On the relationship between conservation laws and invariance groups of nonlinear field equations in Hamilton’s canonical form

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Abstract
It is shown that whenever fields governed by the equations ∂/∂tpα=−δH/δqα, ∂/∂tqα =δH/δpα allow a conservation law of the form ∂ρ/∂t+divJ=0, there exists a corresponding Lie–Bäcklund infinitesimal contact transformation which leaves the Hamiltonian equations invariant. A condition that an invariant Lie–Bäcklund infinitesimal contact transformation gives rise to a conservation law is established. Each such transformation, which may involve derivatives of arbitrary order, yields a one‐parameter local Lie group of invariance transformations. The results are established with the aid of a Lie bracket formalism for Hamiltonian fields. They account for a number of recently discovered conservation laws associated with nonlinear time evolution equations.