A Hamilton–Jacobi theory for implicit differential systems

Abstract

In this paper, we propose a geometric Hamilton-Jacobi theory for systems of implicit differential equations.In particular, we are interested in implicit Hamiltonian systems, described in terms of Lagrangian submanifolds of T T * Q generated by Morse families. The implicit character implies the nonexistence of a Hamiltonian function describing the dynamics. This fact is here amended by a generating family of Morse functions which plays the role of a Hamiltonian. A Hamilton-Jacobi equation is obtained with the aid of this generating family of functions. To conclude, we apply our results to singular Lagrangians by employing the construction of special symplectic structures.

Contact Dynamics: Legendrian and Lagrangian Submanifolds

“…Hamilton-Jacobi theory for (explicit) Hamiltonian contact dynamics is recently examined in [72,76]. Hamilton-Jacobi theory for implicit symplectic dynamics is discussed in [77,78].…”

A Hamilton–Jacobi formalism for higher order implicit Lagrangians

“…In this paper, we construct a Hamilton-Jacobi theory for higher order mechanical systems described through implicit differential equations. More concretely, we will generalize the geometry that we have proposed in [24]. As we have mentioned previously, in [24], two ideas are proposed.…”

“…More concretely, we will generalize the geometry that we have proposed in [24]. As we have mentioned previously, in [24], two ideas are proposed. In the present work, we generalize the first idea by considering a higher order implicit differential equation as a submanifold S of T T * T k−1 Q generated by a Morse function F defined on the Whitney sum T * T k−1 Q × T k−1 Q T k Q and the projected submanifold is S γ in T T k−1 Q.…”

“…As a result, we arrive at a vector field X σ that satisfies Hamilton-like equations. We will show details in forthcoming sections applying this for implicit higher order systems [24].…”

“…It is important to remark that the classical HJ theory only deals with explicit Hamiltonian systems. In [24], we formulated an idea to deal with the implicit character. We considered implicit first order differential equations as a submanifold S of T T * Q.…”