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A kind of non-conservative Hamilton system solved by the Hamilton-Jacobi method

Wang Yong Mei Feng-Xiang Xiao Jing Guo Yong-Xin

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A kind of non-conservative Hamilton system solved by the Hamilton-Jacobi method

Wang Yong, Mei Feng-Xiang, Xiao Jing, Guo Yong-Xin
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  • The Hamilton-Jacobi equation is an important nonlinear partial differential equation. In particular, the classical Hamilton-Jacobi method is generally considered to be an important means to solve the holonomic conservative dynamics problems in classical dynamics. According to the classical Hamilton-Jacobi theory, the classical Hamilton-Jacobi equation corresponds to the canonical Hamilton equations of the holonomic conservative dynamics system. If the complete solution of the classical Hamilton-Jacobi equation can be found, the solution of the canonical Hamilton equations can be found by the algebraic method. From the point of geometry view, the essential of the Hamilton-Jacobi method is that the Hamilton-Jacobi equation promotes the vector field on the cotangent bundle T* M to a constraint submanifold of the manifold T* M R, and if the integral curve of the promoted vector field can be found, the projection of the integral curve in the cotangent bundle T* M is the solution of the Hamilton equations. According to the geometric theory of the first order partial differential equations, the Hamilton-Jacobi method may be regarded as the study of the characteristic curves which generate the integral manifolds of the Hamilton 2-form . This means that there is a duality relationship between the Hamilton-Jacobi equation and the canonical Hamilton equations. So if an action field, defined on UI (U is an open set of the configuration manifold M, IR), is a solution of the Hamilton-Jacobi equation, then there will exist a differentiable map from MR to T* MR which defines an integral submanifold for the Hamilton 2-form . Conversely, if * =0 and H1(UI)=0 (H1(UI) is the first de Rham group of U I), there will exist an action field S satisfying the Hamilton-Jacobi equation. Obviously, the above mentioned geometric theory can not only be applicable to the classical Hamilton-Jacobi equation, but also to the general Hamilton-Jacobi equation, in which some first order partial differential equations correspond to the non-conservative Hamiltonian systems. The geometry theory of the Hamilton-Jacobi method is applied to some special non-conservative Hamiltonian systems, and a new Hamilton-Jacobi method is established. The Hamilton canonical equations of the non-conservative Hamiltonian systems which are applied with non-conservative force Fi = (t)pi can be solved with the new method. If a complete solution of the corresponding Hamilton-Jacobi equation can be found, all the first integrals of the non-conservative Hamiltonian system will be found. The classical Hamilton-Jacobi method is a special case of the new Hamilton-Jacobi method. Some examples are constructed to illustrate the proposed method.
      Corresponding author: Guo Yong-Xin, yxguo@lnu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11572145, 11272050, 11572034) and the Natural Science Foundation of Guangdong Province, China (Grant No. 2015A030310127).
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    Fedkiw R P, Aslam T, Merrima B, Osher S 1999J.Comput.Phys. 152 457

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    Yang S Z, Lin K 2010Sci.China 40 507(in Chinese)[杨树政, 林恺2010中国科学40 507]

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    Yang S Z, Lin K 2013Acta Phys.Sin. 62 060401(in Chinese)[杨树政, 林恺2013物理学报62 060401]

    [7]

    Kim J H, Lee H W 2000Can.J.Phys. 77 411

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    Arnold V I.1978Mathematical Methods of Classical Mechanics(New York:Spriner-Verlag) pp161-271

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    Mei F X 2013Analytical Mechanics(Vol.1)(Beijing:Beijing Institute of Technology Press) pp272-287(in Chinese)[梅凤翔2013分析力学(上册)(北京:北京理工大学出版社)第272-287页]

    [11]

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    [12]

    Guo Y X, Luo S K, Mei F X 2004Adv.Mech. 34 477(in Chinese)[郭永新, 罗绍凯, 梅凤翔2004力学进展34 477]

    [13]

    Guo Y X, Liu S X, Liu C, Luo S K, Wang Y 2007J.Math.Phys. 48 082901

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    Marmo G, Morandi G, Mukunda N 1990La Rivista del Nuovo Cimento 13 1

    [15]

    Wang H 2013 arXiv:1305.3457v2[math.SG]

    [16]

    Westenholtz C N 1981Differential Forms in Mathematical Physics(Amsterdam:North-Horland Publishing Company) pp389-439

    [17]

    Barbero-Linn M, de Len M, Martin de Diego D 2012Monatsh.Math. 171 269

    [18]

    Marmo G, Morandi G, Mukunda N 2009J.Geom.Mech. 1 317

    [19]

    Vitagliano L 2012Int.J.Geom.Methods Mod.Phys. 9 1260008

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    de Len M, Vilario S 2014Int.J.Geom.Methods Mod.Phys. 11 1450007

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    Ohsawa T, Bloch A M 2009J.Geom.Mech. 1 461

  • [1]

    Benamou J 1996J.Comput.Phys.128 463

    [2]

    Fleming W H, Rishel R 1975Deterministic and Stochastic Optimal Control(Berlin:Spinger) pp80-105

    [3]

    Feng C J, Wang P, Wang X M 2015Acta Phys.Sin. 64 030502(in Chinese)[封晨洁, 王鹏, 王旭明2015物理学报64 030502]

    [4]

    Fedkiw R P, Aslam T, Merrima B, Osher S 1999J.Comput.Phys. 152 457

    [5]

    Yang S Z, Lin K 2010Sci.China 40 507(in Chinese)[杨树政, 林恺2010中国科学40 507]

    [6]

    Yang S Z, Lin K 2013Acta Phys.Sin. 62 060401(in Chinese)[杨树政, 林恺2013物理学报62 060401]

    [7]

    Kim J H, Lee H W 2000Can.J.Phys. 77 411

    [8]

    Joulin G, Mitani T 1981Comb.Flame. 40 235

    [9]

    Arnold V I.1978Mathematical Methods of Classical Mechanics(New York:Spriner-Verlag) pp161-271

    [10]

    Mei F X 2013Analytical Mechanics(Vol.1)(Beijing:Beijing Institute of Technology Press) pp272-287(in Chinese)[梅凤翔2013分析力学(上册)(北京:北京理工大学出版社)第272-287页]

    [11]

    Courant R, Hilbert D 1989Methods of Mathematical Physics(Vol.2)(New York:John WileySons) pp62-153

    [12]

    Guo Y X, Luo S K, Mei F X 2004Adv.Mech. 34 477(in Chinese)[郭永新, 罗绍凯, 梅凤翔2004力学进展34 477]

    [13]

    Guo Y X, Liu S X, Liu C, Luo S K, Wang Y 2007J.Math.Phys. 48 082901

    [14]

    Marmo G, Morandi G, Mukunda N 1990La Rivista del Nuovo Cimento 13 1

    [15]

    Wang H 2013 arXiv:1305.3457v2[math.SG]

    [16]

    Westenholtz C N 1981Differential Forms in Mathematical Physics(Amsterdam:North-Horland Publishing Company) pp389-439

    [17]

    Barbero-Linn M, de Len M, Martin de Diego D 2012Monatsh.Math. 171 269

    [18]

    Marmo G, Morandi G, Mukunda N 2009J.Geom.Mech. 1 317

    [19]

    Vitagliano L 2012Int.J.Geom.Methods Mod.Phys. 9 1260008

    [20]

    de Len M, Vilario S 2014Int.J.Geom.Methods Mod.Phys. 11 1450007

    [21]

    Ohsawa T, Bloch A M 2009J.Geom.Mech. 1 461

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Publishing process
  • Received Date:  18 August 2016
  • Accepted Date:  03 December 2016
  • Published Online:  05 March 2017

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