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Global minimizers of autonomous Lagrangians. (English) Zbl 0957.37065
22\(^o\) Colóquio Brasileiro de Matemática. Rio de Janeiro: Instituto de Matemática Pura e Aplicada (IMPA). 148 p. (1999).
The action of a differentiable curve \(\gamma:[0,T]\to M\), on a smooth, boundaryless \(n\)-dimensional complete Riemannian manifold \(M\) is defined to be the integral (called the action integral), \[ A(\gamma)= \int^T_0 L\bigl( \gamma(t), \dot\gamma(t) \bigr) dt. \] One of the main problems of variational calculus is to find the curves that minimize the action. The authors study: the minimizing curves with free time interval using the special action potential function at the critical level; ergodic properties of the critical value of the action functional defined by the Borel probability measure on the cotangent bundle \(TM\); the Aubry-Mather theory of minimal action measures for positive-definite Lagrangian systems; the Tonelli, globally minimizing orbits and constructibility of curves that realize the action potential.
In general, they report on the Mañé theory [R. Mañé, Proc. Int. Congr. Math. Zürich 1994, Vol. 2, 1216-1220 (1995; Zbl 0848.58035)] of global and ergodic variational methods in conservative dynamics. The authors consider also the Hamilton-Jacobi equation in their variational framework studying the dynamics on the prescribed energy level and especially on Anosov energy levels.
Finally they prove that for the generic Lagrangians (in the sense of Mañé) there is a uniquely minimizing measure, which is uniquely ergodic and if the measure is supported on a periodic orbit then this orbit is hyperbolic.

37J50 Action-minimizing orbits and measures (MSC2010)
58E30 Variational principles in infinite-dimensional spaces
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
70H20 Hamilton-Jacobi equations in mechanics
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37A99 Ergodic theory