Dynamics of globally minimizing orbits in contact Hamiltonian systems

Xu, Yang; Yan, Jun; Zhao, Kai

doi:10.1090/proc/17138

Dynamics of globally minimizing orbits in contact Hamiltonian systems
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by Yang Xu, Jun Yan and Kai Zhao;

Proc. Amer. Math. Soc. 153 (2025), 2527-2538

DOI: https://doi.org/10.1090/proc/17138

Published electronically: April 8, 2025

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Abstract:

In this paper, we study the asymptotic behavior of globally minimizing orbits of contact Hamiltonian systems. Under some assumptions, we prove that the $\omega$-limit set of globally minimizing orbits is contained in the set of semi-static orbits.

References

Patrick Bernard, Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 5, 1533–1568 (English, with English and French summaries). MR 1935556, DOI 10.5802/aif.1924
Patrick Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc. 21 (2008), no. 3, 615–669. MR 2393423, DOI 10.1090/S0894-0347-08-00591-2
Gonzalo Contreras, Jorge Delgado, and Renato Iturriaga, Lagrangian flows: the dynamics of globally minimizing orbits. II, Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2, 155–196. MR 1479500, DOI 10.1007/BF01233390
Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8
Hitoshi Ishii, Kaizhi Wang, Lin Wang, and Jun Yan, Hamilton-Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown, Comm. Partial Differential Equations 47 (2022), no. 2, 417–452. MR 4378613, DOI 10.1080/03605302.2021.1983598
Liang Jin and Jun Yan, On the dynamics of contact Hamiltonian systems: I. Monotone systems, Nonlinearity 34 (2021), no. 5, 3314–3336. MR 4260796, DOI 10.1088/1361-6544/abe833
Pierre-Louis Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667669
Ricardo Mañé, Lagrangian flows: the dynamics of globally minimizing orbits, Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2, 141–153. MR 1479499, DOI 10.1007/BF01233389
Stefano Marò and Alfonso Sorrentino, Aubry-Mather theory for conformally symplectic systems, Comm. Math. Phys. 354 (2017), no. 2, 775–808. MR 3663623, DOI 10.1007/s00220-017-2900-3
Xiang Shu, Jun Yan, and Kai Zhao, Viscosity subsolutions of Hamilton-Jacobi equations and invariant sets of contact Hamilton systems, J. Math. Phys. 63 (2022), no. 10, Paper No. 102702, 12. MR 4492821, DOI 10.1063/5.0078226
Xifeng Su, Lin Wang, and Jun Yan, Weak KAM theory for Hamilton-Jacobi equations depending on unknown functions, Discrete Contin. Dyn. Syst. 36 (2016), no. 11, 6487–6522. MR 3543596, DOI 10.3934/dcds.2016080
Kaizhi Wang, Lin Wang, and Jun Yan, Implicit variational principle for contact Hamiltonian systems, Nonlinearity 30 (2017), no. 2, 492–515. MR 3604353, DOI 10.1088/1361-6544/30/2/492
Kaizhi Wang, Lin Wang, and Jun Yan, Aubry-Mather theory for contact Hamiltonian systems, Comm. Math. Phys. 366 (2019), no. 3, 981–1023. MR 3927084, DOI 10.1007/s00220-019-03362-2
Kaizhi Wang, Lin Wang, and Jun Yan, Variational principle for contact Hamiltonian systems and its applications, J. Math. Pures Appl. (9) 123 (2019), 167–200 (English, with English and French summaries). MR 3912650, DOI 10.1016/j.matpur.2018.08.011
Kaizhi Wang, Lin Wang, and Jun Yan, Aubry-Mather theory for contact Hamiltonian systems II, Discrete Contin. Dyn. Syst. 42 (2022), no. 2, 555–595. MR 4369021, DOI 10.3934/dcds.2021128
Kaizhi Wang and Jun Yan, Ergodic problems for contact Hamilton-Jacobi equations, arXiv:2107.11554, 2021.

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Dynamics of globally minimizing orbits in contact Hamiltonian systems
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Abstract: