变分法反问题不定常时的Helmholtz条件和对称性
Helmholtz conditions and symmetries for the time dependent case of the inverse problem of the calculus of variations
摘要：
本文利用Frölicher-Nijenhuis理论，在第一喷流束$J^1π$上，给出了二阶常微分方程不定常系统变分反问题的一个新的形式.证明了一个由半基本1-形式构成的不定常SODE系统是Lagrangian的当且仅当一个特殊的半基本1-形式类$S$不为空.给出了半基本1-形式类$Λ^1_S(J^1π)$的整体Helmholtz条件.每一个这样的类都包含某个拉格朗日函数的Poincaré-Cartan 1-形式。证明了如果在$Λ^1_S(J^1π)$中存在一个非Poincaré-Cartan 1-形式的半基本1-形式，则它决定了给定SODE系统的一个对偶对称和一个首次积分.
英文摘要：
We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Frölicher-Nijenhuis theory on the first jet bundle, $J^1π$. We prove that a system of time dependent SODE, identified with a semispray $S$, is Lagrangian if and only if a special class, $Λ^1_S(J^1π)$, of semi-basic 1-forms is not empty. We provide global Helmholtz conditions to characterize the class $Λ^1_S(J^1π)$ of semi-basic 1-forms. Each such class contains the Poincaré-Cartan 1-form of some Lagrangian function. We prove that if there exists a semi-basic 1-form in $Λ^1_S(J^1π)$, which is not a Poincaré-Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE.
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