Radu Miron: The geometry of higher-order Hamilton spaces

Abstract

Radu Miron: The geometry of higher-order Hamilton spacesKluwer Academic Publishers, Dordrecht, 2003, 264 p., This monograph is the sixth one resulting from 50 years of research activity of the prominent Romanian school on Finsler geometry, Lagrange-Hamilton spaces, and their higher-order generalizations [1][2][3][4][5]. The new book presents an overview of the higher-order Hamilton spaces with applications to higher-order mechanics following the canonical non-linear connection (N-connection) and (semi-)spray formalism and the geometry of induced almost complex or contact Riemann-Cartan structures. The variational principles for higher-order Hamiltonians, HamiltonJacobi equations and related conservation laws and symmetries are described and investigated.In this review, I would like to discuss some relations of this original approach to geometry and physics and reflect on applications to modern gravity and string theory.There are two general directions towards geometrization of mechanics on the tangent or cotangent bundle and their higher-order extensions: The first one is based on the idea of describing classical mechanics and classical field theory in terms of symplectic geometry following certain procedures of geometrization of the Euler-Lagrange and Hamilton equations by using the (multi-)symplectic formalism, differential forms, jets, etc. (see, for instance, [6,7]). In a quite different way, in the second direction Lagrange and Hamilton mechanics was geometrized using Riemann-Finsler spaces and, in a more general context, certain types of non-holonomic manifolds enabled with non-integrable distributions defined by N-connections. The general ideas and methods originate from E. Cartan [8] and A. Kawaguchi [9, 10] who were the first to consider metric-compatible models of Finsler spaces and N-connection structures. In this way, the geometric constructions are not related to certain particular properties of the Euler-Lagrange, or

On axiomatic formulation of gravity and matter field theories with MDRs and Finsler-Lagrange-Hamilton geometry on (co)tangent Lorentz bundles

“…In the past, the author of this article cited and highly appreciated and cited R. Miron and company works in Refs. [296,320,299,308,325] but have to reconsider his former scientific evaluation on Finsler geometry and applications in Romania. Beginning 2008, one have been published some files and newspapers in Western Countries and Romania how Ceauşescu's secret service supervised scientific activity via their agents at the Academy of Science and Universities administration.…”

Nonholonomic Clifford and Finsler Structures, Non-Commutative Ricci Flows, and Mathematical Relativity

“…(d) Noncommutative gauge gravity. 4 One should be mentioned here some series of "not less" important contributions containing various preliminary, or alternative, ideas and results published in Romania [65]- [69] and Republic of Moldova [76]- [80]; monographs [70]- [72], chapters and sections in collections of works [81]- [83], reviews in journals and encyclopedia [73,74], articles in Proceedings of Conferences [84]- [91] and some recent electronic preprints with reviews, computation details and proofs, see [132,133].…”

Finsler-Lagrange Geometries and Standard Theories in Physics: New Methods in Einstein and String Gravity

“…A brief summary of R. Miron school's results on Finsler, Lagrange, Hamilton and higher order generalizations and further perspectives in gravity and field theories is given in Ref. [200].…”

“…[91] as a summary of alternative directions. The nonlinear connection formalism was developed for higher order (co) tangent bundles which resulted in a series of monographs on higher order Lagrange-Finsler and Hamilton-Cartan spaces [103,104,112,106], see also a recent work [38] and a brief review [200]. Such higher order geometric mechanical constructions are naturally adapted to corresponding (semi) spray configurations, from which canonical nonlinear and distinguished connections can be derived.…”