Abstract
This paper mainly studies the contact extension of conservative or dissipative systems, including some old and new results for wholeness. Then extension of contact system is corresponding to the symplectification of contact Hamiltonian system. This is a reciprocal process and the relation between symplectic system and contact system has been discussed. We have an interesting discovery that by adding a pure variable p, the slope of the tangent of the orbit, every differential system can be regarded as an independent subsystem of contact Hamiltonian system defined on the projection space of contact phase space.
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References
Arnold, V. I. Mathematical Methods of Classical Mechanics. Springer-Verlag, New York, 1978
Bryant, J. The contact transformation groups of the Extended Hamiltonian System. Celest. Mech. Dyn. Astr., 25: 41–49 (1981)
Bravetti, A., Cruz, H., Tapias, D. Contact Hamiltonian mechanics. Ann. Phys., 376: 17–39 (2016)
Carathéodory, C. Calculus of Variations and Partial Differential Equations of First Order: Second Edition, Amer. Math. Soc., Providence, RI, 2000. Translated by Robert B. Dean Julius J. Brandstatter, Translating Editor
Eisenhart, L. P. Contact transformations. Ann. Math., 30: 211–249 (1928–1929)
Eisenhart, L. P. Invariant theory of homogeneous contact transformations. Ann. Math., 37: 747–765 (1936)
Giaquinta, M., Hildebrandt, S. Calculus of Variations II. Springer-Verlag, Berlin Heidelberg, 2004
Gizatullin, M. Klein’s conjecture for contact automorphisms of the three-dimensional affine space. Mich. Math. J., 56: 89–98 (2008)
Gray, J. W. Global properties of contact structures. Ann. Math., 69: 421–450 (1959)
Ince, E. L. Ordinary differential equations. Dover Publications, New York, 1956
Klein, F. Vorlesungen über höhere Geometrie, dritte Auflage, bearbeitet und herausgegeben von W. Blashke, Verlag von Julius Springer, Berlin, 1926. Russian translation 1939. The first lithographic edition, Einleitung in die höhere Geometrie
Liu, Q., Torres, P. J., Chao, W. Contact Hamiltonian dynamics: Variational principles, invariants, completeness and periodic behavior. Ann. Phys., 395: 26–44 (2018)
Liu, Q., Torres, P. J. Orbital dynamics on invariant sets of contact Hamiltonian systems. Discrete Cont. Dyn.-B, 27: 5821–5844 (2022) (2022)
Rajeev, S. G. A Hamilton-Jacobi formalism for thermodynamics. Ann. Phys., 323: 2265–2285 (2008)
Wang, K., Wang, L., Yan, J. Implicit variational principle for contact Hamiltonian systems. Nonlinearity, 30: 492–515 (2017)
Wang, K., Wang, L., Yan, J. Variational principle for contact Hamiltonian systems and its applications. J. Math. Pures et Appl., 123: 167–200 (2019)
Wang, K., Wang, L., Yan, J. Aubry-mather theory for contact Hamiltonian systems. Comm. Math. Phys., 366: 981–1023 (2019)
Wang, Y., Yan, J. A variational principle for contact Hamiltonian systems. J. Differential Equations, 267: 4047–4088 (2019)
Zadra, F., Bravetti, A., Seri, M. Geometric numerical integration of Liénard systems via a contact Hamiltonian approach. Mathematics, 9: 1–26 (2021)
Acknowledgments. The authors appreciate the referees and editors for good suggestions to improve this paper.
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This paper is supported by the National Natural Science Foundation of China (Nos. 11771105, 12071410), the Natural Science Foundation of Guangxi Province (Nos. 2017GXNSFFA198012) and Guangxi Distinguished Expert Project.
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Liu, Qh., Xie, A. & Wang, C. Contact Extension and Symplectification. Acta Math. Appl. Sin. Engl. Ser. 39, 962–971 (2023). https://doi.org/10.1007/s10255-023-1093-0
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DOI: https://doi.org/10.1007/s10255-023-1093-0