Kaloshin Vadim and Zhang Ke
- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691202525
- eISBN:
- 9780691204932
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202525.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter describes weak Kolmogorov-Arnold-Moser (KAM) theory and forcing relation. One change from the standard presentation is that one needs to modify the definition of Tonelli Hamiltonians to ...
More
This chapter describes weak Kolmogorov-Arnold-Moser (KAM) theory and forcing relation. One change from the standard presentation is that one needs to modify the definition of Tonelli Hamiltonians to allow different periods in the t component. The chapter points out an alternative definition of the alpha function, namely, one can replace the class of minimal measures with the class of closed measures. It then considers a dual setting which corresponds to forward dynamic. It also looks at elementary solutions, static classes, and Peierls barrier. In many parts of the proof, the chapter studies the hyperbolic property of a minimizing orbit, for which the concept of Green bundles is very useful.Less
This chapter describes weak Kolmogorov-Arnold-Moser (KAM) theory and forcing relation. One change from the standard presentation is that one needs to modify the definition of Tonelli Hamiltonians to allow different periods in the t component. The chapter points out an alternative definition of the alpha function, namely, one can replace the class of minimal measures with the class of closed measures. It then considers a dual setting which corresponds to forward dynamic. It also looks at elementary solutions, static classes, and Peierls barrier. In many parts of the proof, the chapter studies the hyperbolic property of a minimizing orbit, for which the concept of Green bundles is very useful.
Kaloshin Vadim and Zhang Ke
- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691202525
- eISBN:
- 9780691204932
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202525.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter explores perturbation aspects of the weak Kolmogorov-Arnold-Moser (KAM) theory. By perturbative weak KAM theory, we mean two things. How do the weak KAM solutions and the Mather, Aubry, ...
More
This chapter explores perturbation aspects of the weak Kolmogorov-Arnold-Moser (KAM) theory. By perturbative weak KAM theory, we mean two things. How do the weak KAM solutions and the Mather, Aubry, and Mañé sets respond to limits of the Hamiltonian? How do the weak KAM solutions change when we perturb a system, in particular, what happens when we perturb (1) completely integrable systems, and (2) autonomous systems by a time-periodic perturbation? The chapter states and proves results in both aspects, as a technical tool for proving forcing equivalence. It derives a special Lipshitz estimate of weak KAM solutions for perturbations of autonomous systems. The proof relies on semi-concavity of weak KAM solution.Less
This chapter explores perturbation aspects of the weak Kolmogorov-Arnold-Moser (KAM) theory. By perturbative weak KAM theory, we mean two things. How do the weak KAM solutions and the Mather, Aubry, and Mañé sets respond to limits of the Hamiltonian? How do the weak KAM solutions change when we perturb a system, in particular, what happens when we perturb (1) completely integrable systems, and (2) autonomous systems by a time-periodic perturbation? The chapter states and proves results in both aspects, as a technical tool for proving forcing equivalence. It derives a special Lipshitz estimate of weak KAM solutions for perturbations of autonomous systems. The proof relies on semi-concavity of weak KAM solution.
