Periodic Trajectories near Degenerate Equilibria in the Hénon-Heiles and Yang-Mills Hamiltonian Systems

Abstract

This paper deals with connected branches of nonstationary periodic trajectories of Hamilton equations

$$ {\dot q}_i = {\partial H \over \partial p_{i}}, \quad {\dot p}_i = - {\partial H \over \partial q_{i}}, \quad i = 1, \ldots ,n $$

emanating from the degenerate stationary point $x_0 = (p_0 ,q_0) \in (\nabla H)^{ - 1} (\{ 0\} )$ for H being the generalized Hénon-Heiles (HH) Hamiltonian:

$$ H(p_1, p_2, q_1, q_2 ) = {1 \over 2} 1^2 + p_2^2 + aq_1^2 + bq_2^2 ) + cq_1^2 q_2 + {1 \over 3} dq_2^3 + eq_1 q_2^2 $$

or the generalized Yang-Mills (YM) Hamiltonian:

$$\eqalign{ H(p_1, p_2, q_1, q_2) = {1 \over 2}(p_1^2 + p_2^2 + aq_1^2 + bq_2^2 ) + {1 \over 4}cq_1^4 + {1 \over 4}dq_2^4 + {1 \over 2}edq_1^2 q_2^2\cr \quad \qquad+ kq_1 q_2^3 + lq_1^3 q_2}. $$

The existence of such branches has been proved. Minimal periods of searched trajectories near x₀ have been described.