Higher-order Lagrangian systems: Geometric structures, dynamics, and constraints

Abstract

In order to study the connections between Lagrangian and Hamiltonian formalisms constructed from a-perhaps singular-higher-order Lagrangian, some geometric structures are constructed. Intermediate spaces between those of Lagrangian and Hamiltonian formalisms, partial OstrogradskiI's transformations and unambiguous evolution operators connecting these spaces are intrinsically defined, and some of their properties studied. Equations of motion, constraints, and arbitrary functions of Lagrangian and Hamiltonian formalisms are thoroughly studied. In particular, all the Lagrangian constraints are obtained from the Hamiltonian ones. Once the gauge transformations are taken into account, the true number of degrees of freedom is obtained, both in the Lagrangian and Hamiltonian formalisms, and also in all the "intermediate formalisms" herein defined.

Path-integral approach to ’t Hooft’s derivation of quantum physics from classical physics

“…[39,41] and citations therein). In passing, we should mention that the currently popular Hamilton-Jacobi [47] and Legendre-Ostrogradskiȋ [48] approaches for a treatment of constrained systems, though highly convenient in certain cases (e.g., in higher-order Lagrangian systems), have not found as yet any particular utility in the present context.…”

Non-Standard Lagrangians with Higher-Order Derivatives and the Hamiltonian Formalism

“…However, it is notable that in the standard approach, the Hamiltonian approach with higherorder derivatives has been formulated by Ostrogradiski [38] and discussed in many papers [39][40][41][42][43][44]. In Ostrogradiski's approach, the Hamiltonian is defined by H ¼ P n P m n À1 a n _ q ða n Þ n p n;a n À Lðq n ; _ q n ;...;q ðm n Þ n Þ;m n !1;n ¼ 1;2;... where p n;a n obeys the recursion formula p n;i nÀ1 ¼ oL .…”

Untitled

“…Observe that relations (29), (30) and (31) arise because the commutativity of diagram (32) demands it.…”

“…Most of these results have also been generalized for higher-order Lagrangian systems [10], [27], [30], [31], and for the case of more general types of singular differential equations on manifolds (implicit systems of equations) [25]. Finally, although a covariant description of this operator was not available, it has also been used to study several characteristics of some physical models in field theory, namely the bosonic string [1], [2], [26].…”