Helmholtz conditions and symmetries for the time dependent case of the inverse problem of the calculus of variations

被引:4
作者
Bucataru, Ioan [1 ]
Constantinescu, Oana [1 ]
机构
[1] Alexandru Ioan Cuza Univ, Fac Math, Iasi 700506, Romania
关键词
Semi-basic forms; Poincare lemma; Helmholtz conditions; Inverse problem; Dual symmetry; First integral; 2ND-ORDER DIFFERENTIAL-EQUATIONS; LAGRANGIAN DYNAMICS; ADJOINT SYMMETRIES; LINEAR CONNECTIONS; GEOMETRY;
D O I
10.1016/j.geomphys.2010.06.016
中图分类号
O29 [应用数学];
学科分类号
070104 ;
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 Frolicher-Nijenhuis theory on the first jet bundle, J(1)pi. We prove that a system of time dependent SODE, identified with a semispray S, is Lagrangian if and only if a special class, Lambda(1)(S)(J(1)pi) of semi-basic 1-forms is not empty. We provide global Helmholtz conditions to characterize the class Lambda(1)(S)(J(1)pi) of semi-basic 1-forms. Each such class contains the Poincare-Cartan 1-form of some Lagrangian function. We prove that if there exists a semibasic 1-form in Lambda(1)(S)(J(1)pi) , which is not a Poincare-Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE. (c) 2010 Elsevier B.V. All rights reserved.
引用
收藏
页码:1710 / 1725
页数:16
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