[Paper Review] On the existence of orbits satisfying periodic or conormal boundary conditions for Euler-Lagrange flows

Luca Asselle|arXiv (Cornell University)|Nov 24, 2015

Quantum chaos and dynamical systems9 references3 citations

TL;DR

This Ph.D. thesis establishes the existence of periodic and conormal boundary condition orbits for Euler-Lagrange flows on compact manifolds using variational methods. By analyzing the free-time action functional and applying minimax principles, the author proves critical existence results for supercritical and subcritical energies, generalizing the Lusternik-Fet theorem and extending to magnetic flows on the 2-torus.

ABSTRACT

Let $(M,g)$ be a closed Riemannian manifold and $L:TM ightarrow \mathbb R$ be a Tonelli Lagrangian. In this thesis we study the existence of orbits of the Euler-Lagrange flow associated with $L$ satisfying suitable boundary conditions. We first look for orbits connecting two given closed submanifolds of $M$ satisfying the conormal boundary conditions: We introduce the Mañé critical value that is relevant for the problem and prove existence results for supercritical and subcritical energies; we also complement these with counterexamples, thus showing the sharpness of our results. We then move to the problem of finding periodic orbits: We provide an existence result of periodic orbits for non-aspherical manifolds generalizing the Lusternik-Fet Theorem, and a multiplicity result in case the configuration space is the 2-torus.

Motivation & Objective

• To establish the existence of orbits connecting two submanifolds $ Q_0 $ and $ Q_1 $ under conormal boundary conditions using variational techniques.
• To generalize the Lusternik-Fet theorem to the setting of conormal boundary conditions and energy levels.
• To investigate periodic orbits for magnetic flows on the 2-torus $ \mathbb{T}^2 $, particularly under oscillating magnetic fields.
• To extend existence results to the weakly-exact and closed but non-exact case of the magnetic form $ \sigma $.
• To provide counterexamples in subcritical energy regimes where such orbits may fail to exist.

Proposed method

• Formalize the free-time action functional $ \mathbb{A}_k $ on the Hilbert manifold $ \mathcal{M}_Q = H^1_Q([0,1],M) \times (0,\infty) $, identifying paths via time rescaling.
• Apply the Palais-Smale condition to ensure convergence of minimizing sequences in the variational setting.
• Use MaÑé critical values to classify energy levels and distinguish supercritical from subcritical regimes.
• Construct minimax classes via homotopy theory and relative homotopy groups to detect critical points of the action functional.
• Employ a Struwe-type monotonicity argument to control the behavior of Palais-Smale sequences in the minimax setting.
• Leverage the fact that conormal bundles are Lagrangian submanifolds of $ T^*M $, and use the Liouville 1-form vanishing on them to ensure geometric compatibility.

Experimental results

Research questions

• RQ1 Under what conditions does a solution exist to the Euler-Lagrange equation connecting two submanifolds $ Q_0 $ and $ Q_1 $ under conormal boundary conditions?
• RQ2 How do the MaÑé critical values influence the existence of such orbits, particularly in supercritical versus subcritical energy regimes?
• RQ3 Can the Lusternik-Fet theorem be generalized to cover orbits satisfying conormal boundary conditions?
• RQ4 What is the behavior of periodic orbits under oscillating magnetic fields on $ \mathbb{T}^2 $, and how does the magnetic form $ \sigma $ affect existence?
• RQ5 Are there cases where conormal boundary condition orbits fail to exist, and if so, under what conditions?

Key findings

• For supercritical energies, the existence of orbits satisfying conormal boundary conditions is guaranteed via the minimax method on the free-time action functional.
• In the subcritical regime, the existence of such orbits is not guaranteed, and the author provides counterexamples to demonstrate non-existence.
• The generalized Lusternik-Fet theorem is established, ensuring the existence of periodic orbits under conormal boundary conditions when the energy level is above the MaÑé critical value.
• For oscillating magnetic fields on $ \mathbb{T}^2 $, the existence of local minimizers is proven, and a Struwe-type monotonicity argument ensures convergence of minimizing sequences.
• The minimax class used in the proof is constructed via relative homotopy groups of the pair $ (D^k, \partial D^k) $, ensuring non-triviality of the critical class.
• The conormal bundle $ N^*Q $ is shown to be a Lagrangian submanifold of $ T^*M $, and the Liouville 1-form vanishes identically on it, which is essential for the variational structure.