Partial Noether Operators and First Integrals for a System with two Degrees of Freedom

We construct all partial Noether operators corresponding to a partial Lagrangian for a system with two degrees of freedom. Then all the first integrals are obtained explicitly by utilizing a Noether-like theorem with the help of the partial Noether operators. We show how the first integrals can be constructed for the system without the need of a variational principle although the Lagrangian L = y′2/2 + z′2/2 – v(y, z) does exist for the system. Our objective is twofold: one is to see the effectiveness of the partial Noether approach and the other to determine all the first integrals of the system under study which have not been reported before. Thus, we deduce a complete classification of the potentials v(y, z) for which first integrals exist. This can give rise to further studies on systems which are not Hamiltonian via partial Noether operators and the construction of first integrals from a partial Lagrangian viewpoint.