Hamiltonian systems

In section 2.4 we were introduced to Hamiltonian systems. If H is a C2 function of the 2n variables p _i , q _i , i = 1, … n, H : ℝ2n → ℝ, then the equations of Hamilton(15-x) $${\dot p}_i = -{\partial H\over \partial q_i}, {\dot q}_i = {\partial H\over \partial p_i}, i = 1, “ots , n.$$Now we have for the orbital derivative L _t H = 0, so H(p, g) is a first integral of the equations 15.1. We have seen a number of examples where n = 1; in this case the integral H(p, q) = constant describes the orbits in the phase-plane completely.