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.