Invariant Lagrange submanifolds of dissipative systems

TL;DR

This paper investigates invariant Lagrange submanifolds within dissipative systems by analyzing solutions to modified Hamilton-Jacobi equations on compact manifolds.

Contribution

It introduces a novel approach to understanding invariant structures in dissipative systems through modified Hamilton-Jacobi equations.

Findings

Identification of conditions for invariance of Lagrange submanifolds

Existence results for solutions to the modified Hamilton-Jacobi equations

Insights into the stability properties of these solutions

Abstract

We study solutions of modified Hamilton-Jacobi equations H(du/dq,q) + cu(q) = 0, q \in M, on a compact manifold M .

Equations

$$H(z)=\frac{1}{2}⟨I_q(z+{\omega }_q),z+{\omega }_q⟩+V(q),z\in T_q^{\ast }M.$$

$$H(p,q)=\frac{1}{2}(p+\omega (q){)}^{\ast }{I}_q(p+\omega (q))+V(q),$$

$$B_H\stackrel{def}{=}\{z\in T^{\ast }M:H(z-{\omega }_{\pi (z)})\le \underset{q\in M}{\max }H(0_q)\},$$

$$\xi \mapsto \mathfrak{R}(\xi ,I_{q}z)I_{q}z+({\nabla }_q^{2}V)\xi ,\xi \in T_{q}M,$$

$$r=\max \{\rho (z):z\in B_H\}.$$

$${\mathfrak{I}}_{\alpha }(\gamma )=\underset{0}{\overset{+\infty }{\int }}{e}^{-\alpha t}\left(\frac{1}{2}⟨I_{\gamma (t)}^{-1}\dot{\gamma }(t),\dot{\gamma }(t)⟩-V(\gamma (t))+⟨{\omega }_{\gamma (t)},\dot{\gamma }(t)⟩\right)dt,\gamma \in {\Omega }_{\alpha }.$$

$$H(p,q)=\frac{1}{2}|p+a{|}^2+V(q),$$

$$\dot{q}=p+a,\dot{p}=-\frac{dV}{dq}-\alpha p.$$

Full text

Let $M$ be a compact Riemannian manifold of class $C^{k},\ k\geq 2,$ with the Riemannian structure $(\xi,\eta)\mapsto\langle I_{q}^{-1}\xi,\eta\rangle,\ \xi,\eta\in T_{q}M,\ q\in M$, where $I_{q}:T^{*}_{q}M\to T_{q}M$ is a self-adjoint linear map such that the quadratic form $z\mapsto\langle z,I_{q}z\rangle,\ z\in T_{q}^{*}M,$ is positive definite.

Let $V\in C^{k}(M)$ and $\omega$ be a closed differential 1-form on $M$ of class $C^{k}$ such that $\nabla\omega=0$, where $\nabla\omega$ is the covariant derivative of $\omega$. We consider the Hamiltonian function $H\in C^{k}(T^{*}M)$ defined by the formula:

[TABLE]

Let $\vec{H}$ be the Hamiltonian vector field on $T^{*}M$ associated to $H$ and $\ell$ be the “vertical” Euler vector field of the vector bundle $T^{*}M\to M$. In local coordinates, $z=(p,q),\ p,q\in\mathbb{R}^{n},\ T^{*}_{q}M=(\mathbb{R}^{n},q)$,

[TABLE]

$\vec{H}(p,q)=\sum\limits_{i}\left(\frac{\partial H}{\partial p^{i}}\frac{\partial}{\partial q^{i}}-\frac{\partial H}{\partial q^{i}}\frac{\partial}{\partial p^{i}}\right),\ \ell(p,q)=\sum\limits_{i}p^{i}\frac{\partial}{\partial p^{i}}.$

We study the dissipative system $\dot{z}=\vec{H}(z)-\alpha\ell(z),$ where $\alpha$ is a positive constant. It is not hard to see that any bounded trajectory of this system is contained in the set

[TABLE]

where $0_{q}$ is the origin of $T^{*}_{q}M$ and $\pi:T^{*}M\to M,\ \pi(T^{*}_{q}M)=q$.

Given $z\in T^{*}M$, we denote by $\rho(z)$ the maximal eigenvalue of the symmetric operator

[TABLE]

where $\mathfrak{R}$ is the Riemannian curvature. Finally, we set

[TABLE]

Let $\Omega^{\alpha}$ be the set of all absolutely continuous curves $\gamma:[0,+\infty)\to M$ such that the integral $\int\limits_{0}^{+\infty}e^{-\alpha t}\langle I^{-1}_{\gamma(t)}\dot{\gamma}(t),\dot{\gamma}(t)\rangle\,dt$ converges. We introduce the discounted action functional

[TABLE]

• •

$u\in C^{k}(M)$* and the map $(H,\alpha)\mapsto u$ is continuous in the $C^{2}$-topology.*

• •

The function $u$ satisfies the modified Hamilton–Jacobi equation $H(du)+\alpha u=0$ and $\{d_{q}u:q\in M\}\subset T^{*}M$ is an invariant submanifold of the system $\dot{z}=\vec{H}(z)-\alpha\ell(z)$.

• •

There exists a containing 0 neighborhood $\mathcal{O}$ of $u$ in $C^{2}(M)$ such that $\forall v_{0}\in\mathcal{O}$ the classical solution $v_{t}$ of the Cauchy problem $\frac{\partial u_{t}}{\partial t}+H(du_{t})+\alpha u_{t}=0,\ u_{0}=v_{0}$, is defined for all $t\geq 0$ and $\|dv_{t}-du\|_{C^{1}}\to 0$ as $t\to+\infty$ with the exponential convergence rate.

Remark. Theorem 1 is applied to the Hamiltonians in $\mathbb{R}^{n}\times\mathbb{R}^{n}$ of the form

[TABLE]

where $a\in\mathbb{R}^{n}$ is a constant vector and $V$ is a smooth periodic potential. Then $r$ is the maximum of the eigenvalues of the matrices $\frac{d^{2}V}{dq^{2}},\ q\in\mathbb{R}^{n}$. If $r<\frac{\alpha^{2}}{4}$, then the equation $\frac{1}{2}|\frac{du}{dq}+a|^{2}+V(q)+\alpha u=0$ has a periodic $C^{k}$-solution $u$, where $k$ is maximal integer that is strictly smaller than $\frac{2}{1-\sqrt{1-\frac{4r}{\alpha^{2}}}}$. Moreover, $\{(\frac{du}{dq},q):q\in\mathbb{R}^{n}\}$ is an invariant submanifold of the system

[TABLE]

The proof of Theorem 1 can be derived from [1] and [2]. Indeed, Theorem 1 is an improvement of results of paper [1]. The improvement concerns more general Hamiltonians (nonzero forms $\omega$ are available), better smoothness of $u$, and stability properties. One can check that $\omega$ does not affect the canonical connection and the curvature operators; hence more general Hamiltonians do not require essential changes in the proof.

The better smoothness and stability follow from [2, Th.4.1]. Indeed, Prop. 1 in [1] implies that $\{d_{q}u:q\in M\}$ is a normally hyperbolic invariant submanifold (see [2] for the definition) of the flow generated by the vector field $\vec{H}(z)-\alpha\ell(z)$. Moreover, this normally hyperbolic invariant submanifold has zero unstable subbundle and can be actually called the “normally stable” invariant submanifold. Theorem 4.1 in [2] contains the estimate for the smoothness class of the normally hyperbolic invariant submanifold in terms of the Lyapunov exponents while the analysis of the proof of Prop. 1 in [1] gives explicit estimates for the Lyapunov exponents in terms of $r$ and $\alpha$.