Kinetic energy and the stable set (Q2431070)

A classical result of Lagrange-Dirichlet states that in the theory of \(C^k\) mechanical systems, \(k\geq 2\), the strict minimum point for the potential energy \(\pi(q)\) is necessarily stable in the sense of Lyapunov. The first author and \textit{S. V. Bolotin} have proved the existence of two potential energies \(T_1\), \(T_2\) and a potential energy \(\pi\) such that the equilibrium (origin) is stable for \(H_1 = T_1 + \pi\) but unstable for \(H_2 = T_2 +\pi\) [Arch. Ration. Mech. Anal. 152, No. 1, 65--79 (2000; Zbl 0968.70016)]. Later, the last two authors provided the explicit example of this fact, where \(\pi \in C^k\) instead of \(C^{\infty}\) and \(T_1\), \(T_2\) are supposed to be analytic in contrast to the previous result [``Stability of equilibrium of conservative systems with two degrees of freedom'', J. Differ. Equations 194, No. 2, 364--381 (2003; Zbl 1134.70317)]; also, it was showed there that the kinetic energy can have an influence on the dimension of the stable set. Here it is shown that if the potential energy satisfies a simple arithmetical criteria, the dimension of the stable set of the origin for \(H_1\) is different from the dimension of the stable set of the origin for \(H_2\).