Connections on tangent bundles of higher order associated to regular Lagrangians

Abstract

Given a regular Lagrangian L of order k on M we show that there exists a canonical connection F L on T 2k-tM whose paths are the solutions of the Euler-Lagrange equations for L. IfL is the kinetic energy defined by a Riernannian metric then F L is the Riemannian connection.

Lie algebroids generated by cohomology operators

“…It is well known (see [1,9]) that for any semispray S, the endomorphism Γ = −L S J is a connection on T M , with associated projectors Remark 9. This intimate relation between semisprays and connections has been used in Physics to study such topics as the inverse problem for Lagrangian dynamics [7], degenerate Lagrangian systems [3], or the geometry and mechanics of higher order tangent bundles [8].…”

“…Remark 9. This intimate relation between semisprays and connections has been used in Physics to study such topics as the inverse problem for Lagrangian dynamics [7], degenerate Lagrangian systems [3], or the geometry and mechanics of higher order tangent bundles [8].…”

Lagrangians on iterated tangent bundles and the Hamiltonization

“…Example. We take 3 w and m h I with local coordinates xY y 100 h yY y 010 h Y y 110 h Y y 001 h $Y y 101 h Y y 011 h ¤Y y 111 h r. Given is also the Lagrangian L h FPy g G g r 2 .…”

“…Over the past years, results have been published of de León et al, [3], dealing with semisprays and mutual relations on r w and motivated by mechanics. The second generalization is represented by higher order iterated tangent bundles.…”