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Published online by Cambridge University Press: 24 October 2008
It is pointed out that the equations of motion for any field obtained by varying a Lagrangian subject to auxiliary conditions are exactly equivalent to a certain set of canonical equations and that the commutation relations between the dynamical variables for the latter equations are Lorentz-invariant. By extending the theory to Lagrangians containing higher derivatives of the field quantities, it is shown that any given set of field equations can be put into the canonical form, though it is not derived from variational principles. The question of Lagrangians with missing momenta is also considered. It is shown that if the Lagrangian is ‘gauge-invariant’, some of the p's must be missing and the corresponding Eulerian equations can be replaced by equations containing no q and then can be replaced by initial conditions. The commutation relations between gauge-invariant quantities are Lorentz-invariant. For Lagrangians which are not gauge-invariant but are such as to have missing momenta, the passage to quantum theory will in general give rise to non-Lorentz-invariant commutation relations. In both cases, the equations of motion can be cast in canonical forms.
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† q r will sometimes not be written out explicitly. Thus f(q) may mean f(q, q r). We shall meet later p r ≡ ∂p/∂x r, which will also not be written out explicitly.
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† Such cases happen, for example, in the Lagrangian of a Maxwell field interacting with matter, where
‡ All g may contain space derivatives of p, which are not written out.
† More precisely, it follows from (16) for the other q and (29) for Q l. This means, of course, that equations (20) are not independent.
