Keywords: chaos, Riemannian geometry, pendulum.

In this paper the geometrical approach is applied to describe dynamics of a two degrees-of-freedom mechanical system. This approach has been recently widely used to study chaos in Hamiltonian systems [1,2,3,4]. The crucial point of this method is that trajectories of a system can be viewed as geodesics of a Riemannian manifold endowed with a metric coming from dynamics of the analysed system. The instability of the system is linked with instability properties of geodesics through the Jacobi-Levi-Civitta (JLC) equation, which governs the evolution of geodesics separation. Owing to Riemann tensor occurence in the JLC equation, instability properties are completely determined by curvature properties of the underlying manifold. It is well-known that manifolds with negative curvature induce chaos. However, a chaotic behaviour may also occur in systems with positive curvatures manifolds due to curvature fluctuations along geodesics. This technique is still under development and other classes of dynamical systems can be also analysed in this manner e.g. systems with velocity-dependent potentials are described by Finsler geometry [4]. It is believed that geometrical approach may provide a good explanation of the onset of chaos in Hamiltonian systems. Note that a few manifolds can serve as a Riemannian one, for example configuration manifold with Jacobi metric, enlarged space-time manifold with Eisenhart metric or tangent bundle of configuration manifold with Sasaki metric. In this paper we use configuration manifold with Jacobi metric as the Riemannian manifold, since our system is conservative. We show how the Riemannian geometry tools govern the behavior of our two degrees-of-freedom mechanical system with metric dependent on coordinates. The analysed system consists of elastic swinging pendulum with nonlinear stiffness (see Figure 1).

The investigated system is governed by the following non-dimensional equations:

and the relations between dimensional (see Figure 1) and non-dimensional parameters read

Since the system possesses two degrees-of-freedom, the Riemannian manifold is two-dimensional and the Riemann tensor has only one independent nonzero component. From the Jacobi-Levi-Civitta equation the Hill equation is obtained, which is further solved numerically.

1

L. Casetti, C. Clementi, M. Pettini: Phys. Rev. E, 54 (6), 1996. doi:10.1103/PhysRevE.54.5969

2

L. Casetti, M. Pettini: Phys, Rev. E, 48 (6), 1993. doi:10.1103/PhysRevE.48.4320

3

M. Cerruti-Sola, M. Pettini: Phys. Rev. E, 51 (1), 1995. doi:10.1103/PhysRevE.51.53

4

M. Di Bari, P. Cipriani: Planet. Space Sci. Vol. 46 (11/12), 1998.

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