Unified formalism for nonautonomous mechanical systems

Abstract

We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of regular and non-regular time-dependent mechanical systems, which is based on the approach of Skinner and Rusk [18]. The dynamical equations of motion and their compatibility and consistency are carefully studied, making clear that all the characteristics of the Lagrangian and the Hamiltonian formalisms are recovered in this formulation. As an example, it is studied a semidiscretization of the nonlinear wave equation proving the applicability of the proposed formalism.

Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control

“…Thus, the symplectic section Ωg is ; and a curve on E is admissible if its tangent vectors are admissible. The set of admissible elements on E will be denote E (2) .…”

“…If (x i , pA) are the local coordinates on E * associated with the local basis {e A } of Γ( E * ), then (x i , pA, y a ) are local coordinates on W0 and we may consider the local basis { e (1) A , ( e A ) (2) , e , ( e A ) (2) , where (ě, e * ) ∈ W0 and ν(ě, e * ) = x. If ([[·, ·]] ν , ρ ν ) is the Lie algebroid structure on T ν E, we have that…”

“…The idea of this formulation was to obtain a common framework for both regular and singular dynamics, obtaining simultaneously the Hamiltonian and Lagrangian formulations of the dynamics. Over the years, however, Skinner and Rusk's framework was extended in many directions: It was originally developed for first-order autonomous mechanical systems [70], and later generalized to non-autonomous dynamical systems [2,24,68], control systems [4] and, more recently to classical field theories [12,28].…”

Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

“…This formalism has been generalized for non-autonomous mechanics [7,20,45] and recently for classical field theories [24,71]. The main features of this formulation are explained in Section 4.…”

Quasivelocities and Optimal Control for underactuated Mechanical Systems

“…In Section 2 we describe the concept of quasivelocities from a geometric point of view and we introduce the Euler-Lagrange equations on quasivelocities (called Hamel equations). In section 3 we describe the conditions of optimality for optimal control problem using the Skinner and Rusk formulation [10,1].…”