Symplectic maps: from generating functions to Liouvillian forms

In this article we introduce a new method for constructing implicit symplectic maps using special symplectic manifolds and Liouvillian forms. This method extends, in a natural way, the method of generating functions to 1-forms which are globally defined on the symplectic manifold. The maps constructed by this method, are related to the symplectic Cayley's transformation and belong to a continuous space of dimension n(2n+1). Applying the implicit map to the discrete Hamilton equations we obtain the generalized symplectic Euler scheme. We show the relations of the elements of this family with other discrete symplectic mapping, in particular 1) with the mappings obtained by generating functions of type I, II, and III and IV; 2) with the symplectic Euler methods A and B; and 3) with the mid-point rule. Moreover, we show the corresponding symplectic diffeomorphisms and their Liouvillian forms on the product symplectic manifold. We illustrate the details of the method in constructing two different families of implicit symplectic maps for $n=1$. This is a geometrical method which overcomes the difficulties of the Hamilton-Jacobi theory and generating functions.