The Geometry of Higher-Order Lagrange Spaces

This monograph is mostly devoted to the problem of the geome trizing of Lagrangians which depend on higher order accelerations. It naturally prolongs the theme of the monograph "The Geometry of La grange spaces: Theory and Applications", written together with M. Anastasiei and published by Kluwer Academic Publishers in 1994. The existence of Lagrangians of order k 1 has been contemplated by mechanicists and physicists for a long time. Einstein had grasped their presence in connection with the Brownian motion. They are also present in relativistic theories based on metrics which depend on speeds and accelerations of particles or in the Hamiltonian formulation of non linear systems given by Korteweg-de Vries equations. There resulted from here the methods to be adopted in their theoretical treatment. One is based on the variational problem involving the integral action of the Lagrangian. A second one is derived from the axioms of Analytical Mechanics involving the Poincare-Cartan forms. The geometrical methods based on the study of the geometries of higher order could invigorate the whole theory. This is the way adopted by us in defining and studying the Lagrange spaces of higher order. The problems raised by the geometrization of Lagrangians of order k 1 investigated by many scholars: Ch. Ehresmann, P. Libermann, J. Pommaret; J.T. Synge, M. Crampin, P. Saunders; G.S. Asanov, P.Aringazin; I. Kolar, D. Krupka; M. de Leon, W. Sarlet, P. Cantrjin, H. Rund, W.M. Tulczyjew, A. Kawaguchi, K. Yano, K. Kondo, D.

Klappentext

This monograph is mostly devoted to the problem of the geome­ trizing of Lagrangians which depend on higher order accelerations. It naturally prolongs the theme of the monograph "The Geometry of La­ grange spaces: Theory and Applications", written together with M. Anastasiei and published by Kluwer Academic Publishers in 1994. The existence of Lagrangians of order k > 1 has been contemplated by mechanicists and physicists for a long time. Einstein had grasped their presence in connection with the Brownian motion. They are also present in relativistic theories based on metrics which depend on speeds and accelerations of particles or in the Hamiltonian formulation of non­ linear systems given by Korteweg-de Vries equations. There resulted from here the methods to be adopted in their theoretical treatment. One is based on the variational problem involving the integral action of the Lagrangian. A second one is derived from the axioms of Analytical Mechanics involving the Poincare-Cartan forms. The geometrical methods based on the study of the geometries of higher order could invigorate the whole theory. This is the way adopted by us in defining and studying the Lagrange spaces of higher order. The problems raised by the geometrization of Lagrangians of order k > 1 investigated by many scholars: Ch. Ehresmann, P. Libermann, J. Pommaret; J.T. Synge, M. Crampin, P. Saunders; G.S. Asanov, P.Aringazin; I. Kolar, D. Krupka; M. de Leon, W. Sarlet, P. Cantrjin, H. Rund, W.M. Tulczyjew, A. Kawaguchi, K. Yano, K. Kondo, D.

Inhalt
Preface. 1. Lagrange Spaces of Order 1. 2. The Geometry of 2-Osculator Bundle. 3. N-Linear Connections. 4. Lagrangians of Second Order. Variational Problem. Nöther Type Theorems. 5. Second Order Lagrange Spaces. 6. Geometry of the k-Osculator Bundle. 7. Linear Connections of OsckM. 8. Lagrangians of Order k. Applications to Higher-Order Analytical Mechanics. 9. Prolongation of the Riemannian, Finslerian and Lagrangian Structures to the k-Osculator Bundle. 10. Higher Order Lagrange Spaces. 11. Subspaces in Higher Order Lagrange Spaces. 12. Gauge Theory in the Higher Order Lagrange Spaces. References. Index.