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.
Pavˇ siˇ c, International Journal of Geometric Methods in Modern Physics13, 1630015 (2016)
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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.