Introduction
What is the best way to implement the laws of physics on a computer? Typical implementations of Newton's law,
F = m a (Euler's
method or Runge-Kutta schemes for example) result in discretization
errors which violate the conservation of energy. Moreover, such
implementations are not coordinate invariant. Discrete mechanics
implementations attempt to overcome such limitations.
Discrete-Time Hamiltonian dynamics (DTH dynamics) is a discrete
mechanics which exactly conserves energy and linear and angular
momentum and has a coordinate-invariant formulation based on a discrete
variational principle. To my knowledge, DTH dynamics is the only known
example of what is called a symplectic-energy-momentum integrator.
Background
My
work on 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. In my 1994 paper, 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 explored other ways for using conservation of energy to
adapt the step size of the midpoint scheme. In my 1997 paper, I
characterized the asymptotic behavior of time in DTH dynamics.
In my
2005 paper, I explain how to regularize the points described in my 1994
paper in
a manner which reserves the symplectic-energy-momentum properties of
DTH dynamics. In my 2006 paper, I generalized the existence and
uniqueness results given in my 1992 Ph. D. Thesis.
In my 2009 paper, I describe efficient iteration schemes for DTH
dynamics and demonstrate that even though DTH dynamics is implicit, its
adaptive capabilities make it competitive with a widely used explicit
scheme - the leap frog
method - for a three-body, near collision, trajectory.
Papers
- Discrete-Time Hamiltonian Dynamics, Ph. D. Thesis, U. of Texas at Arlington, 1992.
- Time-Discretization of Hamiltonian Dynamcial Systems, Computers and Mathematics with Applications, 28 (10-12): 123-145, 1994.
- A Variable Time-Step Midpoint Scheme for Hamiltonian Systems, RHIT Mathematics Technical Report, MS TR 95-03, May 1995.
- A Discrete-Time Formulation of Hamiltonian Dynamics, June 28, 1997. (unpublished)
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 integrtor 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.
- How to Regularize a Symplectic-Energy- Momentum Integrator, arXiv:math/0507483, July 2005.
Animation of a one parameter family of DTH trajectories crossing psi = 0.
- Is Symplectic-Energy-Momentum Integration Well-Posed? arXiv:math-ph/0608016, August 2006.
- Efficient Symplectic-Energy-Momentum Integration, Proceedings of the 2009 International Conference on Scientific Computing. Las Vegas, Nevada, July 13-16, 2009.
Animation of a near-collision, three-body trajectory.
- Leap Frog Method
- DTH Dynamics
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Papers
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