A NOTE ON INTEGRABLE TWO-DEGREES-OF-FREEDOM HAMILTONIAN SYSTEMS WITH A SECOND INTEGRAL QUARTIC IN THE MOMENTA

摘要

Two of the simplest integrable Hamiltonians H(x, y, p(x), p(y)) = (p(x)(2) + p(y)(2))/2 + V(x, y) with a second integral quartic in the momenta are those with potentials V-3(x, y) = by(3x(2) + 16y(2)) + d(x(2) + 16y(2)) + eta y and V-4(x,y) = a(x(4) + 6x(2)y(2) + 8 y(4)) + c(x(2) + 4y(2)) + vy(-2). We show how V-3 can be obtained from V-4. In the process we obtain a new potential of the class, V-N, that includes both V-3 and V-4 as particular cases. For this potential we give the second integral of motion, separating variables, a Lax representation and a bi-Hamiltonian structure, thus synthesizing the corresponding results for potentials V-3 and V-4. The integrable extension V-N + mu x(-2) is also discussed. [References: 13]