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This monograph is devoted to the problem of the geometrizing of Lagrangians which depend on higher-order accelerations.
It presents a construction of the geometry of the total space of the bundle of the accelerations of order k>=1. A geometrical study of the notion of the higher-order Lagrange space is conducted, and the old problem of prolongation of Riemannian spaces to k-osculator manifolds is solved. Also, the geometrical ground for variational calculus on the integral of actions involving higher-order Lagrangians is dealt with. Applications to higher-order analytical mechanics and theoretical physics are included as well.
Audience: This volume will be of interest to scientists whose work involves differential geometry, mechanics of particles and systems, calculus of variation and optimal control, optimization, optics, electromagnetic theory, and biology.
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.