scholarly journals Aubry-Mather theory for contact Hamiltonian systems II

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kaizhi Wang ◽  
Lin Wang ◽  
Jun Yan
Keyword(s):  
Hamiltonian Systems ◽  
Viscosity Solution ◽  
Invariant Set ◽  
Negative Direction ◽  
Mather Theory ◽  
Graph Property ◽  
Aubry Set ◽  
New Phenomena ◽  
Comparison Property ◽  
Weak Kam Solutions

<p style='text-indent:20px;'>In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems <inline-formula><tex-math id="M1">\begin{document}$ H(x,u,p) $\end{document}</tex-math></inline-formula> with certain dependence on the contact variable <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula>. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set <inline-formula><tex-math id="M3">\begin{document}$ \tilde{\mathcal{S}}_s $\end{document}</tex-math></inline-formula> consists of <i>strongly</i> static orbits, which coincides with the Aubry set <inline-formula><tex-math id="M4">\begin{document}$ \tilde{\mathcal{A}} $\end{document}</tex-math></inline-formula> in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show <inline-formula><tex-math id="M5">\begin{document}$ \tilde{\mathcal{S}}_s\subsetneqq\tilde{\mathcal{A}} $\end{document}</tex-math></inline-formula> in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of <inline-formula><tex-math id="M6">\begin{document}$ H $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M7">\begin{document}$ u $\end{document}</tex-math></inline-formula> fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the <i>minimal</i> viscosity solution and <i>non-minimal</i> ones.</p>

scholarly journals A GEOMETRIC DEFINITION OF THE AUBRY–MATHER SET

2010 ◽  
Vol 02 (03) ◽  
pp. 385-393 ◽  
Author(s):  
PATRICK BERNARD ◽  
JOANA OLIVEIRA DOS SANTOS
Keyword(s):  
Hamiltonian Systems ◽  
Symplectic Geometry ◽  
Point Of View ◽  
Mather Theory ◽  
Aubry Set ◽  
Definition Of ◽  
Geometric Definition

We study the Aubry set which appears in Mather theory of convex Hamiltonian systems from the point of view of symplectic geometry.

Aubry–Mather Theory for Contact Hamiltonian Systems

2019 ◽  
Vol 366 (3) ◽  
pp. 981-1023 ◽  
Author(s):  
Kaizhi Wang ◽  
Lin Wang ◽  
Jun Yan
Keyword(s):  
Hamiltonian Systems ◽  
Mather Theory

scholarly journals Global propagation of singularities for discounted Hamilton-Jacobi equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Cui Chen ◽  
Jiahui Hong ◽  
Kai Zhao
Keyword(s):  
Viscosity Solution ◽  
Jacobi Equation ◽  
Technical Difficulty ◽  
Singular Locus ◽  
Locally Lipschitz ◽  
Propagation Of Singularities ◽  
Aubry Set ◽  
Fixed Constant ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

<p style='text-indent:20px;'>The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE333"> \begin{document}$ \begin{align} \lambda v(x)+H( x, Dv(x) ) = 0 , \quad x\in \mathbb{R}^n. \quad\quad\quad (\mathrm{HJ}_{\lambda})\end{align} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with fixed constant <inline-formula><tex-math id="M1">\begin{document}$ \lambda\in \mathbb{R}^+ $\end{document}</tex-math></inline-formula>. We reduce the problem for equation <inline-formula><tex-math id="M2">\begin{document}$(\mathrm{HJ}_{\lambda})$\end{document}</tex-math></inline-formula> into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of <inline-formula><tex-math id="M3">\begin{document}$(\mathrm{HJ}_{\lambda})$\end{document}</tex-math></inline-formula> propagate along locally Lipschitz singular characteristics <inline-formula><tex-math id="M4">\begin{document}$ {{\bf{x}}}(s):[0,t]\to \mathbb{R}^n $\end{document}</tex-math></inline-formula> and time <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula> can extend to <inline-formula><tex-math id="M6">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula>. Essentially, we use <inline-formula><tex-math id="M7">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-compactness of the Euclidean space which is different from the original construction in [<xref ref-type="bibr" rid="b4">4</xref>]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of <inline-formula><tex-math id="M8">\begin{document}$ u $\end{document}</tex-math></inline-formula> and the complement of Aubry set using the basic idea from [<xref ref-type="bibr" rid="b9">9</xref>].</p>

Deterministic Mechanism of Irreversibility

2018 ◽  
Vol 14 (3) ◽  
pp. 5708-5733 ◽  
Author(s):  
Vyacheslav Michailovich Somsikov
Keyword(s):  
Internal Energy ◽  
Hamiltonian Systems ◽  
Classical Mechanics ◽  
Motion Equation ◽  
Holonomic Constraints ◽  
Dissipative Processes ◽  
Motion Energy ◽  
Analytical Review ◽  
History Of ◽  
Material Points

The analytical review of the papers devoted to the deterministic mechanism of irreversibility (DMI) is presented. The history of solving of the irreversibility problem is briefly described. It is shown, how the DMI was found basing on the motion equation for a structured body. The structured body was given by a set of potentially interacting material points. The taking into account of the body’s structure led to the possibility of describing dissipative processes. This possibility caused by the transformation of the body’s motion energy into internal energy. It is shown, that the condition of holonomic constraints, which used for obtaining of the canonical formalisms of classical mechanics, is excluding the DMI in Hamiltonian systems. The concepts of D-entropy and evolutionary non-linearity are discussed. The connection between thermodynamics and the laws of classical mechanics is shown. Extended forms of the Lagrange, Hamilton, Liouville, and Schrödinger equations, which describe dissipative processes, are presented.

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