First integrals of some two-dimensional integrable Hamiltonian systems

Abstract In this paper, we discuss some results on integrable Hamiltonian systems with two degrees of freedom. We revisit the much-studied problem of the two-dimensional harmonic oscillator and discuss its (super)integrability in the light of a canonical transformation which can map the anisotropic oscillator to a corresponding isotropic one. Following this, we discuss the computation of first integrals for integrable two-dimensional systems using the framework of the Jacobi last multiplier. Using the latter, we describe some novel physical examples, namely, the classical Landau problem with a scalar-potential-induced hyperbolic mode, the two-dimensional Kepler problem, and a problem involving a linear curl force.