doi: 10.3934/dcdsb.2021297
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Orbital dynamics on invariant sets of contact Hamiltonian systems

1. 

School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin 541002, China

2. 

Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, Granada 18071, Spain

*Corresponding author: Pedro J. Torres

Received  June 2021 Revised  October 2021 Early access December 2021

Fund Project: The first author is supported by National Natural Science Foundation of China (Grant No. 11771105), Guangxi Natural Science Foundation (Grants No. 2017GXNSFFA198012 and 2018GXNSFAA138177)

In this paper, we shall give new insights on dynamics of contact Hamiltonian flows, which are gaining importance in several branches of physics as they model a dissipative behaviour. We divide the contact phase space into three parts, which are corresponding to three differential invariant sets $ \Omega_\pm, \Omega_0 $. On the invariant sets $ \Omega_\pm $, under some geometric conditions, the contact Hamiltonian system is equivalent to a Hamiltonian system via the Hölder transformation. The invariant set $ \Omega_0 $ may be composed of several equilibrium points and heteroclinic orbits connecting them, on which contact Hamiltonian system is conservative. Moreover, we have shown that, under general conditions, the zero energy level domain is a domain of attraction. In some cases, such a domain of attraction does not have nontrivial periodic orbits. Some interesting examples are presented.

Citation: Qihuai Liu, Pedro J. Torres. Orbital dynamics on invariant sets of contact Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021297
References:
[1] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989.  doi: 10.1007/978-1-4757-2063-1.
[2]

A. Ashtekar, Introduction to loop quantum gravity and cosmology, Quantum Gravity and Quantum Cosmology, (eds. G. Calcagni, L. Papantonopoulos, G. Siopsis and N. Tsamis), Lecture Notes in Physics, Berlin, Heidelberg, 863 (2013).

[3]

C. P. Boyer, Completely integrable contact Hamiltonian systems and toric contact ctructures on $\mathbb{S}^2\times \mathbb{S}^3$, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 1-22.  doi: 10.3842/SIGMA.2011.058.

[4]

A. Bravetti, Contact Hamiltonian dynamics: The concept and its use, Entropy, 19 (2017), 535.  doi: 10.3390/e19100535.

[5]

A. BravettiH. Cruz and D. Tapias, Contact Hamiltonian mechanics, Annals Physics, 376 (2017), 17-39.  doi: 10.1016/j.aop.2016.11.003.

[6]

A. BravettiM. de LeónJ. C. Marrero and E. Padrón, Invariant measures for contact Hamiltonian systems: Symplectic sandwiches with contact bread, J. Phys. A, 53 (2020), 455205.  doi: 10.1088/1751-8121/abbaaa.

[7]

A. Bravetti, M. Seri and F. Zadra, New directions for contact integrators, Geometric Science of Information, (eds. F. Nielsen and F. Barbaresco), Lecture Notes in Computer Sciences, Springer, 2021.

[8]

A. Bravetti and D. Tapias, Thermostat algorithm for generating target ensembles, Phys. Rev. E, 93 (2016), 022139.  doi: 10.1103/PhysRevE.93.022139.

[9]

P. Cannarsa, W. Cheng, K. Wang and J. Yan, Herglotz' generalized variational principle and contact type Hamilton-Jacobi equations, Trends in Control Theory and Partial Differential Equations, (eds. F. Alabau-Boussouira, F. Ancona, A. Porretta and C. Sinestrari), Springer INdAM Series, 32 (2019), 39–67.

[10]

Y. ChenQ. Liu and H. Su, Generalized Hamiltonian forms of dissipative mechanical systems via a unified approach, J. Geom. Phys., 160 (2021), 103976.  doi: 10.1016/j.geomphys.2020.103976.

[11]

F. CiagliaH. Cruz and G. Marmo, Contact manifolds and dissipation, classical and quantum, Ann. Physics, 398 (2018), 159-179.  doi: 10.1016/j.aop.2018.09.012.

[12]

A. DaviniA. FathiR. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted Hamilton-Jacobi equation, Invent. Math., 206 (2016), 29-55.  doi: 10.1007/s00222-016-0648-6.

[13]

M. de León and M. Lainz Valcázar, Contact hamiltonian systems, J. Math. Phys., 60 (2019), 102902.  doi: 10.1063/1.5096475.

[14]

M. de León and M. Lainz Valcázar, Infinitesimal symmetries in contact Hamiltonian systems, J. Geom. Phys., 153 (2020), 103651.  doi: 10.1016/j.geomphys.2020.103651.

[15]

M. de LeónM. Lainz and A. Muñiz-Brea, The Hamilton–Jacobi theory for contact Hamiltonian systems, Mathematics, 9 (2021), 1993. 

[16]

J. GasetX. GràciaM. C. Muñoz-LecandaX. Rivas and N. Rom án-Roy, New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries, Int. J. Geom. Methods Mod. Phys., 17 (2020), 2050090.  doi: 10.1142/S0219887820500905.

[17] M. Giaquinta and S. Hildebrandt, Calculus of Variations I, Springer-Verlag, Berlin, 1996. 
[18]

S. Grillo and E. Padrón, Extended Hamilton–Jacobi theory, contact manifolds, and integrability by quadratures, J. Math. Phys., 61 (2020), 012901.  doi: 10.1063/1.5133153.

[19]

M. Grmela and H. C. Őttinger, Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Phys. Rev. E, 56 (1997), 6620-6632.  doi: 10.1103/PhysRevE.56.6620.

[20]

S. Gryb and D. Sloan, When scale is surplus, Synthese, (2021).  doi: 10.1007/s11229-021-03443-7.

[21]

S. Gryb and D. Sloan, New action for cosmology, Phys. Rev. D, 103 (2021), 043524.  doi: 10.1103/physrevd.103.043524.

[22]

J. HongW. ChengS. Hu and K. Zhao, Representation formulas for contact type Hamilton-Jacobi equations, J. Dyn. Diff. Equat., (2021).  doi: 10.1007/s10884-021-09960-w.

[23]

R. Huang, A qd-type method for computing generalized singular values of BF matrix pairs with sign regularity to high relative accuracy, Math. Comp., 89 (2020), 229-252.  doi: 10.1090/mcom/3444.

[24]

A. L. Kholodenko, Applications of Contact Geometry and Topology in Physics, World Scientific, Singapore, 2013. doi: 10.1142/8514.

[25]

Q. LiuX. Li and D. Qian, An abstract theorem on the existence of periodic motions of non-autonomous Lagrange systems, J. Differential Equations, 261 (2016), 5784-5802.  doi: 10.1016/j.jde.2016.08.010.

[26]

Q. LiuP. J. Torres and W. Chao, Contact Hamiltonian dynamics: Variational principles, invariants, completeness and periodic behavior, Ann. Physics, 395 (2018), 26-44.  doi: 10.1016/j.aop.2018.04.035.

[27]

S. G. Rajeev, A Hamilton-Jacobi formalism for thermodynamics, Ann. Physics, 323 (2008), 2265-2285.  doi: 10.1016/j.aop.2007.12.007.

[28]

H. RamirezB. Maschke and D. Sbarbaro, Partial stabilization of input-output contact systems on a Legendre submanifold, IEEE Trans. Automat. Control, 62 (2017), 1431-1437.  doi: 10.1109/TAC.2016.2572403.

[29]

D. Sloan, Scale symmetry and friction, Symmetry, 13 (2021).  doi: 10.3390/sym13091639.

[30]

D. Sloan, Dynamical similarity, Phys. Rev. D, 97 (2018), 123541.  doi: 10.1103/physrevd.97.123541.

[31]

K. WangL. Wang and J. Yan, Implicit variational principle for contact Hamiltonian systems, Nonlinearity, 30 (2017), 492-515.  doi: 10.1088/1361-6544/30/2/492.

[32]

K. WangL. Wang and J. Yan, Variational principle for contact Hamiltonian systems and its applications, J. Math. Pures Appl., 123 (2019), 167-200.  doi: 10.1016/j.matpur.2018.08.011.

[33]

K. WangL. Wang and J. Yan, Aubry-Mather theory for contact Hamiltonian systems, Comm. Math. Phys., 366 (2019), 981-1023.  doi: 10.1007/s00220-019-03362-2.

[34]

Y. Wang and J. Yan, A variational principle for contact Hamiltonian systems, J. Differential Equations, 267 (2019), 4047-4088.  doi: 10.1016/j.jde.2019.04.031.

show all references

References:
[1] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989.  doi: 10.1007/978-1-4757-2063-1.
[2]

A. Ashtekar, Introduction to loop quantum gravity and cosmology, Quantum Gravity and Quantum Cosmology, (eds. G. Calcagni, L. Papantonopoulos, G. Siopsis and N. Tsamis), Lecture Notes in Physics, Berlin, Heidelberg, 863 (2013).

[3]

C. P. Boyer, Completely integrable contact Hamiltonian systems and toric contact ctructures on $\mathbb{S}^2\times \mathbb{S}^3$, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 1-22.  doi: 10.3842/SIGMA.2011.058.

[4]

A. Bravetti, Contact Hamiltonian dynamics: The concept and its use, Entropy, 19 (2017), 535.  doi: 10.3390/e19100535.

[5]

A. BravettiH. Cruz and D. Tapias, Contact Hamiltonian mechanics, Annals Physics, 376 (2017), 17-39.  doi: 10.1016/j.aop.2016.11.003.

[6]

A. BravettiM. de LeónJ. C. Marrero and E. Padrón, Invariant measures for contact Hamiltonian systems: Symplectic sandwiches with contact bread, J. Phys. A, 53 (2020), 455205.  doi: 10.1088/1751-8121/abbaaa.

[7]

A. Bravetti, M. Seri and F. Zadra, New directions for contact integrators, Geometric Science of Information, (eds. F. Nielsen and F. Barbaresco), Lecture Notes in Computer Sciences, Springer, 2021.

[8]

A. Bravetti and D. Tapias, Thermostat algorithm for generating target ensembles, Phys. Rev. E, 93 (2016), 022139.  doi: 10.1103/PhysRevE.93.022139.

[9]

P. Cannarsa, W. Cheng, K. Wang and J. Yan, Herglotz' generalized variational principle and contact type Hamilton-Jacobi equations, Trends in Control Theory and Partial Differential Equations, (eds. F. Alabau-Boussouira, F. Ancona, A. Porretta and C. Sinestrari), Springer INdAM Series, 32 (2019), 39–67.

[10]

Y. ChenQ. Liu and H. Su, Generalized Hamiltonian forms of dissipative mechanical systems via a unified approach, J. Geom. Phys., 160 (2021), 103976.  doi: 10.1016/j.geomphys.2020.103976.

[11]

F. CiagliaH. Cruz and G. Marmo, Contact manifolds and dissipation, classical and quantum, Ann. Physics, 398 (2018), 159-179.  doi: 10.1016/j.aop.2018.09.012.

[12]

A. DaviniA. FathiR. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted Hamilton-Jacobi equation, Invent. Math., 206 (2016), 29-55.  doi: 10.1007/s00222-016-0648-6.

[13]

M. de León and M. Lainz Valcázar, Contact hamiltonian systems, J. Math. Phys., 60 (2019), 102902.  doi: 10.1063/1.5096475.

[14]

M. de León and M. Lainz Valcázar, Infinitesimal symmetries in contact Hamiltonian systems, J. Geom. Phys., 153 (2020), 103651.  doi: 10.1016/j.geomphys.2020.103651.

[15]

M. de LeónM. Lainz and A. Muñiz-Brea, The Hamilton–Jacobi theory for contact Hamiltonian systems, Mathematics, 9 (2021), 1993. 

[16]

J. GasetX. GràciaM. C. Muñoz-LecandaX. Rivas and N. Rom án-Roy, New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries, Int. J. Geom. Methods Mod. Phys., 17 (2020), 2050090.  doi: 10.1142/S0219887820500905.

[17] M. Giaquinta and S. Hildebrandt, Calculus of Variations I, Springer-Verlag, Berlin, 1996. 
[18]

S. Grillo and E. Padrón, Extended Hamilton–Jacobi theory, contact manifolds, and integrability by quadratures, J. Math. Phys., 61 (2020), 012901.  doi: 10.1063/1.5133153.

[19]

M. Grmela and H. C. Őttinger, Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Phys. Rev. E, 56 (1997), 6620-6632.  doi: 10.1103/PhysRevE.56.6620.

[20]

S. Gryb and D. Sloan, When scale is surplus, Synthese, (2021).  doi: 10.1007/s11229-021-03443-7.

[21]

S. Gryb and D. Sloan, New action for cosmology, Phys. Rev. D, 103 (2021), 043524.  doi: 10.1103/physrevd.103.043524.

[22]

J. HongW. ChengS. Hu and K. Zhao, Representation formulas for contact type Hamilton-Jacobi equations, J. Dyn. Diff. Equat., (2021).  doi: 10.1007/s10884-021-09960-w.

[23]

R. Huang, A qd-type method for computing generalized singular values of BF matrix pairs with sign regularity to high relative accuracy, Math. Comp., 89 (2020), 229-252.  doi: 10.1090/mcom/3444.

[24]

A. L. Kholodenko, Applications of Contact Geometry and Topology in Physics, World Scientific, Singapore, 2013. doi: 10.1142/8514.

[25]

Q. LiuX. Li and D. Qian, An abstract theorem on the existence of periodic motions of non-autonomous Lagrange systems, J. Differential Equations, 261 (2016), 5784-5802.  doi: 10.1016/j.jde.2016.08.010.

[26]

Q. LiuP. J. Torres and W. Chao, Contact Hamiltonian dynamics: Variational principles, invariants, completeness and periodic behavior, Ann. Physics, 395 (2018), 26-44.  doi: 10.1016/j.aop.2018.04.035.

[27]

S. G. Rajeev, A Hamilton-Jacobi formalism for thermodynamics, Ann. Physics, 323 (2008), 2265-2285.  doi: 10.1016/j.aop.2007.12.007.

[28]

H. RamirezB. Maschke and D. Sbarbaro, Partial stabilization of input-output contact systems on a Legendre submanifold, IEEE Trans. Automat. Control, 62 (2017), 1431-1437.  doi: 10.1109/TAC.2016.2572403.

[29]

D. Sloan, Scale symmetry and friction, Symmetry, 13 (2021).  doi: 10.3390/sym13091639.

[30]

D. Sloan, Dynamical similarity, Phys. Rev. D, 97 (2018), 123541.  doi: 10.1103/physrevd.97.123541.

[31]

K. WangL. Wang and J. Yan, Implicit variational principle for contact Hamiltonian systems, Nonlinearity, 30 (2017), 492-515.  doi: 10.1088/1361-6544/30/2/492.

[32]

K. WangL. Wang and J. Yan, Variational principle for contact Hamiltonian systems and its applications, J. Math. Pures Appl., 123 (2019), 167-200.  doi: 10.1016/j.matpur.2018.08.011.

[33]

K. WangL. Wang and J. Yan, Aubry-Mather theory for contact Hamiltonian systems, Comm. Math. Phys., 366 (2019), 981-1023.  doi: 10.1007/s00220-019-03362-2.

[34]

Y. Wang and J. Yan, A variational principle for contact Hamiltonian systems, J. Differential Equations, 267 (2019), 4047-4088.  doi: 10.1016/j.jde.2019.04.031.

Figure 1.  In $ \mathbb{R}^3 $, the invariant set $ \Omega_0 $ of system (10) consists of two equilibrium points $ (0,0,\pm\sqrt{2}) $ and infinitely many heteroclinic orbits, where we take $ h = 1 $
Figure 2.  The invariant set $ \Omega_0 $ of system (11) consists of four equilibrium points $ (0,0,\pm(R\pm r)) $ and infinitely many heteroclinic orbits, where we take $ R = 2, r = 1 $. Three different curves with red, blue and green colors correspond to three heteroclinic orbits of system (11) starting from the initial values $ (-0.01,0.01,-2.999933331759) $, $ (0.01,-0.01,-1.0000000025) $, $ (-0.01,0.01,1.0000000025) $, respectively
Figure 3.  The periodic orbits are located on the planes perpendicular to the $ xOp $-plane
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