Calculus of variations with hyperdifferential operators from Tabasaki-Takebe-Toda lattice arguments

We introduce the calculus of variations characterized by a Lagrangian containing both classical derivatives and hyperdifferential operators inspired from Tabasaki-Takebe-Toda lattice arguments. Necessary optimality conditions of the Euler-Lagrange type are obtained and proved. The Hamiltonian formalism is discussed and the isoperimetric problem is considered as well.