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The principle of least action in geometry and dynamics. (English) Zbl 1060.37048

Lecture Notes in Mathematics 1844. Berlin: Springer (ISBN 3-540-21944-7/pbk). xii, 128 p. (2004).
This book is about the symplectic nature of the minimal action. The first two chapters contain introductions to the Aubry-Mather theory of a minimal action for monotone twist maps, the Mather theory of the minimal action for convex Lagrangians, and the theory of Mané’s critical value. The following part of the book is devoted mostly to results of the author. It is showed how to associate the minimal action to planar convex billiards and to a generic elliptic fixed-point on a plane. It is proved that in the first case, this action is invariant with respect to continuous deformations of a domain preserving the length spectrum and that in the second case, this action is a symplectic invariant. To a generic elliptic closed geodesic on a two-dimensional Riemannian manifold there associates the germ of the minimal action which is the length spectrum invariant under continuous deformations of the metric.
Some other results relate to high-dimensional dynamical systems. For a Hamiltonian diffeomorphism of the unit ball cotangent bundle to an \(n\)-dimensional torus generated by a convex Hamiltonian, the author establishes a lower estimate on the Hofer distance between the diffeomorphism and the identity map in terms of the minimal action associated to the Hamiltonian.
In the last chapter of the book, the author exposes his joint results with G. Paternain and L. Polterovich which, in particular, imply that the stable norm of a Riemannian metric on a torus and the minimal action of a convex Lagrangian on the tangent bundle to a torus both admit a symplectically invariant description.

MSC:

37J50 Action-minimizing orbits and measures (MSC2010)
37E40 Dynamical aspects of twist maps
53D05 Symplectic manifolds (general theory)
58E30 Variational principles in infinite-dimensional spaces
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
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