Let J^1(R^n, R) be the space of 1-jets of a function from R^n to R with contact 1-form Cu = du - u_i dx^i.  A vector field X on J^1(R^n, R) such that LieDerivative(X, Cu) = F Cu is called an infinitesimal contact transformation or contact vector field.

There is a formula which assigns to each real-valued function S on J^1(R^n, R) a contact vector field X.  The function S is called the generating function for the contact vector field X.  The explicit formula for X in terms of S is given for n = 1, 2, 3 in Example 1.  The formula in the general case can be found in P. J. Olver, Equivalence, Invariants and Symmetry, page 131

The command GeneratingFunctionToContactVector(S) returns the contact vector field defined by the function S.

The command GeneratingFunctionToContactVector is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form GeneratingFunctionToContactVector(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-GeneratingFunctionToContactVector(...).