Ruprecht-Karls-Universität Heidelberg
Variational Methods for Convex Hamiltonian Systems
Summer Semester 2020 - Master Seminar


An important problem in mechanics is to determine physical trajectories satisfying given boundary conditions (for example trajectories which are periodic or connect two fixed point). For Hamiltonian systems these trajectories bear a special meaning: they are the stationary points of the action functional. In particular, we will concentrate on stationary points that globally minimize the action and see that they build interesting invariant sets (so-called Aubry-Mather Sets) for the dynamics. In the first part of the seminar, we will look at a toy model of this theory given by monotone twist maps, minimal geodesics on the two-torus and the Frenkel-Kontorova model following a paper by Bangert. In the second part, we will deal with the general picture and investigate its relationship with Weak KAM Theory, which studies weak solutions of the Hamilton-Jacobi equation.

Plan of the seminar

Follow the link for detailed information on each talk.

Part I: Basics on Tonelli Lagrangians
  1. 20.04.2020 (Alex Arnhold): The action functional (Notes and Slides)
  2. 27.04.2020 (Christian Alber): Tonelli's Theorem (Notes)
  3. 04.05.2020 (Camillo Tissot): Weierstrass Theorem and global minimizers (Slides)
  4. 11.05.2020 (Gabriele Benedetti): The Hamilton-Jacobi method (sketch) (Notes and Slides)
Part II: A discrete model for minimizing orbits
  1. 11.05.2020 (Gabriele Benedetti): Motivation: minimizing geodesics on oriented surfaces (Notes)
  2. 18.05.2020 (Gabriele Benedetti): The discrete model: existence of periodic minimizers and order properties of minimizers (Notes)
  3. 25.05.2020 (Gabriele Benedetti): Structure of the set of minimal orbits: circle homeomorphisms and rotation number (Notes)
  4. 08.06.2020 (Raphael Schlarb): Application to monotone twist maps (Notes)
Part III: Minimizing orbits for Tonelli Lagrangians - Mañé Theory
  1. 15.06.2020 (Maximilian Schmahl): Aubry and Mañé sets (Notes)
  2. 22.06.2020 (Levin Maier): Graph property for the Aubry set and generalities on probability measures (Notes)
Part IV: Minimizing orbits for Tonelli Lagrangians - Mather Theory
  1. 29.06.2020 (Valerio Assenza): Mather measures I: the example of KAM tori and the alpha function (Notes)
  2. 06.07.2020 (Valerio Assenza): Mather measures II: the beta function and the example of the pendulum (Notes)
Part V: Minimizing orbits for Tonelli Lagrangians -- Weak KAM Theory
  1. 13.07.2020 (Johanna Bimmermann): Weak KAM solutions and dynamics (Notes)
  2. 20.07.2020 (Leon Happ): Lax-Oleinik semigroup and weak KAM solutions (Notes)
  3. 27.07.2020 (Gabriele Benedetti): Complements about Mather measures and Global Tonelli minimizers vs free-time minimizers (Notes)
  1. [Arn] Marie-Claude Arnaud, On a theorem due to Birkhoff. Geom. Funct. Anal. 20 (2010), no. 6, 1307-1316.
  2. [Ban] Victor Bangert, Mather sets for twist maps and geodesics on tori. Dynamics reported, Vol. 1, 1-56, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester, 1988.
  3. [CI] Gonzalo Contreras and Renato Iturriaga, Global minimizers of autonomous Lagrangians, 22 Coloquio Brasileiro de Matematica, IMPA, 1999.
  4. [Fat] Albert Fathi, Weak KAM Theorem in Lagrangian Dynamics, Preliminary Version, June 2008.
  5. [Fat'] Albert Fathi, Weak KAM theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, Proc. of the ICM-Seoul 2014. Vol III, 597-621, 2014.
  6. [KH] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, 1995.
  7. [Maz] Marco Mazzucchelli, Critical point theory for Lagrangian systems. Progress in Mathematics, 293. Birkhäuser/Springer Basel AG, Basel, 2012.
  8. [Sib] Karl Friedrich Siburg, The Principle of Least Action in Geometry and Dynamics, Lecture notes in Mathematics 1844, Springer, 2004.
  9. [Sor] Alfonso Sorrentino, Action-minimizing methods in Hamiltonian dynamics. An introduction to Aubry-Mather theory. Mathematical Notes, 50. Princeton University Press, 2015.