[Paper Review] On commuting Tonelli Hamiltonians: Time-periodic case

Xiaojun Cui|arXiv (Cornell University)|Jan 8, 2010

Quantum chaos and dynamical systems5 references4 citations

TL;DR

This paper establishes that two commuting time-periodic Tonelli Hamiltonians share identical Aubry sets, Mañé sets, and barrier functions, extending results from the autonomous case to the time-periodic setting. Despite the absence of an energy integral, the authors prove that the dynamical structures—such as weak minimizers, critical subsolutions, and action functionals—are preserved under commutation, using weak KAM theory and time-dependent Legendre transforms.

ABSTRACT

We show that the Aubry sets, the Mañé sets and Mather's barrier functions are the same for two commuting time-periodic Tonelli Hamiltonians.

Motivation & Objective

• To extend the theory of commuting autonomous Tonelli Hamiltonians to the time-periodic case, where the energy integral is absent and the Lax-Oleinik semigroup fails to converge.
• To investigate how the absence of time-independent energy conservation affects the structure of invariant sets such as the Aubry and Mañé sets in time-periodic systems.
• To establish the invariance of key dynamical objects—Aubry sets, Mañé sets, barrier functions—under the commutation condition $[H_1, H_2] = 0$ in the time-periodic setting.
• To explore the implications for the dynamics of the sum Hamiltonian $H_1 + H_2$, which cannot be directly inferred from the individual dynamics due to time dependence.
• To formulate a conjecture on the existence of a common $C^{1,1}$ critical subsolution for two commuting time-periodic Hamilton-Jacobi equations.

Proposed method

• Utilizes weak KAM theory and Mather theory to analyze the dynamics of time-periodic Tonelli Hamiltonians on a closed Riemannian manifold.
• Defines the commutation condition via the bracket $[H_1, H_2] = \{H_1, H_2\} + \partial_t H_1 - \partial_t H_2 = 0$, which generalizes the Poisson bracket to time-dependent systems.
• Applies the Legendre transformation $\mathcal{L}_H$ to relate Lagrangian and Hamiltonian dynamics, mapping weak minimizers and invariant measures between tangent and cotangent bundles.
• Introduces action functionals $h_H^{T}$ and their limit $h_H$, which define the barrier function $\rho_H$ via $\rho_H(q_1,[t_1], q_2,[t_2]) = h_H((q_1,[t_1]),(q_2,[t_2])) + h_H((q_2,[t_2]),(q_1,[t_1]))$.
• Uses the space of subsolutions $\mathcal{S}^+_H$ and $\mathcal{S}^-_H$ to define barrier functions via infima over pairs of subsolutions with matching asymptotic behavior.
• Applies the time-periodic version of the weak KAM theorem, relying on results from Massart and Bernard on $C^1$ and $C^{1,1}$ critical subsolutions.

Experimental results

Research questions

• RQ1 Do two commuting time-periodic Tonelli Hamiltonians share the same Aubry set and Mañé set?
• RQ2 How does the lack of an energy integral affect the convergence of the Lax-Oleinik semigroup and the structure of invariant measures?
• RQ3 Can the barrier functions and action functionals of two commuting time-periodic Hamiltonians be shown to be identical?
• RQ4 Is there a common $C^{1,1}$ critical subsolution for two commuting time-periodic Hamilton-Jacobi equations?
• RQ5 How do the dynamical structures of the sum Hamiltonian $H_1 + H_2$ relate to those of $H_1$ and $H_2$ when $[H_1, H_2] = 0$?

Key findings

• The Aubry sets $\stackrel{{\scriptstyle\ast}}{{A}}_{H_1}$ and $\stackrel{{\scriptstyle\ast}}{{A}}_{H_2}$ coincide for two commuting time-periodic Tonelli Hamiltonians.
• The Mañé sets $\stackrel{{\scriptstyle\ast}}{{N}}_{H_1}$ and $\stackrel{{\scriptstyle\ast}}{{N}}_{H_2}$ are identical under the commutation condition $[H_1, H_2] = 0$.
• The barrier functions $b_{H_1}$ and $b_{H_2}$ are equal, as are the action functionals $\rho_{H_1}$ and $\rho_{H_2}$, implying isometric structures on the projected phase space.
• The sets of weak minimizers $\dot{W}_{H_1}$ and $\dot{W}_{H_2}$, and their images under the Legendre transform, are identical in $TM \times \mathbb{T}$ and $T^*M \times \mathbb{T}$, respectively.
• The projected Aubry and Mañé sets $A_{H_1} = A_{H_2}$ and $N_{H_1} = N_{H_2}$ are equal in $M \times \mathbb{T}$.
• The paper conjectures that two commuting time-periodic Tonelli Hamiltonians admit at least one common $C^{1,1}$ critical subsolution for their respective Hamilton-Jacobi equations.