Abstract
Abstract In this work, we study the dynamics of two-dimensional convex billiards with time-periodically perturbed boundaries. Due to the non-conservation of the particle’s energy, the billiard map becomes a four-dimensional twist map. Rather than analyzing this high-dimensional twist map directly, we develop a novel variational approach to address this problem. Additionally, we present and prove a Lagrangian version of Moser’s twist map theorem in higher dimensions to support our variational method. As a result, we demonstrate the existence of invariant tori near the boundary of a convex billiard when the time-periodic perturbation is sufficiently small. In addition, for a time-periodic perturbed circular billiard, the invariant tori are not necessarily located near the boundary.
Published Version
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