524 ダイナミカルシステムの基本構造 : シンプレクティック構造とラグランジュ・ハミルトン系

概要

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This paper illustrates fundamental structures in Lagrangian and Hamiltonian dynamical systems. It is shown that the canonical symplectic structure can be naturally defined in the momentum phase space, namely, the cotangent bundle of a configuration manifold and also that a Hamiltonian system can be defined in the context of the canonical symplectic structure. As its dual structure, it is demonstrated that a symplectic structure called the Lagrangian two form on the velocity phase space, i.e., the tangent bundle can be induced from the canonical symplectic structure on the momentum phase space by the Legendre transformation for the case in which a given Lagrangian is regular. Then, it is shown that a regular Lagrangian system can be defined in the context of the induced symplectic structure.
2007-09-25