Jet bundle geometry, dynamical connections, and the inverse problem of lagrangian mechanics

Some aspects of the geometry of jet-bundles, especially relevant for the formulation of Classical Mechanics, are investigated. The main result is the construction of a tensor analysis on the first jet extension of the configuration space-time, based on a suitable linear connection, determined entirely by the dynamics of the system. The significance of this "dynamical connection" in the geometrization of Classical Mechanics is discussed, paying a special attention to two particular aspects, namely the implementation of the concept of "relative time derivative" in the Lagrangian framework, and the derivation of the Helmholtz conditions for the inverse problem of Lagrangian Dynamics. Key-words : Lagrangian Dynamics, inverse problem, linear and affine connections. 02.40.+m, 03.20.+i 1991 Mathematical subject Classification : 70 D 10, 34 A 55, 53 B 05 This research was partly supported by the National Group for Mathematical Physics of the Italian Research Council (CNR), and by the Italian Ministry for Public Education, through the research project "Metodi Geometrici e Probabilistici in Sistems Dinamici, Meccanica Statistica, Relativita e Teoria dei Campi". Annales de l’Institut Henri Poincaré Physique theorique 0246-0211 Vol. 61/94/01/$ 4.00/@ Gauthier-Villars 18 E. MASSA AND E. PAGANI Dans cet article on etudie des aspects de la geometric des espaces des jets qui sont importants pour une formulation geometrique de la mecanique classique. On construit une « analyse tensorielle » sur la premiere extension des jets de l’espace-temps des configurations moyennant une connexion lineaire determinee completement par la dynamique du systeme. On met en evidence Ie role joue par cette « connexion dynamique » dans la geometrisation de la mecanique classique ; en particulier on introduit Ie concept de « derivee temporelle relative » dans Ie contexte lagrangien et les conditions de Helmholtz pour Ie probleme inverse de la dynamique lagrangienne.

[1]  M. Crampin,et al.  Evading the inverse problem for second-order ordinary differential equations by using additional variables , 1987 .

[2]  G. Marmo,et al.  The inverse problem in the calculus of variations and the geometry of the tangent bundle , 1990 .

[3]  M. Crampin.  Tangent bundle geometry Lagrangian dynamics , 1983 .

[4]  M. Crampin,et al.  Reduction of degenerate Lagrangian systems , 1986 .

[5]  N. Mukunda,et al.  Dynamics and symmetry for constrained systems: a geometrical analysis , 1983 .

[6]  K. Nomizu,et al.  Foundations of Differential Geometry , 1963 .

[7]  Geometric theory of the equivalence of Lagrangians for constrained systems , 1985 .

[8]  W. Sarlet.  Note on equivalent Lagrangians and symmetries , 1983 .

[9]  M. Henneaux,et al.  Lagrangians for spherically symmetric potentials , 1982 .

[10]  Willy Sarlet,et al.  Adjoint symmetries for time-dependent second-order equations , 1990 .

[11]  E. Massa,et al.  Classical dynamics of non-holonomic systems : a geometric approach , 1991 .

[12]  J. Douglas.  Solution of the Inverse Problem of the Calculus of Variations. , 1939, Proceedings of the National Academy of Sciences of the United States of America.

[13]  Charles-Michel Marle,et al.  Symplectic geometry and analytical mechanics , 1987 .

[14]  Joseph Grifone,et al.  Structure presque tangente et connexions I , 1972 .

[15]  W. Sarlet,et al.  The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics , 1982 .

[16]  M. Crampin,et al.  A new look at second-order equations and Lagrangian mechanics , 1984 .

[17]  D. Saunders.  The Geometry of Jet Bundles , 1989 .

[18]  G. Giachetta,et al.  Jet Methods in Nonholonomic Mechanics , 1992 .

[19]  C. Godbillon.  Géométrie différentielle et mécanique analytique , 1969 .

[20]  A geometrical version of the Helmholtz conditions in time- dependent Lagrangian dynamics , 1984 .

[21]  Helmholtz conditions, covariance, and invariance identities , 1985 .

[22]  J. Cariñena,et al.  Symmetry theory and Lagrangian inverse problem for time-dependent second-order differential equations , 1989 .

[23]  M. Crampin.  On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics , 1981 .

[24]  S. Sternberg.  Lectures on Differential Geometry , 1964 .

[25]  J. Pommaret.  Systems of partial differential equations and Lie pseudogroups , 1978 .