Efficient Symplectic-Energy-Momentum Integration of Hamiltonian Dynamical Systems
نویسنده
چکیده
The implicit equations of a symplectic-energymomentum integrator are not easily solved, especially for small time steps. An inefficient, nested iteration scheme has typically been used to solve these equations. We describe new, more efficient, iteration schemes which avoid nested iterations. We present simulation results comparing five different secondorder, integration methods for two different types of threebody trajectories. Symplectic-energy-momentum integration is shown to be as efficient as the leapfrog method for one of these trajectories.
منابع مشابه
Stability by KAM Confinement of Certain Wild, Nongeneric Relative Equilibria of Underwater Vehicles with Coincident Centers of Mass and Buoyancy
Purely rotational relative equilibria of an ellipsoidal underwater vehicle occur at nongeneric momentum where the symplectic reduced spaces change dimension. The stability of these relative equilibria under momentum changing perturbations is not accessible by Lyapunov functions obtained from energy and momentum. A blow-up construction transforms the stability problem to the analysis of symmetry...
متن کاملStability of Poisson Equilibria and Hamiltonian Relative Equilibria by Energy Methods
We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generalisations of the Energy-Casimir and Energy-Momentum Methods. Using a topological generalisation of Lyapunov’s result that an extremal critical point of a conserved quantity is stable, we show that a Poisson equilibrium ...
متن کامل2 Example : The Nonlinear Pendulum
We provide new existence and uniqueness results for the discrete-time Hamilton (DTH) equations of a symplectic-energy-momentum (SEM) integrator. In particular, we identify points in extended-phase space where the DTH equations of SEM integration have no solution for arbitrarily small time steps. We use the nonlinear pendulum to illustrate the main ideas. 1 Background Is symplectic-energy-moment...
متن کاملEquivariant Constrained Symplectic Integration
We use recent results on symplectic integration of Hamiltonian systems with constraints to construct symplectic integrators on cotangent bundles of manifolds by embedding the manifold in a linear space. We also prove that these methods are equivariant under cotangent lifts of a symmetry group acting linearly on the ambient space and consequently preserve the corresponding momentum. These result...
متن کاملSymplectic Integration of Constrained Hamiltonian Systems by Composition Methods
Recent work reported in the literature suggests that for the long-term integration of Hamiltonian dynamical systems one should use methods that preserve the symplectic structure of the ow. In this paper we investigate the symplecticity of numerical integrators for constrained Hamiltonian systems with holonomic constraints. We will derive the following two results: (i) We show that any rst or se...
متن کامل