[Paper Review] Is Symplectic-Energy-Momentum Integration Well-Posed?

Yosi Shibberu|ArXiv.org|Aug 4, 2006

Numerical methods for differential equations19 references3 citations

TL;DR

This paper investigates the well-posedness of symplectic-energy-momentum (SEM) integrators by analyzing the discrete-time Hamilton (DTH) equations in extended phase space. It proves that for nonlinear Hamiltonian systems, including the nonlinear pendulum, solutions to the DTH equations may fail to exist for arbitrarily small time steps at certain points in phase space, challenging the assumption that SEM integration is always well-posed despite its symplectic and energy-conserving properties.

ABSTRACT

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.

Motivation & Objective

• To investigate whether symplectic-energy-momentum (SEM) integration is well-posed by analyzing the solvability of the discrete-time Hamilton (DTH) equations.
• To identify specific points in extended phase space where the DTH equations of SEM integration have no solution, even for arbitrarily small time steps.
• To extend existence and uniqueness results for DTH dynamics beyond linear systems to nonlinear Hamiltonian systems.
• To clarify the conditions under which the variational formulation of SEM integration fails to produce solutions, despite its theoretical appeal.

Proposed method

• Formulates the discrete-time Hamiltonian (DTH) dynamics using a variational principle in extended phase space, treating time as a generalized coordinate with conjugate momentum.
• Derives the DTH equations from a stationary action principle with constraints on the Hamiltonian at midpoints of time steps.
• Analyzes the solvability of the resulting nonlinear algebraic equations by studying the behavior of a function $ g(s, z_k) $ related to the action and energy constraints.
• Applies the Intermediate Value Theorem and monotonicity arguments to determine existence and uniqueness of solutions for the time-step parameter $ s_k $.
• Uses bounds derived from Taylor expansions and Lipschitz-type estimates to establish regions in phase space where solutions may not exist.
• Employs the nonlinear pendulum as a concrete example to illustrate the failure of solution existence under specific energy and momentum conditions.

Experimental results

Research questions

• RQ1 Under what conditions do the DTH equations of SEM integration fail to have a solution for arbitrarily small time steps?
• RQ2 Are there points in extended phase space where the DTH equations are ill-posed, even when the time step is infinitesimally small?
• RQ3 How do energy and momentum constraints in the DTH formulation affect the solvability of the discrete equations?
• RQ4 Can the failure of solution existence be quantified in terms of phase space variables such as $ \mathcal{H}_k $, $ \psi_k $, and $ \Lambda_k $?
• RQ5 Does the non-existence of solutions contradict the theoretical justification of SEM integrators, particularly in light of Ge’s Theorem?

Key findings

• Solutions to the DTH equations of SEM integration do not exist for arbitrarily small time steps at certain points in extended phase space, even for nonlinear Hamiltonian systems.
• For the nonlinear pendulum, existence fails when $ \mathcal{H}_k / \psi_k > \frac{2}{3}(\psi_k / \psi_k')^2 $, indicating a threshold beyond which no solution exists in $ (0, S_k) $.
• When $ S_k \geq 6 $ and $ \mathcal{H}_k / \psi_k > \frac{2}{3}(\psi_k / \psi_k')^2 $, no solution exists in $ (0, S_k) $, and two solutions may coexist in $ (0, 6/5) $ and $ (6/5, S_k) $ under certain conditions.
• Existence fails on $ (-S_k, 0) $ when $ \mathcal{H}_k / \psi_k > \frac{1}{16}\Lambda_k^2(2 - S_k) $ and $ S_k < 6/5 $, indicating a region of non-solvability.
• The function $ g(s, z_k) $ is strictly monotonic in $ s $, ensuring uniqueness of solutions where they exist, as established via Lemma 10 (ii) and (iii).
• For $ \mathcal{H}_k / \psi_k' < -\frac{1}{16}\Lambda_k^3 $, no solution exists in $ [-\Lambda_k, \Lambda_k] $, demonstrating a regime of ill-posedness in the time-step parameter.