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.