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 On smooth time functions

2011 ◽  
Vol 152 (2) ◽  
pp. 303-339 ◽  
Author(s):  
ALBERT FATHI ◽  
ANTONIO SICONOLFI
Keyword(s):  
Kam Theory ◽  
Time Function ◽  
Lorentzian Manifolds ◽  
Weak Kam Theory ◽  
Continuity Properties ◽  
Integral Curves ◽  
Intrinsic Length ◽  
Definition Of ◽  
General Abstract ◽  
Jacobi Equations

AbstractWe are concerned with the existence of smooth time functions on connected time-oriented Lorentzian manifolds. The problem is tackled in a more general abstract setting, namely in a manifold M where is just defined a field of tangent convex cones (Cx)x ∈ M enjoying mild continuity properties. Under some conditions on its integral curves, we will construct a time function.Our approach is based on the definition of an intrinsic length for curves indicating how a curve is far from being an integral trajectory of Cx. We find connections with topics pertaining to Hamilton–Jacobi equations, and make use of tools and results issued from weak KAM theory.

Weak KAM Theory and Forcing Equivalence

2020 ◽  
pp. 55-65
Author(s):  
Kaloshin Vadim ◽  
Zhang Ke
Keyword(s):  
Kam Theory ◽  
Alternative Definition ◽  
Weak Kam Theory ◽  
Peierls Barrier ◽  
Definition Of ◽  
Standard Presentation ◽  
Alpha Function

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.

scholarly journals Local modal participation analysis of nonlinear systems using Poincaré linearization

2019 ◽  
Vol 99 (1) ◽  
pp. 803-811 ◽  
Author(s):  
Boumediene Hamzi ◽  
Eyad H. Abed
Keyword(s):  
Nonlinear Systems ◽  
Autonomous Systems ◽  
Analytical Formula ◽  
Linear Case ◽  
Local Mode ◽  
Initial Condition ◽  
Initial State ◽  
State Participation ◽  
Symmetry Assumption ◽  
Definition Of

AbstractThe paper studies an extension to nonlinear systems of a recently proposed approach to the definition of modal participation factors. A definition is given for local mode-in-state participation factors for smooth nonlinear autonomous systems. While the definition is general, the resulting measures depend on the assumed uncertainty law governing the system initial condition, as in the linear case. The work follows Hashlamoun et al. (IEEE Trans Autom Control 54(7):1439–1449 2009) in taking a mathematical expectation (or set-theoretic average) of a modal contribution measure over an uncertain set of system initial state. Poincaré linearization is used to replace the nonlinear system with a locally equivalent linear system. It is found that under a symmetry assumption on the distribution of the initial state, the tractable calculation and analytical formula for mode-in-state participation factors found for the linear case persists to the nonlinear setting. This paper is dedicated to the memory of Professor Ali H. Nayfeh.

Erratum: Commun. Math. Phys. 217, 127-163 (2001)

2002 ◽  
Vol 225 (1) ◽  
pp. 219-221 ◽  
Author(s):  
Not Available Not Available
Keyword(s):  
Commun Math Phys ◽  
Math Phys

1870s Agrarian Activism in Southern Illinois: Mediator between Two Eras

1992 ◽  
Vol 16 (3) ◽  
pp. 365-400 ◽  
Author(s):  
Jane Adams
Keyword(s):  
Nineteenth Century ◽  
Specific Class ◽  
Southern Illinois ◽  
Local Society ◽  
Economic Dimension ◽  
Economic Relationships ◽  
Close Analysis ◽  
Union County ◽  
Definition Of ◽  
Political Economic

During the latter part of the nineteenth century, farmers in extreme southern Illinois, along with farmers throughout the state and region, organized politically and economically. The first big upsurge of organization was in 1873 with the organization of farmers’ clubs and granges of the Patrons of Husbandry. In Union County, Illinois, all sectors of the local society appear to have been swept up in the tide of discontent, although a close analysis of those active in the movement and the associations that succeeded it indicates that the movement gave voice and organized expression to a specific class. To use McNall’s (1988) analysis of the later populist movement, the agrarian movement of the mid-1870s was an incipient “class movement,” although it failed to articulate a program that effectively welded farmers into a unit that could contend for political power, even as it provided a vehicle for elite farmers to transform preexisting economic relationships. “A class movement,” McNall (ibid.: 5) writes, “is one in which the participants are involved in a struggle over the very definition of their political, economic, and ideological interests. All class movements have at their core an economic dimension and, like class relationships, are about relationships of power.” The organizations formed in the populist era, he argues, were attempts by farmers to create a “class in and for itself” (ibid.: 12).

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
Sign in / Sign up

Export Citation Format

Share Document