Variational calculus with constraints on general algebroids

Abstract

Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained EulerLagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers majority of first-order Lagrangian systems which are present in the literature and reduces to the standard variational calculus and the Euler-Lagrange equations in Classical Mechanics for E = T M . 2000: 70H03, 70H25, 53D17, 17B66, 53D10. MSC

Hamilton–Jacobi theory for degenerate Lagrangian systems with holonomic and nonholonomic constraints

“…We note that the idea of applying implicit differential equations to nonholonomic systems is found in an earlier work by Ibort et al, 32 see also Grabowska and Grabowski 33 for a generalization to vector bundles with algebroid structures.…”

Brackets

“…The notation is borrowed from [19,21,42] and we refer to these papers for details. In particular, we use affine coordinates (x a , ξ i ) on E * and the dual coordinates (x a , y i ) on E, associated with dual local bases, (e i ) and (e i ), of sections of E and E * , respectively.…”

“…[9,75,76]). In [19,21], in turn, has been observed that its geometric background is actually based on double vector bundle morphisms ε and the Jacobi identity plays no role in the construction of dynamics, that gives a space for further generalizations.…”

“…that can be viewed as Euler-Lagrange equations for the algebroid associated with the orthogonal projection of the tensor Π ε onto C * , according to the orthogonal decomposition E * = C 0 ⊕ C * [19]. Of course, even for E being a Lie algebroid, if C is not a Lie subalgebroid, the the projected tensor is not a Poisson tensor and we deal with mechanics on a general algebroid, in fact, a skew algebroid, since the projected Poisson tensor remains skew-symmetric.…”

Variational constrained mechanics on Lie affgebroids

“…In the same Lie algebroid setting, other authors [16] consider variational constrained mechanical systems. In another direction, a unified approach of nonholonomic and vakonomic mechanics, using general algebroids instead of just Lie algebroids, was developed in [9].…”