On Nonlinear Connections in Higher Order Lagrange Spaces (original) (raw)

Remarks on Symplectic Connections

Letters in Mathematical Physics, 2006

This note contains a short survey on some recent work on symplectic connections: properties and models for symplectic connections whose curvature is determined by the Ricci tensor, and a procedure to build examples of Ricci-flat connections. For a more extensive survey, see [5]. This note also includes a moment map for the action of the group of symplectomorphisms on the space of symplectic connections, an algebraic construction of a large class of Ricci flat symmetric symplectic spaces, and an example of global reduction in a non symmetric case.

A Lagrangian form of tangent forms

Journal of Geometry and Physics, 2014

The aim of the paper is to study some dynamic aspects coming from a tangent form, i.e. a time dependent differential form on a tangent bundle. The action on curves of a tangent form is natural associated with that of a second order Lagrangian linear in accelerations, while the converse association is not unique. An equivalence relation of tangent form, compatible with gauge equivalent Lagrangians, is considered. We express the Euler-Lagrange equation of the Lagrangian as a second order Lagrange derivative of a tangent form, considering controlled and higher order tangent forms. Hamiltonian forms of the dynamics generated are given, extending some quantization formulas given by Lukierski, Stichel and Zakrzewski. Using semi-sprays, local solutions of the E-L equations are given in some special particular cases.

Lagrangian distributions and connections in multisymplectic and polysymplectic geometry

Differential Geometry and its Applications, 2013

We discuss the interplay between lagrangian distributions and connections in symplectic geometry, beginning with the traditional case of symplectic manifolds and then passing to the more general context of poly-and multisymplectic structures on fiber bundles, which is relevant for the covariant hamiltonian formulation of classical field theory. In particular, we generalize Weinstein's tubular neighborhood theorem for symplectic manifolds carrying a (simple) lagrangian foliation to this situation. In all cases, the Bott connection, or an appropriately extended version thereof, plays a central role.

Lagrangian submanifolds in k-symplectic settings

Monatshefte für Mathematik, 2013

In this paper we extend the well-know normal form theorem for Lagrangian submanifolds proved by A. Weinstein in symplectic geometry to the setting of k-symplectic manifolds.

Presymplectic structures and intrinsic Lagrangians

arXiv: High Energy Physics - Theory, 2016

It is well-known that a Lagrangian induces a compatible presymplectic form on the equation manifold (stationary surface, understood as a submanifold of the respective jet-space). Given an equation manifold and a compatible presymplectic form therein, we define the first-order Lagrangian system which is formulated in terms of the intrinsic geometry of the equation manifold. It has a structure of a presymplectic AKSZ sigma model for which the equation manifold, equipped with the presymplectic form and the horizontal differential, serves as the target space. For a wide class of systems (but not all) we show that if the presymplectic structure originates from a given Lagrangian, the proposed first-order Lagrangian is equivalent to the initial one and hence the Lagrangian per se can be entirely encoded in terms of the intrinsic geometry of its stationary surface. If the compatible presymplectic structure is generic, the proposed Lagrangian is only a partial one in the sense that its stat...

On geometric properties of Lagrangian submanifolds in product symplectic spaces

Hokkaido Mathematical Journal, 2006

We study the generic properties of symplectic relations. Local models of symplectic relations are described and the corresponding local symplectic invariants are derived. A stratification of the Lagrangian Grassmannian in the product symplectic space (N × M, π * M ω M − π * N ω N) is constructed and global homological properties of the strata are investigated.

Lagrangian submanifolds in product symplectic spaces

Journal of Mathematical Physics, 2000

We analyze the global structure of Lagrangian Grassmannian in the product symplectic space and investigate the local properties of generic symplectic relations. The cohomological symplectic invariant of discrete dynamical systems is generalized to the class of generalized canonical mappings. Lower bounds for the number of two-point and three-point symplectic invariants for billiard-type dynamical systems are found and several examples of symplectic correspondences encountered from physics are presented.

2 Variational problem for the higher order Lagrangians

2003

A Lagrange space of order k ≥ 1 is the space of accelerations of order k endowed with a regular Lagrangian. For theses spaces we discuss: certain natural geometrical structures, variational problem associated to a given regular Lagrangian and the induced semispray, nonlinear connection, metrical connections. A special attention is paid to the prolongations of the Riemannian and Finslerian structures. In the end we sketch the geometry of time dependent Lagrangian. The geometry, which we have developed, is directed to Mechanicists and Physicists. The paper is a brief survey of our results in the higher order geometry. For details we refer to the monograph [3]. AMS Subject Classification: 53C60, 53C80, 58A20, 58A30.