Hamiltonians and conjugate Hamiltonians of some fourth-order nonlinear ODEs

We first derive the Lagrangians of the reduced fourth-order ordinary differential equations studied by Kudryashov under the assumption that they satisfy the conditions stated by Fels M.E. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 348, 1996, 5007-5029], using Jacobi’s last multiplier technique. In addition we derive the Hamiltonians of these equations using the Jacobi-Ostrogradski theory. Next, we derive the conjugate Hamiltonian equations for such fourth-order equations passing the Painlevé test. Finally, we investigate the conjugate Hamiltonian formulation of certain additional equations belonging to this family.