scholarly journals Minimal period solutions in asymptotically linear Hamiltonian system with symmetries

2019 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Zhiping Fan ◽  
◽  
Duanzhi Zhang ◽  
Keyword(s):  
Hamiltonian System ◽  
Linear Hamiltonian System ◽  
Minimal Period ◽  
Asymptotically Linear

scholarly journals Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Linyu Peng ◽  
Huafei Sun ◽  
Xiao Sun
Keyword(s):  
Hamiltonian System ◽  
Geometric Structure ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Hamiltonian Dynamics ◽  
Jacobi Field ◽  
Linear Hamiltonian System ◽  
Conformal Metric ◽  
Jacobi Equations ◽  
Special Case

We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After investigating the Eisenhart metric, we study the corresponding conformal metric and obtain the geometric structure of the classical Hamiltonian dynamics. Furthermore, the equations for the conformal geodesics, for the Jacobi field along the geodesics, and the equations for a certain flow constrained in a family of conformal equivalent nondegenerate metrics are obtained. At last the conformal curvatures, the geodesic equations, the Jacobi equations, and the equations for the flow of the famous models, anNdegrees of freedom linear Hamiltonian system and the Hénon-Heiles model are given, and in a special case, numerical solutions of the conformal geodesics, the generalized momenta, and the Jacobi field along the geodesics of the Hénon-Heiles model are obtained. And the numerical results for the Hénon-Heiles model show us the instability of the associated geodesic spreads.

scholarly journals Periodic solutions with prescribed minimal period for second order even Hamiltonian systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Juhong Kuang ◽  
Weiyi Chen ◽  
Zhiming Guo
Keyword(s):  
Hamiltonian System ◽  
Periodic Solutions ◽  
Convergent Subsequence ◽  
Step Length ◽  
Continuous Functions ◽  
Second Order ◽  
Limit Function ◽  
Minimal Period ◽  
Nonnegative Solutions ◽  
Convexity Assumptions

<p style='text-indent:20px;'>In this paper, we develop a new method to study Rabinowitz's conjecture on the existence of periodic solutions with prescribed minimal period for second order even Hamiltonian system without any convexity assumptions. Specifically, we first study the associated homogenous Dirichlet boundary value problems for the discretization of the Hamiltonian system with given step length and obtain a sequence of nonnegative solutions corresponding to different step lengths by using discrete variational methods. Then, using the sequence of nonnegative solutions, we construct a sequence of continuous functions which can be shown to be precompact. Finally, by utilizing the limit function of convergent subsequence and the symmetry of the potential, we will obtain the desired periodic solution. In particular, we prove Rabinowitz's conjecture in the case when the potential satisfies a certain symmetric assumption. Moreover, our main result greatly improves the related results in the literature in the case where <inline-formula><tex-math id="M1">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula>.</p>

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