SEMI-BASIC 1-FORMS AND HELMHOLTZ CONDITIONS FOR THE INVERSE PROBLEM OF THE CALCULUS OF VARIATIONS

被引：30

作者：

Bucataru, Ioan ^{[1
]}

Dahl, Matias F. ^{[2
]}

机构：

[1] Alexandru Ioan Cuza Univ, Fac Math, Iasi 700506, Romania

[2] Aalto Univ, Inst Math, Helsinki 02015, Finland

来源：

JOURNAL OF GEOMETRIC MECHANICS | 2009年 / 1卷 / 02期

基金：

芬兰科学院;

关键词：

Poincare-Cartan; 1-form; Helmholtz conditions; inverse problem; projective metrizability; Finsler metrizability; TANGENT BUNDLE; GEOMETRY; EQUATIONS; FORMS;

D O I：

10.3934/jgm.2009.1.159

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We use Frolicher-Nijenhuis theory to obtain global Helmholtz conditions, expressed in terms of a semi-basic 1-form, that characterize when a semispray is a Lagrangian vector field. We also discuss the relation between these Helmholtz conditions and their classic formulation written using a multiplier matrix. When the semi-basic 1-form is 1-homogeneous (0-homogeneous) we show that two (one) of the Helmholtz conditions are consequences of the other ones. These two special cases correspond to two inverse problems in the calculus of variation: Finsler metrizability for a spray, and projective metrizability for a spray.

引用

页码：159 / 180

页数：22

共 43 条

[1]

AMBROSE W, 1960, AN ACAD BRAS CIENC, V32, P163

[2]

[Anonymous], 1973, COHOMOLOGY DIFFERENT

[3]

[Anonymous], 1986, Foundations of Finsler Geometry and Special Finsler Spaces

[4]

[Anonymous], Q J MATH

[5]

Bucataru I., 2007, Finsler-Lagrange Geometry. Applications to Dynamical Systems

[6] Metric nonlinear connections [J].

Bucataru, Ioan .

DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, 2007, 25 (03) :335-343

[7]

Carinena J. F., 1991, GROUP THEORETICAL ME, VII

[8] TANGENT BUNDLE GEOMETRY FOR LAGRANGIAN DYNAMICS [J].

CRAMPIN, M .

JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 1983, 16 (16) :3755-3772

[9] ON THE DIFFERENTIAL GEOMETRY OF THE EULER-LAGRANGE EQUATIONS, AND THE INVERSE PROBLEM OF LAGRANGIAN DYNAMICS [J].

CRAMPIN, M .

JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 1981, 14 (10) :2567-2575

[10]

Crampin M, 1996, ANN I H POINCARE-PHY, V65, P223

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