Symmetry and resonance in Hamiltonian systems (Q2706084)

The authors consider a two-degree-of-freedom Hamiltonian NEWLINE\[NEWLINEH(p_1,q_1,p_2,q_2)= \textstyle{{1\over 2}} \omega_1(p^2_1+ q^2_1)+ \textstyle{{1\over 2}} \omega_2(p^2_2+ q^2_2)+ H_3+ H_4+\cdots,NEWLINE\]NEWLINE where \(H_k\) \((k\geq 3)\) are homogeneous polynomials of degree \(k\). Using normal forms and introducing in the usual way a small parameter by rescaling the variables, the authors analyze the dynamics of Hamiltonian flow in a neighborhood of equilibrium in the origin of phase space. Due to the presence of a symmetry condition on one of the degrees of freedom, some resonances vanish as lower-order resonances. The authors give a new sharp estimate of the size of resonance domain for higher-order resonances. As an application, the authors discuss one of the classical examples with symmetry -- the elastic pendulum. In this problem, the symmetry assumption produces a new hierarchy of resonances in which, after well-known 2:1 resonance, the 4:1 resonance is the most prominent one. The asymptotic analysis is supplemented by numerical calculations which show excellent agreement.