Chaotic trajectories for natural systems on a torus (Q1811379)

The authors consider natural Lagrangian systems on a torus, and give sufficient conditions for the existence of chaotic trajectories for values of the energy slightly below the maximum of the potential energy. Chaotic trajectories always exist unless the system is ``variationally separable'', i.e. minimizers of the action behave like in separable systems. Let \((M,g)\) be a compact Riemannian manifold of dimension 2; consider a Lagrangian \(L := (1/2) g_{ij} \dot{q}^i \dot{q}^j - V(q)\) of class \(C^3\) on \(M\). Assume \(V\) has a strict nondegenerate maximum at \(p \in M\), say with \(V(p) = 0\). Fix an energy level \(h\), and let \(M_h\) the corresponding (invariant) energy manifold, \( \Sigma_h := \{ (q,\dot{q}): (1/2) g_{ij} \dot{q}^i \dot{q}^j + V(q) = h \}\). The properties of the dynamical system issued by \(L\) depend on the topology of \(M\) and on the energy level \(h\). Define the Jacobi metric on \(M_h := \{ q: V(q)< h \} \subseteq M\) via \( ||\dot{q} ||^2_h := 2 [ h - V(q)] g_{ij} \dot{q}^i \dot{q}^j\); then the length of a curve \(\gamma: [a,b] \to M\) in this metric is given by its Maupertuis action \( J_h (\gamma) = \int_a^b ||\dot{\gamma } (t) ||^2_h d t\), and if \(\gamma\) is the trajectory of a solution to the system issued by \(L\), then it is a geodetic for the Jacobi metric, i.e. an extremal for the functional \(J_h\) restricted to curves with fixed endpoints. Let \(\Omega\) be the set of rectifiable loops in \(M\) through \(p\), and let \(J_0 : \Omega \to {\mathbb R}\) be the Maupertuis action corresponding to \(h = V(p) = 0\). Then connected components of \(\Omega\) are elements of \(G = \pi_1 (M,p) \simeq {\mathbb Z}^2\). It is possible to define a seminorm on \(G\) by \(\|a \|= \inf_{\gamma \in a} J_0 (\gamma) \geq 0 \). This seminorm carries information about the behaviour of the system for \(h\) near to \(V(p)\): in fact, \(\|.\|\) is a norm if the system is separable, and if \(\|.\|\) is not a norm, then the system has chaotic trajectories. Any trajectory homoclinic to \(p\) can be reparametrized so that it corresponds to a \(\gamma \in \Omega\); denote by \([\gamma] \in G\) its homotopy type. One says that \(\gamma\) is minimizing if \(J_0 (\gamma) = \|[\gamma] \|\). The homoclinic loop \(\gamma\) defines two unit vectors in \(TM\), \(\xi (\gamma) \) and \(\eta (\gamma)\), i.e. \[ \xi (\gamma) = \lim_{t \to - \infty}[\dot{\gamma} (t) / |\dot{\gamma} (t)|],\quad\eta (\gamma) = - \lim_{t \to + \infty}[\dot{\gamma} (t) /|\dot{\gamma} (t)|]. \] The main result of this paper is the following: Theorem. Suppose that \(M \simeq {\mathbb T}^2\), and that there is no \(\varepsilon > 0 \) such that the system issued by \(L\) has chaotic trajectories on all energy levels \(\Sigma_h\) for \(V(p) - \varepsilon < h < V(p)\). Then there exist generators \(a_1 , a_2 \in G\) such that \( \|n_1 a_1 + n_2 a_2 \|= |n_1|\cdot \|a_1\|+ |n_2|\cdot \|a_2\|\) for all \((n_1,n_2) \in {\mathbb Z}^2\), and the minimum of \(J\) on \(a_i\) is attained on a homoclinic trajectory \(\gamma_i\) such that \(\xi (\gamma_i) = - \eta (\gamma_i) \) and \(\xi (\gamma_1) \perp \xi (\gamma_2) \). This results generalizes previous results by \textit{S. V. Bolotin} [Mosc. Univ. Mech. Bull. 45, 7-14 (1990; Zbl 0712.70031) and NATO Adv. Sci. Inst. Ser. B Phys. 331, 173-179 (1994)]; \textit{P. H. Rabinowitz} [Topol. Methods Nonlinear Anal. 9, 41-76 (1997; Zbl 0898.34048)]; \textit{S. V. Bolotin} and \textit{P. H. Rabinowitz} [J. Differ. Equations 148, 364-387 (1998; Zbl 0990.37046)]; \textit{S. V. Bolotin} and \textit{P. Negrini} [Russ. J. Math. Phys. 5, 415-436 (1997; Zbl 0951.37029)].