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1 Introduction. 7 11 Lagrangian Dynamics. . . . . . . . . . . . . . . . . . . . . 7
 

Summary: Contents
1 Introduction. 7
1­1 Lagrangian Dynamics. . . . . . . . . . . . . . . . . . . . . 7
1­2 The Euler­Lagrange equation. . . . . . . . . . . . . . . . . 9
1­3 The Energy function. . . . . . . . . . . . . . . . . . . . . . 12
1­4 Hamiltonian Systems. . . . . . . . . . . . . . . . . . . . . 14
1­5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Ma~n'e's critical value. 21
2­1 The action potential and the critical value. . . . . . . . . 21
2­2 Continuity of the critical value. . . . . . . . . . . . . . . . 25
2­3 Holonomic measures. . . . . . . . . . . . . . . . . . . . . . 26
2­4 Ergodic characterization of the critical value. . . . . . . . 41
2­5 The Aubry­Mather Theory. . . . . . . . . . . . . . . . . . 44
2­5.a Homology of measures. . . . . . . . . . . . . . . . . 44
2­5.b The asymptotic cycle. . . . . . . . . . . . . . . . . 44
2­5.c The alpha and beta functions. . . . . . . . . . . . 47
2­6 Coverings. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Globally minimizing orbits. 51
3­1 Tonelli's theorem. . . . . . . . . . . . . . . . . . . . . . . . 51
3­2 A priori compactness. . . . . . . . . . . . . . . . . . . . . 58

  

Source: Acevedo, Renato Iturriaga - Centro de Investigación en Matemáticas (CIMAT)

 

Collections: Mathematics