Marsden and M. Ortiz announcement of a symplectic-energy-momentum integrator in the Journal of Mathematical Physics, 40, July 1999. The manuscript was updated February 25, 1998 to include a discussion of Professor Ernst Hairer’s meta-algorithm for variable-step symplectic integrators. (PDF).

[5] How to Regularize a Symplectic-Energy-Momentum Integrator, July, 2005 (PDF)

Animation of a one parameter family of DTH trajectories crossing ψ = 0. (avi)

[6] Is Symplectic-Energy-Momentum Integration Well-Posed?, August 2006 (PDF)