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.
Whittaker,A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies, 2nd ed
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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.