Hamiltonian Zoo for systems with one degree of freedom

Abstract: We present alternative explicit forms of the standard Hamiltonian for systems with one degree of freedom. This new class of infinite Hamiltonians is called Newton-equivalent Hamiltonian zoo, producing the same equation of motion. These Hamiltonians are directly solved from the Hamilton's equations and come with extra-parameters, which are interpreted as scaling factors for the time evolution on phase space. Moreover, each Hamiltonian in the zoo can be used as a generating function for a Hamiltonian hierarchy.