Hamiltonian Systems and Transformation in Hilbert Space

In recent years the theory of Hilbert space and its linear transformations has come into prominence.'It has been recognized to an increasing extent that many of the most important departments of mathematical physics can be subsumed under this theory.In classical physics, for example in those phenomena which are governed by linear conditions- linear differential or integral equations and the like, in those relating to harmonic analysis, and in many phenomena due to the operation of the laws of chance, the essential r6le is played by certain linear transformations in Hilbert space.And the importance of the theory in quantum me- chanics is known to all.It is the object of this note to outline certain investigations of our own in which the domain of this theory has been extended in such a way as to include classical Hamiltonian mechanics, or, more generally, systems defining a steady n-dimensional flow of a fluid of positive density.Consider the dynamical system of n degrees of freedom, the canonical equations of which are formed from the Hamiltonian H(q, p) = H(ql, * a qny ply .... ps), which we will assume to be single-valued, real, and analytic in a certain 2n-dimensional region R of the real qp-space.The solutions, or equations of motion, are qk = fk(q0, p0, t), Pk = gk(q0, po, t), (k = 1, ..., n), these functions being single-valued, real and analytic for all (q, p) in R and for t in a real interval containing t = 0 dependent on (q, p).It is shown that the transformation St: (q, po) > (q, p) defined by these equations for suitably restricted t has the formal proper- ties: St1S1, = Si, + ,, So = I.The system admits the "integral of energy"