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Constants of motion associated with alternative Hamiltonians

It is shown that if a non-autonomous system of $2n$ first-order ordinary differential equations is expressed in the form of the Hamilton equations in terms of two different sets of coordinates, $(q_{i}, p_{i})$ and $(Q_{i}, P_{i})$, then the determinant and the trace of any power of a certain matrix formed by the Poisson brackets of the $Q_{i}, P_{i}$ with respect to $q_{i}, p_{i}$, are constants of motion.