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Action-Minimizing Curves for Tonelli Lagrangians

Action-Minimizing Curves for Tonelli Lagrangians

(p.48) Chapter Four Action-Minimizing Curves for Tonelli Lagrangians
Action-minimizing Methods in Hamiltonian Dynamics (MN-50)
Alfonso Sorrentino
Princeton University Press

This chapter discusses the notion of action-minimizing orbits. In particular, it defines the other two families of invariant sets, the so-called Aubry and Mañé sets. It explains their main dynamical and symplectic properties, comparing them with the results obtained in the preceding chapter for the Mather sets. The relation between these new invariant sets and the Mather sets is described. As a by-product, the chapter introduces the Mañé's potential, Peierls' barrier, and Mañé's critical value. It discusses their properties thoroughly. In particular, it highlights how this critical value is related to the minimal average action and describes these new concepts in the case of the simple pendulum.

Keywords:   action-minimizing orbits, invariant sets, Aubry set, Mañé set, Maher sets, Mañé's potential, Peierls' barrier, Mañé's critical value

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