For higher-derivative Lagrangians, Ostrogradsky's canonical coordinates and momenta are exactly the coefficients of the variation terms in the boundary term of delta L.
Lanczos,The Variational Principles of Mechanics(Dover Publications, 1986)
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The First Variational Formula and the Ostrogradsky Formalism
For higher-derivative Lagrangians, Ostrogradsky's canonical coordinates and momenta are exactly the coefficients of the variation terms in the boundary term of delta L.