Twisted Legendre transformation

The general framework of Legendre transformation is extended to the case of symplectic groupoids, using an appropriate generalization of the notion of generating function (of a Lagrangian submanifold). 1 Tulczyjew triple and its generalization The general framework of Legendre transformation was introduced by Tulczyjew [1]. It consists in recognition of the following structure, which we call the Tulczyjew triple: T (TQ) α ←− T (T Q) β −→ T (T Q). (1) Here Q is the manifold of configurations (of the system), T Q — its phase space (cotangent bundle), TQ — its velocity space (tangent bundle) and α, β are natural symplectic isomorphisms. From those two isomorphisms, β is the easy one: it is just the vector bundle isomorphism induced by the symplectic form on P = T Q (β exists for any symplectic manifold P ). An explicit construction of α can be found in [1, 2]. Let us mention here how the existence of such an isomorphism follows from the general theory of symplectic groupoids [3, 4]. Cotangent bundles are exactly those symplectic groupoids which have commutative multiplication and connected, simply connected fibers. The tangent bundle to such a symplectic groupoid is again a commutative symplectic groupoid with connected, simply connected fibers, hence a cotangent bundle to its ‘space of units’, which in this case coincides with TQ. The dynamics is described by specifying a Lagrangian submanifold D ∈ T (T Q) (in the case of a section, we just have a Hamiltonian vector field on T Q). The two isomporphisms α and β allow to consider D as a Lagrangian submanifold of a cotangent bundle to TQ and to T Q, respectively. One can speak about generating functions of D in those two ‘control modes’. In the first case the function is called the Lagrangian, and in the second case it is (minus) the Hamiltonian. The passage from one formulation of dynamics to another consists in applying the universal Legendre transformation (1) to the particular case of Lagrangian submanifold. The aim of this paper is to stress the fact that the above scheme can be generalized to the case when the cotangent bundle T Q is replaced by an arbitrary symplectic