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Exact solvability of Hamiltonian systems via Poisson \texorpdfstring{\cinf}{cinf}-structures

A. J. Pan-Collantes, C. Sardón, X. Zhao

TL;DR

The paper introduces Poisson $\mathcal{C}^{\infty}$-structures, an ordered set of $2n-2$ functions $\mathcal{F}$ with $f_0:=H$ that close triangularly under the Poisson bracket, to achieve exact solvability of Hamiltonian systems without requiring conserved quantities. It proves that such a structure induces a $\mathcal{C}^{\infty}$-structure for the Hamiltonian distribution, enabling a constructive integration through $2n-1$ Pfaffian equations via explicitly built 1-forms $\omega_i$ defined from the bracket matrix $\mathbf{F}$. The method yields an explicit, local integration algorithm applicable to time-dependent Hamiltonians by working in an extended phase space, and is illustrated on the two-particle non-periodic Toda lattice and on waterbag reductions of the Vlasov equation, including a detailed $N=2$ analysis that leads to Weierstrass elliptic-function solutions. This framework offers a new path to exact solvability beyond classical first-integral-based integrability, with particular relevance to finite-dimensional reductions in plasma physics.

Abstract

We introduce a framework for the exact integration of Hamiltonian systems based on an ordered family of functions whose Poisson brackets close in a triangular way. In contrast with Liouville--Arnold integrability and noncommutative variants, the functions entering this closure need not be first integrals. The triangular Poisson relations instead generate a \cinf-structure on phase space, given by the Hamiltonian vector fields of the family, and this structure can be exploited algorithmically. We show that the equations of motion can be reduced to and solved by a finite sequence of completely integrable Pfaffian equations. We call the resulting geometric object a Poisson \cinf-structure and provide a systematic Pfaffian integration procedure that applies even when a complete set of conserved quantities is unavailable. The construction is illustrated on two models of physical interest: the two-particle non-periodic Toda lattice and the multi-waterbag reduction of the Vlasov equation. Also, we discuss how the theory extends to time-dependent Hamiltonian systems.

Exact solvability of Hamiltonian systems via Poisson \texorpdfstring{\cinf}{cinf}-structures

TL;DR

The paper introduces Poisson -structures, an ordered set of functions with that close triangularly under the Poisson bracket, to achieve exact solvability of Hamiltonian systems without requiring conserved quantities. It proves that such a structure induces a -structure for the Hamiltonian distribution, enabling a constructive integration through Pfaffian equations via explicitly built 1-forms defined from the bracket matrix . The method yields an explicit, local integration algorithm applicable to time-dependent Hamiltonians by working in an extended phase space, and is illustrated on the two-particle non-periodic Toda lattice and on waterbag reductions of the Vlasov equation, including a detailed analysis that leads to Weierstrass elliptic-function solutions. This framework offers a new path to exact solvability beyond classical first-integral-based integrability, with particular relevance to finite-dimensional reductions in plasma physics.

Abstract

We introduce a framework for the exact integration of Hamiltonian systems based on an ordered family of functions whose Poisson brackets close in a triangular way. In contrast with Liouville--Arnold integrability and noncommutative variants, the functions entering this closure need not be first integrals. The triangular Poisson relations instead generate a \cinf-structure on phase space, given by the Hamiltonian vector fields of the family, and this structure can be exploited algorithmically. We show that the equations of motion can be reduced to and solved by a finite sequence of completely integrable Pfaffian equations. We call the resulting geometric object a Poisson \cinf-structure and provide a systematic Pfaffian integration procedure that applies even when a complete set of conserved quantities is unavailable. The construction is illustrated on two models of physical interest: the two-particle non-periodic Toda lattice and the multi-waterbag reduction of the Vlasov equation. Also, we discuss how the theory extends to time-dependent Hamiltonian systems.
Paper Structure (10 sections, 3 theorems, 98 equations)

This paper contains 10 sections, 3 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symplectic geometry and Hamiltonian systems
  4. cinf-structures of distributions
  5. Poisson cinf-structures and integration
  6. Integration algorithm
  7. Time-dependent Hamiltonian systems
  8. Applications to Vlasov Plasma
  9. Waterbags
  10. Conclusions

Key Result

Theorem 2.1

pancinf-sym Let $\mathcal{D}$ be a distribution of rank $k$ on an $m$-dimensional manifold $M$. If $\mathcal{D}$ admits a $\mathcal{C}^{\infty}$-structure $(X_1, \dots, X_{m-k})$, then the integral manifolds of $\mathcal{D}$ can be obtained by solving a sequence of $m-k$ completely integrable Pfaffi

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 3.2
  • ...and 5 more