Weak KAM Theory in Time-periodic Lagrangian Systems

2013 ◽  
pp. 227-238
Author(s):  
Kaizhi Wang ◽  
Jun Yan
Keyword(s):  
Kam Theory ◽  
Lagrangian Systems ◽  
Weak Kam Theory ◽  
Time Periodic

Some Results on Weak KAM Theory for Time-periodic Tonelli Lagrangian Systems

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Kaizhi Wang ◽  
Yong Li
Keyword(s):  
Kam Theory ◽  
Lagrangian Systems ◽  
Specific Class ◽  
Initial Condition ◽  
Commun Math Phys ◽  
Arbitrary Continuous Function ◽  
Weak Kam Theory ◽  
Definition Of ◽  
Time Periodic ◽  
Math Phys

AbstractThis paper contributes several results on weak KAM theory for time-periodic Tonelli Lagrangian systems. Wang and Yan [Commun. Math. Phys. 309 (2012), 663-691] introduced a new kind of Lax-Oleinik type operator associated with any time-periodic Tonelli Lagrangian. Firstly, using the new operator we give an equivalent definition of the backward weak KAM solution. Then we prove a result on the asymptotic behavior of the new operators with an arbitrary continuous function as initial condition, by taking advantage of the definition mentioned above. Finally, for a specific class of time-periodic Tonelli Lagrangians, we discuss the rate of convergence of the new operators.

Perturbative weak KAM theory

2020 ◽  
pp. 66-76
Author(s):  
Kaloshin Vadim ◽  
Zhang Ke
Keyword(s):  
Integrable Systems ◽  
Autonomous Systems ◽  
Kam Theory ◽  
Periodic Perturbation ◽  
Completely Integrable Systems ◽  
Technical Tool ◽  
Weak Kam Theory ◽  
Completely Integrable ◽  
Time Periodic ◽  
Weak Kam Solutions

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.

scholarly journals Weak KAM Theory topics in the stationary ergodic setting

2011 ◽  
Vol 44 (3-4) ◽  
pp. 319-350 ◽  
Author(s):  
Andrea Davini ◽  
Antonio Siconolfi
Keyword(s):  
Kam Theory ◽  
Weak Kam Theory

New identities for Weak KAM theory

2017 ◽  
Vol 38 (2) ◽  
pp. 379-392 ◽  
Author(s):  
Lawrence Craig Evans
Keyword(s):  
Kam Theory ◽  
Weak Kam Theory
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