First integrals of generalized Ermakov systems via the Hamiltonian formulation

摘要

We obtain first integrals of the generalized two-dimensional Ermakov systems, in plane polar form, via the Hamiltonian approaches. There are two methods used for the construction of the first integrals, viz. the standard Hamiltonian and the partial Hamiltonian approaches. In the first approach, F(theta) and G(theta) in the Ermakov system are related as G(theta) + F' (theta)/2 = 0. In this case, we deduce four first integrals (three of which are functionally independent) which correspond to the Lie algebra sl(2, R) circle plus A(1) in a direct constructive manner. We recover the results of earlier work that uses the relationship between symmetries and integrals. This results in the complete integrability of the Ermakov system. By use of the partial Hamiltonian method, we discover four new cases: F(theta) = G(theta)(c(1) sin theta + c(3) cos)/(c(1) cos theta - c(3) sin theta) with c(2)c(3) = c(1)c(4), c1 not equal 0, c(3) not equal 0; F(theta) = G(theta) (c(2) sin theta + c(4) cos theta)/(c(2) cos theta - c(4) sin theta) with c(1) = c(3) = 0, c2 not equal 0, c(4) not equal 0; F(theta) = -G(theta) cot theta with c(1) = c(2) = 0, c(3), c(4) arbitrary and F(theta) = G(theta) tan theta with c(3) = c(4) = 0, c(1), c(2) arbitrary, where the c(i)s are constants in all cases. In the last two cases, we find that there are three operators each which give rise to three first integrals each. In both these cases, we have complete integrability of the Ermakov system. The first two cases each result in two first integrals each. For every case, both for the standard and partial Hamiltonian, the angular momentum type first integral arises and this is a consequence of the operator which depends on a momentum coordinate which is a generalized symmetry in the Lagrangian context.