Constants of motion and the variational equations

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Abstract
For field equations of Hamiltonian form the relation between constants of motion and solutions of the linearized equation is discussed. A known result is that the Poisson bracket of a constant of motion with the field variable solves the linearized equation. Here the following converse result is obtained: If a δu which satisfies the linear equation is of the form δu=[u,T], then T is a constant of motion. Further, ∂T/∂t is also a constant. The sequence [T,∂T/∂t], [[T,∂T/∂t],T],. . . is shown to produce other nontrivial constants.

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