On Lagrangians and Hamiltonians of some fourth-order nonlinear Kudryashov ODEs

We derive the Lagrangians of the reduced fourth-order ordinary differential equations studied by Kudryashov, when they satisfy the conditions stated by Fels [Fels ME, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations. Trans Am Math Soc 1996;348:5007–29] using Jacobi’s last multiplier technique. In addition the Hamiltonians of these equations are derived via Jacobi–Ostrogradski’s theory. In particular, we compute the Lagrangians and Hamiltonians of fourth-order Kudryashov equations which pass the Painlevé test.

Research highlights
► The Jacobi Last Multiplier is a useful tool for deriving the Lagrangian of such equations provided the Fels conditions are satisfied.
► Kudryashov derived two hierarchies of fourth-order ODEs which pass the Painlevé test.
► The Hamiltonization of such equations is considered using Ostrogradski’s theory.
► These contributes to the understanding of higher-order ODEs in general.