The Geometry of Higher-Order Hamilton Spaces Applications to Hamiltonian Mechanics - Miron, Radu (Springer Netherlands, 2003)
As is known, the Lagrange and Hamilton geometries have appeared relatively recently [76, 86]. Since 1980 these geometries have been intensively studied by mathematicians and physicists from Romania, Canada, Germany, Japan, Russia, Hungary, U.S.A. etc. Prestigious scientific meetings devoted to Lagrange and Hamilton geometries and their applications have been organized in the above mentioned countries and a number of books and monographs have been published by specialists in the field: R. Miron [94, 95], R. Miron and M. Anastasiei [99, 100], R. Miron, D. Hrimiuc, H. Shimada and S. Sabau [115], P.L. Antonelli, R. Ingarden and M. Matsumoto [7]. Finsler spaces, which form a subclass of the class of Lagrange spaces, have been the subject of some excellent books, for example by: M. Matsumoto [76], M. Abate and G. Patrizio [1], D. Bao, S.S. Chernand Z. Shen [17] and A. Bejancu and H.R. Farran [20]. Also, we would like to point out the monographs of M.
Crampin [34], O. Krupkova [72] and D. Opri~, I. Butulescu [125], D. Saunders [144], which contain pertinent applications in analytical mechanics and in the theory of partial differential equations. Applications in mechanics, cosmology, theoretical physics and biology can be found in the well known books of P.L. Antonelli and T. Zawstaniak [11], G.S. Asanov [14], S. Ikeda [59], M. de Leone and P. Rodrigues [73]. The importance of Lagrange and Hamilton geometries consists of the fact that variational problems for important Lagrangians or Hamiltonians have numerous applications in various fields, such as mathematics, the theory of dynamical systems, optimal control, biology, and economy. In this respect, P.L. Antonelli's remark is interesting: "There is now strong evidence that the symplectic geometry of Hamiltonian dynamical systems is deeply connected to Cartan geometry, the dual of Finsler geometry", (see V.I. Arnold, I.M. Gelfand and V.S. Retach [13]).
The above mentioned applications have also imposed the introduction x Radu Miron of the notions of higher order Lagrange spaces and, of course, higher order Hamilton spaces. The base manifolds of these spaces are bundles of accelerations of superior order. The methods used in the construction of these geometries are the natural extensions of the classical methods used in the edification of Lagrange and Hamilton geometries. These methods allow us to solve an old problem of differential geometry formulated by Bianchi and Bompiani [94] more than 100 years ago, namely the problem of prolongation of a Riemannian structure g defined on the base manifold M, to the tangent k bundle T M, k> 1. By means of this solution of the previous problem, we can construct, for the first time, good examples of regular Lagrangians and Hamiltonians of higher order