Global Minimizers of Autonomous Lagrangians
Gonzalo Contreras
& Renato Iturriaga
1. Introduction.
1-1. Lagrangian Dynamics.
1-2. The Euler-Lagrange equation.
1-3. The Energy function.
1-4. Hamiltonian Systems.
1-5. Examples.
2. Mañé critical value.
2-1. The action potential and the critical value.
2-2. Continuity of the critical value.
2-3. Holonomic measures.
2-4. Invariance of minimizing measures.
2-5. Ergodic characterization of the critical value.
2-6. The Aubry-Mather Theory.
2-6.a. Homology of measures.
2-6.b. The asymptotic cycle.
2-6.c. The alpha and beta functions.
2-6. Coverings.
3. Globally minimizing orbits.
3-1. Tonelli's theorem.
3-2. A priori compactness.
3-3. Energy of time-free minimizers.
3-4. The finite-time potential.
3-5. Global Minimizers.
3-6. Characterization of minimizing measures.
3-7. The Peierls barrier.
3-8. Graph Properties.
3-9. Coboundary Property.
3-10. Covering Properties.
3-11. Recurrence Properties.
4. The Hamiltonian viewpoint.
4-1. The Hamilton-Jacobi equation.
4-2. Dominated functions.
4-3. Weak solutions of the Hamilton-Jacobi equation.
4-4. Lagrangian graphs.
4-5. Finsler metrics.
4-6. Anosov energy levels.
4-7. The weak KAM Theory.
4-8. Construction of weak KAM solutions.
4-8.a. Finite Peierls barrier.
4-8.b. The compact case.
4-8.c. Busemann weak KAM solutions.
4-9. Higher energy levels.
4-10. The Lax-Oleinik semigroup.
4-11. The extended static classes.
5. Examples.
5-1. Riemannian Lagrangians.
5-2. Mechanic Lagrangians.
5-3. Symmetric Lagrangians.
5-4. Simple Pendulum.
5-5. The flat Torus T^n.
5-6. Flat domain for the beta function.
5-7. A Lagrangian with infinite Peierls barrier.
5-8. Horocycle flow.
6. Generic Lagrangians.
6-1. Generic Lagrangians.
6-2. Homoclinic Orbits.
Appendix.
A. Absolutely continuous functions.
B. Measure Theory.
C. Convex functions.
D. The Frenshel and Legendre Transforms.
E. Symplectic Linear Algebra.
Bibliography.
Index.