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Cosymplectic geometry, reductions, and energy-momentum methods with applications

J. de Lucas, A. Maskalaniec, B. M. Zawora

TL;DR

The paper develops a comprehensive cosymplectic framework for energy-momentum methods applicable to time-dependent Hamiltonian systems, enabling reductions and stability analysis without requiring Ad^*-equivariant momentum maps. It introduces general momentum maps, a cosymplectic Marsden–Weinstein reduction, and a novel cosymplectic-to-symplectic reduction, along with the new notion of gradient relative equilibrium points. Through quantum-system examples (two- and n-level Schrödinger dynamics) and a reduced circular three-body problem, the authors illustrate how reduced dynamics and Hessian-based stability criteria can be obtained without explicitly forming reduced spaces. This work broadens the geometric toolkit for non-autonomous systems, with potential impact on quantum control and celestial mechanics by handling time as a structural coordinate rather than a parameter.

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 ${\rm Ad}^*$-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.

Cosymplectic geometry, reductions, and energy-momentum methods with applications

TL;DR

The paper develops a comprehensive cosymplectic framework for energy-momentum methods applicable to time-dependent Hamiltonian systems, enabling reductions and stability analysis without requiring Ad^*-equivariant momentum maps. It introduces general momentum maps, a cosymplectic Marsden–Weinstein reduction, and a novel cosymplectic-to-symplectic reduction, along with the new notion of gradient relative equilibrium points. Through quantum-system examples (two- and n-level Schrödinger dynamics) and a reduced circular three-body problem, the authors illustrate how reduced dynamics and Hessian-based stability criteria can be obtained without explicitly forming reduced spaces. This work broadens the geometric toolkit for non-autonomous systems, with potential impact on quantum control and celestial mechanics by handling time as a structural coordinate rather than a parameter.

Abstract

Classical energy-momentum methods study the existence and stability properties of solutions of -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 -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 -equivariance of momentum maps. Eigenfunctions of -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.
Paper Structure (15 sections, 29 theorems, 193 equations, 1 figure)

This paper contains 15 sections, 29 theorems, 193 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basics on symplectic and cosymplectic geometry
  3. Cosymplectic, symplectic, and Poisson manifolds
  4. Momentum maps
  5. General momentum maps
  6. Cosymplectic Marsden--Weinstein reduction
  7. Lyapunov stability
  8. Characterisation of relative equilibrium points
  9. Stability on the reduced manifold
  10. Stability, reduced manifold, and relative equilibrium points
  11. A two-state quantum system
  12. Relative equilibrium points of sstate quantum system
  13. Cosymplectic-to-symplectic reduction and gradient relative equilibrium points
  14. A reduced circular three-body problem
  15. Conclusions and Outlook

Key Result

Proposition 2.1

The gradient vector field of $f\in C^\infty(M)$ relative to $(M,\omega,\eta)$ reads $\nabla \,f=X_f+(Rf)R$. If $Rf=0$, then $[R,X_f]=0$.

Figures (1)

  • Figure 1: Example of solutions of Hamilton equations on a cosymplectic manifold $(\mathbb{S}^1\times T^*\mathbb{R},\omega_{T^*\mathbb{R}},\eta_{\mathbb{S}^1})$ for $\mathbb{S}^1$ being the circle of radius one and centred at zero. Just the coordinates of solutions in $\mathbb{S}^1\times \mathbb{R}$ are represented

Theorems & Definitions (54)

  • Proposition 2.1
  • proof
  • Lemma 2.1
  • proof
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.1
  • proof
  • Definition 3.3
  • Proposition 4.1
  • ...and 44 more