Approximate Hamiltonian symmetries and related first integrals

We study approximate Hamiltonian symmetries and related first integrals for perturbed Hamiltonian systems. The approximate Hamiltonian symmetry expressed in evolutionary form gives rise to an approximate first integral provided it satisfies certain criteria. We provide formulas to deduce an approximate first integral for any given approximate Hamiltonian symmetry in three ways. As applications, mainly from mechanics, the perturbed orbit equation, two-dimensional deformed galaxy model, a perturbed second-order ODE containing an arbitrary function and an equation arising from fluid mechanics are analysed for approximate first integrals. •We study approximate Hamiltonian symmetries and related first integrals for perturbed Hamiltonian systems.•We provide formulas to deduce an approximate first integral for any given approximate Hamiltonian symmetry in three ways.•The theory is applicable for models from fluids.