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Hamilton equations are such that the relation, between the coordinates and momenta at time t and at time t 0, is a contact transformation. Methods of integration of Hamilton equations, which do preserve the contact transformation property are given here. These methods are of first and second order. They are given, for the equation x= f(x,t), then for the case of one degree of freedom, then for the general case. Some of the formulae are implicit.
René De Vogelaere (Mon,) studied this question.