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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.
[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 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 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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[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 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 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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