Classification of Hamiltonians for One‐Dimensional Particle Systems

Representative(s) of equivalence class(es) of maximally symmetric Hamiltonian(s) for one‐dimensional particle systems which admit Hamiltonian symmetry(ies) is (are) classified. We in fact obtain Hamiltonians for each of the equivalence classes of dynamical equations that admit 1, 2, 3, and 8 Lie point symmetries. Moreover, we also derive the first integrals that correspond to the Hamiltonian symmetries by means of the Hamiltonian version of the Noether's theorem.