February  2022, 42(2): 555-595. doi: 10.3934/dcds.2021128

Aubry-Mather theory for contact Hamiltonian systems II

1. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

2. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

3. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

* Corresponding author: Lin Wang

Received  December 2020 Revised  June 2021 Published  February 2022 Early access  September 2021

In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems $ H(x,u,p) $ with certain dependence on the contact variable $ u $. 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 $ \tilde{\mathcal{S}}_s $ consists of strongly static orbits, which coincides with the Aubry set $ \tilde{\mathcal{A}} $ in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show $ \tilde{\mathcal{S}}_s\subsetneqq\tilde{\mathcal{A}} $ in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of $ H $ on $ u $ fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the minimal viscosity solution and non-minimal ones.

Citation: Kaizhi Wang, Lin Wang, Jun Yan. Aubry-Mather theory for contact Hamiltonian systems II. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 555-595. doi: 10.3934/dcds.2021128
References:
[1]

V. Arnold, Mathematical Methods of Classical Mechanics, $2^{nd}$ edition, Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[2]

P. Bernard, Existence of $C^{1, 1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds, Ann. Sci. École Norm. Sup., 40 (2007), 445-452.  doi: 10.1016/j.ansens.2007.01.004.

[3]

P. Bernard, Symplectic aspects of Mather theory, Duke Math. J., 136 (2007), 401-420.  doi: 10.1215/S0012-7094-07-13631-7.

[4]

P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc., 21 (2008), 615-669.  doi: 10.1090/S0894-0347-08-00591-2.

[5]

A. Bravetti, Contact Hamiltonian dynamics: The concept and its use, Entropy, 19 (2017), 12 pp. doi: 10.3390/e19100535.

[6]

A. Bravetti, Contact geometry and thermodynamics, Int. J. Geom. Meth. Mod. Phys., 16 (2019), 1940003.  doi: 10.1142/S0219887819400036.

[7]

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

[8]

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, Springer INdAM Ser., Springer, Cham, 32 (2019), 39–67.

[9]

P. CannarsaW. ChengL. JinK. Wang and J. Yan, Herglotz' variational principle and Lax-Oleinik evolution, J. Math. Pures Appl., 141 (2020), 99-136.  doi: 10.1016/j.matpur.2020.07.002.

[10]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, In Nonlinear Differential Equations and Their Applications, 58, Birkhäuser Boston, Inc., Boston, MA, 2004.

[11]

Q. Chen, Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function, Adv. Calc. Var., Published online.

[12]

Q. ChenW. ChengH. Ishii and K. Zhao, Vanishing contact structure problem and convergence of the viscosity solutions, Comm. Partial Differential Equations, 44 (2019), 801-836.  doi: 10.1080/03605302.2019.1608561.

[13]

G. ContrerasJ. Delgado and R. Iturriaga, Lagrangian flows: The dynamics of globally minimizing orbits. II, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 155-196.  doi: 10.1007/BF01233390.

[14]

M. CrandallH. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[15]

G. Contreras and R. Iturriaga, Global Minimizers of Autonomous Lagrangians, 22nd Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999.

[16]

M. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.  doi: 10.1090/S0002-9947-1983-0690039-8.

[17]

M. de León and C. Sardón, Cosymplectic and contact structures for time-dependent and dissipative Hamiltonian systems, J. Phys. A, 50 (2017), 255205.  doi: 10.1088/1751-8121/aa711d.

[18]

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

[19]

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

[20]

A. Davini and L. Wang, On the vanishing discount problems from the negative direction, Discrete Contin. Dyn. Syst., 41 (2021), 2377-2389.  doi: 10.3934/dcds.2020368.

[21]

A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, preliminary version 10, Lyon, unpublished (2008).

[22]

A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388.  doi: 10.1007/s00222-003-0323-6.

[23]

D. Gomes, Generalized Mather problem and selection principles for viscosity solutions and Mather measures, Adv. Calc. Var., 1 (2008), 291-307.  doi: 10.1515/ACV.2008.012.

[24]

D. GomesH. Mitake and H. Tran, The selection problem for discounted Hamilton-Jacobi equations: Some non-convex cases, J. Math. Soc. Jpn., 70 (2018), 345-364.  doi: 10.2969/jmsj/07017534.

[25]

G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen, Göttingen, 1930.

[26]

R. Iturriaga and H. Sanchez-Morgado, Limit of the infinite horizon discounted Hamilton-Jacobi equation, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 623-635.  doi: 10.3934/dcdsb.2011.15.623.

[27]

H. IshiiH. Mitake and H. Tran, The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus, J. Math. Pures Appl., 108 (2017), 125-149.  doi: 10.1016/j.matpur.2016.10.013.

[28]

H. IshiiH. Mitake and H. Tran, The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems, J. Math. Pures Appl., 108 (2017), 261-305.  doi: 10.1016/j.matpur.2016.11.002.

[29]

W. JingH. Mitake and H. Tran, Generalized ergodic problems: Existence and uniqueness structures of solutions, J. Differential Equations, 268 (2020), 2886-2909.  doi: 10.1016/j.jde.2019.09.046.

[30]

P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi Equations, preprint.

[31]

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

[32]

S. Marò and A. Sorrentino, Aubry-Mather theory for conformally symplectic systems, Commun. Math. Phys., 354 (2017), 775-808.  doi: 10.1007/s00220-017-2900-3.

[33]

J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.  doi: 10.1007/BF02571383.

[34]

J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.  doi: 10.5802/aif.1377.

[35]

R. Mañé, Lagrangain flows: The dynamics of globally minimizing orbits, Bol. Soc. Brasil. Math., 28 (1997), 141-153.  doi: 10.1007/BF01233389.

[36]

H. Mitake and K. Soga, Weak KAM theory for discounted Hamilton-Jacobi equations and its application, Calc. Var. Partial Differential Equations, 57 (2018), 57-78.  doi: 10.1007/s00526-018-1359-1.

[37]

H. Mitake and H. Tran, Selection problems for a discount degenerate viscous Hamilton-Jacobi equation, Adv. Math., 306 (2017), 684-703.  doi: 10.1016/j.aim.2016.10.032.

[38]

K. Soga, Selection problems of $\mathbb{Z}^2$ -periodic entropy solutions and viscosity solutions, Calc. Var. Partial Differ. Equ., 56 (2017), 30pp. doi: 10.1007/s00526-017-1208-7.

[39]

X. SuL. Wang and J. Yan, Weak KAM theory for Hamilton-Jacobi equations depending on unkown functions, Discrete Contin. Dyn. Syst., 36 (2016), 6487-6522.  doi: 10.3934/dcds.2016080.

[40]

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

[41]

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.

[42]

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.

[43]

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

[44]

K. WangL. Wang and J. Yan, Weak KAM solutions of Hamilton-Jacobi equations with decreasing dependence on unknown functions, J. Differential Equations, 286 (2021), 411-432.  doi: 10.1016/j.jde.2021.03.030.

[45]

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.

[46]

Y. WangJ. Yan and J. Zhang, Convergence of viscosity solutions of generalized contact Hamilton-Jacobi equations, Arch. Rational Mech. Anal., 241 (2021), 885-902.  doi: 10.1007/s00205-021-01667-y.

[47]

K. Zhao and W. Cheng, On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem, Discrete Contin. Dyn. Syst., 39 (2019), 4345-4358.  doi: 10.3934/dcds.2019176.

show all references

References:
[1]

V. Arnold, Mathematical Methods of Classical Mechanics, $2^{nd}$ edition, Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[2]

P. Bernard, Existence of $C^{1, 1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds, Ann. Sci. École Norm. Sup., 40 (2007), 445-452.  doi: 10.1016/j.ansens.2007.01.004.

[3]

P. Bernard, Symplectic aspects of Mather theory, Duke Math. J., 136 (2007), 401-420.  doi: 10.1215/S0012-7094-07-13631-7.

[4]

P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc., 21 (2008), 615-669.  doi: 10.1090/S0894-0347-08-00591-2.

[5]

A. Bravetti, Contact Hamiltonian dynamics: The concept and its use, Entropy, 19 (2017), 12 pp. doi: 10.3390/e19100535.

[6]

A. Bravetti, Contact geometry and thermodynamics, Int. J. Geom. Meth. Mod. Phys., 16 (2019), 1940003.  doi: 10.1142/S0219887819400036.

[7]

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

[8]

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, Springer INdAM Ser., Springer, Cham, 32 (2019), 39–67.

[9]

P. CannarsaW. ChengL. JinK. Wang and J. Yan, Herglotz' variational principle and Lax-Oleinik evolution, J. Math. Pures Appl., 141 (2020), 99-136.  doi: 10.1016/j.matpur.2020.07.002.

[10]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, In Nonlinear Differential Equations and Their Applications, 58, Birkhäuser Boston, Inc., Boston, MA, 2004.

[11]

Q. Chen, Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function, Adv. Calc. Var., Published online.

[12]

Q. ChenW. ChengH. Ishii and K. Zhao, Vanishing contact structure problem and convergence of the viscosity solutions, Comm. Partial Differential Equations, 44 (2019), 801-836.  doi: 10.1080/03605302.2019.1608561.

[13]

G. ContrerasJ. Delgado and R. Iturriaga, Lagrangian flows: The dynamics of globally minimizing orbits. II, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 155-196.  doi: 10.1007/BF01233390.

[14]

M. CrandallH. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[15]

G. Contreras and R. Iturriaga, Global Minimizers of Autonomous Lagrangians, 22nd Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999.

[16]

M. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.  doi: 10.1090/S0002-9947-1983-0690039-8.

[17]

M. de León and C. Sardón, Cosymplectic and contact structures for time-dependent and dissipative Hamiltonian systems, J. Phys. A, 50 (2017), 255205.  doi: 10.1088/1751-8121/aa711d.

[18]

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

[19]

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

[20]

A. Davini and L. Wang, On the vanishing discount problems from the negative direction, Discrete Contin. Dyn. Syst., 41 (2021), 2377-2389.  doi: 10.3934/dcds.2020368.

[21]

A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, preliminary version 10, Lyon, unpublished (2008).

[22]

A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388.  doi: 10.1007/s00222-003-0323-6.

[23]

D. Gomes, Generalized Mather problem and selection principles for viscosity solutions and Mather measures, Adv. Calc. Var., 1 (2008), 291-307.  doi: 10.1515/ACV.2008.012.

[24]

D. GomesH. Mitake and H. Tran, The selection problem for discounted Hamilton-Jacobi equations: Some non-convex cases, J. Math. Soc. Jpn., 70 (2018), 345-364.  doi: 10.2969/jmsj/07017534.

[25]

G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen, Göttingen, 1930.

[26]

R. Iturriaga and H. Sanchez-Morgado, Limit of the infinite horizon discounted Hamilton-Jacobi equation, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 623-635.  doi: 10.3934/dcdsb.2011.15.623.

[27]

H. IshiiH. Mitake and H. Tran, The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus, J. Math. Pures Appl., 108 (2017), 125-149.  doi: 10.1016/j.matpur.2016.10.013.

[28]

H. IshiiH. Mitake and H. Tran, The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems, J. Math. Pures Appl., 108 (2017), 261-305.  doi: 10.1016/j.matpur.2016.11.002.

[29]

W. JingH. Mitake and H. Tran, Generalized ergodic problems: Existence and uniqueness structures of solutions, J. Differential Equations, 268 (2020), 2886-2909.  doi: 10.1016/j.jde.2019.09.046.

[30]

P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi Equations, preprint.

[31]

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

[32]

S. Marò and A. Sorrentino, Aubry-Mather theory for conformally symplectic systems, Commun. Math. Phys., 354 (2017), 775-808.  doi: 10.1007/s00220-017-2900-3.

[33]

J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.  doi: 10.1007/BF02571383.

[34]

J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.  doi: 10.5802/aif.1377.

[35]

R. Mañé, Lagrangain flows: The dynamics of globally minimizing orbits, Bol. Soc. Brasil. Math., 28 (1997), 141-153.  doi: 10.1007/BF01233389.

[36]

H. Mitake and K. Soga, Weak KAM theory for discounted Hamilton-Jacobi equations and its application, Calc. Var. Partial Differential Equations, 57 (2018), 57-78.  doi: 10.1007/s00526-018-1359-1.

[37]

H. Mitake and H. Tran, Selection problems for a discount degenerate viscous Hamilton-Jacobi equation, Adv. Math., 306 (2017), 684-703.  doi: 10.1016/j.aim.2016.10.032.

[38]

K. Soga, Selection problems of $\mathbb{Z}^2$ -periodic entropy solutions and viscosity solutions, Calc. Var. Partial Differ. Equ., 56 (2017), 30pp. doi: 10.1007/s00526-017-1208-7.

[39]

X. SuL. Wang and J. Yan, Weak KAM theory for Hamilton-Jacobi equations depending on unkown functions, Discrete Contin. Dyn. Syst., 36 (2016), 6487-6522.  doi: 10.3934/dcds.2016080.

[40]

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

[41]

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.

[42]

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.

[43]

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

[44]

K. WangL. Wang and J. Yan, Weak KAM solutions of Hamilton-Jacobi equations with decreasing dependence on unknown functions, J. Differential Equations, 286 (2021), 411-432.  doi: 10.1016/j.jde.2021.03.030.

[45]

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.

[46]

Y. WangJ. Yan and J. Zhang, Convergence of viscosity solutions of generalized contact Hamilton-Jacobi equations, Arch. Rational Mech. Anal., 241 (2021), 885-902.  doi: 10.1007/s00205-021-01667-y.

[47]

K. Zhao and W. Cheng, On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem, Discrete Contin. Dyn. Syst., 39 (2019), 4345-4358.  doi: 10.3934/dcds.2019176.

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