HAMILTON-JACOBI EQUATIONS FOR A REGULAR CONTROLLED HAMILTONIAN SYSTEM AND ITS REDUCED SYSTEMS*

PDF (PC)

HAMILTON-JACOBI EQUATIONS FOR A REGULAR CONTROLLED HAMILTONIAN SYSTEM AND ITS REDUCED SYSTEMS^*

HAMILTON-JACOBI EQUATIONS FOR A REGULAR CONTROLLED HAMILTONIAN SYSTEM AND ITS REDUCED SYSTEMS^*

HAMILTON-JACOBI EQUATIONS FOR A REGULAR CONTROLLED HAMILTONIAN SYSTEM AND ITS REDUCED SYSTEMS^*

HAMILTON-JACOBI EQUATIONS FOR A REGULAR CONTROLLED HAMILTONIAN SYSTEM AND ITS REDUCED SYSTEMS^*

摘要/Abstract

参考文献

Metrics

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 3

Metrics

本文评价

推荐阅读 10

doi:10.1007/s10473-023-0221-5

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (2): 855-906.doi: 10.1007/s10473-023-0221-5

Hong Wang

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

收稿日期:2021-11-15 修回日期:2022-04-07 出版日期:2023-03-25 发布日期:2023-04-12
作者简介:Hong Wang, E-mail: hongwang@nankai.edu.cn
基金资助:
This work was partially supported by the Nankai University 985 Project, the Key Laboratory of Pure Mathematics and Combinatorics, Ministry of Education, China and the NSFC (11531011).

Hong Wang

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Received:2021-11-15 Revised:2022-04-07 Online:2023-03-25 Published:2023-04-12
About author:Hong Wang, E-mail: hongwang@nankai.edu.cn
Supported by:
This work was partially supported by the Nankai University 985 Project, the Key Laboratory of Pure Mathematics and Combinatorics, Ministry of Education, China and the NSFC (11531011).

摘要： In this paper, we give the geometric constraint conditions of a canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian (RCH) system and its regular reduced systems, which are called the Type I and Type II Hamilton-Jacobi equations. First, we prove two types of Hamilton-Jacobi theorems for an RCH system on the cotangent bundle of a configuration manifold by using the canonical symplectic form and its dynamical vector field. Second, we generalize the above results for a regular reducible RCH system with symmetry and a momentum map, and derive precisely two types of Hamilton-Jacobi equations for the regular point reduced RCH system and the regular orbit reduced RCH system. Third, we prove that the RCH-equivalence for the RCH system, and the RpCH-equivalence and RoCH-equivalence for the regular reducible RCH systems with symmetries, leave the solutions of corresponding Hamilton-Jacobi equations invariant. Finally, as an application of the theoretical results, we show the Type I and Type II Hamilton-Jacobi equations for the $R_p$ -reduced controlled rigid body-rotor system and the $R_p$ -reduced controlled heavy top-rotor system on the generalizations of the rotation group ${SO}(3)$ and the Euclidean group ${SE}(3)$ , respectively. This work reveals the deeply internal relationships of the geometrical structures of phase spaces, the dynamical vector fields and the controls of the RCH system.

关键词: regular controlled Hamiltonian system, Hamilton-Jacobi equation, regular point reduction, regular orbit reduction, RCH-equivalence

Abstract: In this paper, we give the geometric constraint conditions of a canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian (RCH) system and its regular reduced systems, which are called the Type I and Type II Hamilton-Jacobi equations. First, we prove two types of Hamilton-Jacobi theorems for an RCH system on the cotangent bundle of a configuration manifold by using the canonical symplectic form and its dynamical vector field. Second, we generalize the above results for a regular reducible RCH system with symmetry and a momentum map, and derive precisely two types of Hamilton-Jacobi equations for the regular point reduced RCH system and the regular orbit reduced RCH system. Third, we prove that the RCH-equivalence for the RCH system, and the RpCH-equivalence and RoCH-equivalence for the regular reducible RCH systems with symmetries, leave the solutions of corresponding Hamilton-Jacobi equations invariant. Finally, as an application of the theoretical results, we show the Type I and Type II Hamilton-Jacobi equations for the $R_p$ -reduced controlled rigid body-rotor system and the $R_p$ -reduced controlled heavy top-rotor system on the generalizations of the rotation group ${SO}(3)$ and the Euclidean group ${SE}(3)$ , respectively. This work reveals the deeply internal relationships of the geometrical structures of phase spaces, the dynamical vector fields and the controls of the RCH system.

Key words: regular controlled Hamiltonian system, Hamilton-Jacobi equation, regular point reduction, regular orbit reduction, RCH-equivalence

Hong Wang. HAMILTON-JACOBI EQUATIONS FOR A REGULAR CONTROLLED HAMILTONIAN SYSTEM AND ITS REDUCED SYSTEMS^*[J]. 数学物理学报(英文版), 2023, 43(2): 855-906.

Hong Wang. HAMILTON-JACOBI EQUATIONS FOR A REGULAR CONTROLLED HAMILTONIAN SYSTEM AND ITS REDUCED SYSTEMS^*[J]. Acta mathematica scientia,Series B, 2023, 43(2): 855-906.

[1] Abraham R. Marsden J E.Foundations of Mechanics. Reading, MA: Addison-Wesley, 1978
[2] Abraham R, Marsden J E, Ratiu T S.Manifolds, Tensor Analysis and Applications. New York: Springer- Verlag, 1988
[3] Arnold V I.Mathematical Methods of Classical Mechanics. New York: Springer-Verlag, 1989
[4] Marsden J E, Ratiu T S.Introduction to Mechanics and Symmetry. New York: Springer-Verlag, 1999
[5] Woodhouse N M J. Geometric Quantization. Oxford: Clarendon Press, 1992
[6] Ge Z, Marsden J E.Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory. Phys Lett A, 1988, 133: 134-139
[7] Lázaro-Camí J A. Ortega J P. The stochastic Hamilton-Jacobi equation. J Geom Mech, 2009, 1: 295-315
[8] Marsden J E, Wang H, Zhang Z X.Regular reduction of controlled Hamiltonian system with symplectic structure and symmetry. Diff Geom Appl, 2014, 33(3): 13-45
[9] Cariñena J F, Gràcia X, Marmo G, et al. Geometric Hamilton-Jacobi theory. Int J Geom Methods Mod Phys, 2006, 3: 1417-1458
[10] Cariñena J F, Gràcia X, Marmo G, et al. Geometric Hamilton-Jacobi theory for nonholonomic dynamical systems. Int J Geom Methods Mod Phys, 2010, 7: 431-454
[11] Wang H.Hamilton-Jacobi theorems for regular reducible Hamiltonian system on a cotangent bundle. J Geom Phys, 2017, 119: 82-102
[12] Libermann P, Marle C M.Symplectic Geometry and Analytical Mechanics. Dordrecht: Springer, 1987
[13] Marsden J E.Lectures on Mechanics. Cambridge: Cambridge University Press, 1992
[14] Ortega J P, Ratiu T S.Momentum Maps and Hamiltonian Reduction. Boston: Birkhäuser, 2004
[15] León M, Rodrigues P R.Methods of Differential Geometry in Analytical Mechanics. Amsterdam: North- Holland, 1989
[16] Marsden J E, Misiolek G, Ortega J P, et al.Hamiltonian Reduction by Stages. Berlin: Springer, 2007
[17] Marsden J E, Montgomery R, Ratiu T S.Reduction, Symmetry and Phases in Mechanics. Providence, RI: American Mathematical Society, 1990
[18] Marsden J E, Perlmutter M.The orbit bundle picture of cotangent bundle reduction. C R Math Acad Sci Soc R Can, 2000, 22: 33-54
[19] Marsden J E, Weinstein A.Reduction of symplectic manifolds with symmetry. Rep Math Phys, 1974, 5: 121-130
[20] Meyer K R.Symmetries and integrals in mechanics//Peixoto M. Dynamical Systems. New York: Academic Press, 1973: 259-273
[21] Nijmeijer H, Van der Schaft A J. Nonlinear Dynamical Control Systems. New York: Springer-Verlag, 1990
[22] Wang H, Zhang Z X.Optimal reduction of controlled Hamiltonian system with Poisson structure and symmetry. J Geom Phys, 2012, 62(5): 953-975
[23] Ratiu T S, Wang H.Poisson reduction by controllability distribution for a controlled Hamiltonian system. arXiv: 1312.7047
[24] Wang H.Some developments of reduction theory for controlled Hamiltonian system with symmetry
(in Chinese). Sci Sin Math, 2018, 48: 1-12
[25] Wang H.Reductions of controlled Hamiltonian system with symmetry//Bai C M, Gazeau J P, Ge M L, et al. Symmetries and Groups in Contemporary Physics. Singapore: World Scientific, 2013: 639-642
[26] Marle C M.Symplectic manifolds, dynamical groups and Hamiltonian mechanics//Cahen M, Flato M. Differential Geometry and Relativity. Dordrecht: Reidel Publ Co, 1976: 249-269
[27] Kazhdan D, Kostant B, Sternberg S.Hamiltonian group actions dynamical systems of Calogero type. Comm Pure Appl Math, 1978, 31: 481-508
[28] Krishnaprasad P S, Marsden J E.Hamiltonian structure and stability for rigid bodies with flexible attachments. Arch Rational Mech Anal, 1987, 98: 71-93
[29] León M, Wang H. Hamilton-Jacobi equations for nonholonomic reducible Hamiltonian systems on a cotangent bundle. arXiv:1508.07548(v3)
[30] Wang H.Hamilton-Jacobi equations for nonholonomic magnetic Hamiltonian systems. Commun Math Res, 2022, 38(3): 351-388
[31] Wang H.Dynamical equations of the controlled rigid spacecraft with a rotor. arXiv: 2005.02221
[32] Wang H. Symmetric reduction and Hamilton-Jacobi equations for the controlled underwater vehicle-rotor system. arXiv:1310.3014(v3)

Viewed

Full text

	From	local

	Times	7
	Rate	100%

Abstract

Just accepted	Online first	Issue

0	0	45

	From	Others

	Times	45
	Rate	100%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed

Just accepted

Online first

Just accepted

Online first

[1]	仇金鸟, 魏文宁. UNIQUENESS OF VISCOSITY SOLUTIONS OF STOCHASTIC HAMILTON-JACOBI EQUATIONS[J]. 数学物理学报(英文版), 2019, 39(3): 857-873.
[2]	张正策, 李振杰. A NOTE ON GRADIENT BLOWUP RATE OF THE INHOMOGENEOUS HAMILTON-JACOBI EQUATIONS[J]. 数学物理学报(英文版), 2013, 33(3): 678-686.
[3]	王靖华, 温海瑞, 赵引川. THE LARGE TIME GENERIC FORM OF THE SOLUTION TO HAMILTON-JACOBI EQUATIONS[J]. 数学物理学报(英文版), 2011, 31(6): 2265-2277.