Existence of isotropic complete solutions of the Π-Hamilton-Jacobi equation

JOURNAL OF GEOMETRY AND PHYSICS

卷:148

Article

Grillo, Sergio^1,2 

[1] Univ Nacl Cuyo, Inst Balseiro, Av Bustillo 9500,R8402AGP, San Carlos De Bariloche, Rio Negro, Argentina

[2] Consejo Nacl Invest Cient & Tecn, Av Bustillo 9500,R8402AGP, San Carlos De Bariloche, Rio Negro, Argentina

关键词: Symplectic geometry;    Hamilton-Jacobi theory;    Integrable systems;   

DOI  :  10.1016/j.geomphys.2019.103544

来源: Elsevier

【 摘 要 】

Consider a symplectic manifold M, a Hamiltonian vector field X and a fibration Pi : M -> N. Related to these data we have a generalized version of the (time-independent) Hamilton-Jacobi equation: the Pi-HJE for X, whose unknown is a section sigma : N -> M of Pi. The standard HJE is obtained when the phase space M is a cotangent bundle T*Q (with its canonical symplectic form), Pi is the canonical projection pi(Q) : T*Q -> Q. and the unknown is a closed 1-form dW : Q -> T*Q. The function W is called Hamilton's characteristic function. Coming back to the generalized version, among the solutions of the Pi-HJE, a central role is played by the so-called isotropic complete solutions. This is because, if a solution of this kind is known for a given Hamiltonian system, then such a system can be integrated up to quadratures. The purpose of the present paper is to prove that, under mild conditions, an isotropic complete solution exists around almost every point of M. Restricted to the standard case, this gives rise to an alternative proof for the local existence of a complete family of Hamilton's characteristic functions. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

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