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Mechanical presymplectic structures and Marsden-Weinstein reduction of time-dependent Hamiltonian systems

I. Gutierrez-Sagredo, D. Iglesias Ponte, J. C. Marrero, E. Padrón

TL;DR

This work addresses the reduction of time-dependent Hamiltonian systems beyond the cosymplectic framework. It identifies fundamental limitations of Albert's Marsden–Weinstein reduction for cosymplectic manifolds and introduces mechanical presymplectic structures as a natural generalization. By proving a reduced mechanical presymplectic theorem and then reformulating evolution dynamics as Reeb dynamics through a modified cosymplectic pair $(\omega_H,\eta)$ with a shifted momentum map $J_H$, the authors obtain a robust reduction procedure for time-dependent systems. The theory is illustrated with two physically relevant examples, showing reduced spaces and dynamics that are inaccessible to Albert's method. The proposed framework broadens the applicability of geometric reduction to time-dependent Hamiltonians and observer-dependent descriptions.

Abstract

In 1986, Albert proposed a Marsden-Weinstein reduction process for cosymplectic structures. In this paper, we present the limitations of this theory in the application of the reduction of symmetric time-dependent Hamiltonian systems. As a consequence, we conclude that cosymplectic geometry is not appropriate for this reduction. Motived for this fact, we replace cosymplectic structures by more general structures: mechanical presymplectic structures. Then, we develop Marsden-Weinstein reduction for this kind of structures and we apply this theory to interesting examples of time-dependent Hamiltonian systems for which Albert's reduction method doesn't work.

Mechanical presymplectic structures and Marsden-Weinstein reduction of time-dependent Hamiltonian systems

TL;DR

This work addresses the reduction of time-dependent Hamiltonian systems beyond the cosymplectic framework. It identifies fundamental limitations of Albert's Marsden–Weinstein reduction for cosymplectic manifolds and introduces mechanical presymplectic structures as a natural generalization. By proving a reduced mechanical presymplectic theorem and then reformulating evolution dynamics as Reeb dynamics through a modified cosymplectic pair with a shifted momentum map , the authors obtain a robust reduction procedure for time-dependent systems. The theory is illustrated with two physically relevant examples, showing reduced spaces and dynamics that are inaccessible to Albert's method. The proposed framework broadens the applicability of geometric reduction to time-dependent Hamiltonians and observer-dependent descriptions.

Abstract

In 1986, Albert proposed a Marsden-Weinstein reduction process for cosymplectic structures. In this paper, we present the limitations of this theory in the application of the reduction of symmetric time-dependent Hamiltonian systems. As a consequence, we conclude that cosymplectic geometry is not appropriate for this reduction. Motived for this fact, we replace cosymplectic structures by more general structures: mechanical presymplectic structures. Then, we develop Marsden-Weinstein reduction for this kind of structures and we apply this theory to interesting examples of time-dependent Hamiltonian systems for which Albert's reduction method doesn't work.

Paper Structure

This paper contains 16 sections, 11 theorems, 143 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Marsden-Weinstein reduction of a Hamiltonian system on a symplectic manifold
  3. Cosymplectic and symplectic reduction for time-dependent Hamiltonian systems
  4. Aim and contributions of the paper
  5. Organization of the paper
  6. Cosymplectic Hamiltonian systems and time-dependent Hamiltonian dynamics
  7. The limitations of Albert's reduction for discussing the Reeb and the evolution dynamics
  8. Albert's Theorem
  9. The limitations of Albert's Theorem for Reeb dynamics
  10. The limitations of Albert's Theorem for evolution dynamics
  11. Reduction of mechanical presymplectic manifolds
  12. Reduction of the evolution dynamics of a cosymplectic system
  13. Examples
  14. toc$N$-dimensional harmonic oscillatortoc$N$-dimensional harmonic oscillator seen by an observer that moves with a constant velocity (continuing with Example \ref{['example1']})
  15. Perturbation by a linearly polarized monocromatic plane wave
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $(M,\omega,\eta)$ be a cosymplectic manifold and $H:M\to {\mathbb R}$ be a Hamiltonian function. Then, the pair $(\omega_H,\eta)\in \Omega^2(M)\times \Omega^1(M)$, with is a new cosymplectic structure on $M$ such that the evolution vector field $E^{(\omega,\eta)}_H$ of $H$ with respect to $(\omega,\eta)$ is just the Reeb vector field ${\mathcal{R}}_H$ of $(\omega_H,\eta).$

Figures (1)

  • Figure 1: Cut with $p_3=0, q^2=0,q^3=0$ of the level sets $J_H^{-1} (0)$ for $e A_0 = 1$ (left) and $e A_0 = 0.1$ (right). We have used unit such that $m=c=1$.

Theorems & Definitions (30)

  • Proposition 2.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Theorem 3.4
  • Remark 3.5
  • Example 3.6
  • Proposition 3.7
  • proof
  • ...and 20 more