[Paper Review] Cosymplectic geometry, reductions, and energy-momentum methods with applications

J. de Lucas, A. Maskalaniec+1 more|arXiv (Cornell University)|Feb 12, 2023

Nonlinear Waves and Solitons4 citations

TL;DR

This paper introduces a novel cosymplectic energy-momentum method that generalizes classical energy-momentum techniques to time-dependent Hamiltonian systems by leveraging cosymplectic geometry, enabling the study of relative equilibrium points—including a new class called gradient relative equilibrium points—via a new cosymplectic-to-symplectic reduction. The method removes prior technical constraints like Ad*-equivariance and successfully applies to the restricted circular three-body problem and t-dependent Schrödinger equations.

ABSTRACT

Classical energy-momentum methods study the existence and stability properties of solutions of $t$-dependent Hamilton equations on symplectic manifolds whose evolution is given by their Hamiltonian Lie symmetries. The points of such solutions are called relative equilibrium points. This work devises a new cosymplectic energy-momentum method providing a new and more general framework to study $t$-dependent Hamilton equations. In fact, cosymplectic geometry allows for using more types of distinguished Lie symmetries (given by Hamiltonian, gradient, or evolution vector fields), relative equilibrium points, and reduction methods, than symplectic techniques. To make our work more self-contained and to fill some gaps in the literature, a review of the cosymplectic formalism and the cosymplectic Marsden-Weinstein reduction is included. Known and new types of relative equilibrium points are characterised and studied. Our methods remove technical conditions used in previous energy-momentum methods, like the ${ m Ad}^*$-equivariance of momentum maps. Eigenfunctions of $t$-dependent Schrödinger equations are interpreted in terms of relative equilibrium points in cosymplectic manifolds. A new cosymplectic-to-symplectic reduction is developed and a new associated type of relative equilibrium points, the so-called gradient relative equilibrium points, are introduced and applied to study the Lagrange points and Hill radii of a restricted circular three-body system by means of a not Hamiltonian Lie symmetry of the system.

Motivation & Objective

• To develop a generalized energy-momentum method for time-dependent Hamiltonian systems using cosymplectic geometry, overcoming limitations of classical symplectic approaches.
• To define and characterize new types of relative equilibrium points, including gradient relative equilibrium points, in cosymplectic manifolds.
• To establish a new cosymplectic-to-symplectic reduction technique that applies even when the dynamics is not Hamiltonian, extending standard Marsden–Weinstein reduction.
• To apply the framework to physical systems such as the restricted circular three-body problem and t-dependent Schrödinger equations, providing new dynamical insights.
• To remove technical assumptions like Ad*-equivariance of momentum maps, broadening the applicability of energy-momentum methods.

Proposed method

• Utilizes cosymplectic manifolds (M, ω, η) with closed forms ω and η satisfying ker ω ⊕ ker η = TM, allowing time to be treated as a coordinate.
• Defines three associated vector fields for any smooth function f: Hamiltonian X_f, gradient ∇f, and evolution E_f, with dynamics governed by E_h on M = T × P.
• Introduces a cosymplectic momentum map J^Φ: M → g* whose components are first integrals of the Reeb vector field and Hamiltonian functions of fundamental vector fields.
• Develops a new cosymplectic-to-symplectic reduction by quotienting along the gradient of a Casimir function Υ, yielding a symplectic reduced space.
• Applies the reduction to the restricted circular three-body problem by fixing a rotating frame, transforming the non-autonomous system into an autonomous Hamiltonian system.
• Uses the reduced Hamiltonian k(r, φ′, p_r, p_φ) to describe dynamics in the rotating frame, derived from the original cosymplectic vector field R_TB + X_h.

Experimental results

Research questions

• RQ1 How can energy-momentum methods be generalized to time-dependent systems beyond the symplectic framework?
• RQ2 What new types of relative equilibrium points emerge in cosymplectic geometry, and how do they differ from classical ones?
• RQ3 Can a cosymplectic-to-symplectic reduction be constructed that applies even when the original dynamics is not Hamiltonian?
• RQ4 How can the cosymplectic energy-momentum method be applied to physical systems like the restricted circular three-body problem?
• RQ5 What is the role of gradient vector fields in defining new equilibrium structures in non-autonomous mechanical systems?

Key findings

• The paper introduces a new class of relative equilibrium points—gradient relative equilibrium points—defined as points where π_*(R_TB + X_h) vanishes, which are not captured by standard symplectic methods.
• The cosymplectic-to-symplectic reduction yields a symplectic quotient manifold where the reduced dynamics is Hamiltonian with Hamiltonian function k(r, φ′, p_r, p_φ), explicitly derived in the paper.
• The reduced system for the restricted circular three-body problem becomes autonomous and describes motion in a rotating frame with angular frequency ϖ, enabling analysis of Lagrange points and Hill radii.
• The method removes the need for Ad*-equivariance of momentum maps, broadening the scope of energy-momentum techniques to more general systems.
• Eigenfunctions of t-dependent Schrödinger equations are interpreted as relative equilibrium points in cosymplectic manifolds, linking quantum mechanics to geometric mechanics.
• The approach successfully handles non-Hamiltonian vector fields in reduction, as demonstrated by the non-Hamiltonian nature of R_TB + X_h despite yielding a Hamiltonian reduced system.