Billiard Systems as the Models for the Rigid Body Dynamics
Description of the rigid body dynamics is a complex problem, which goes back to Euler and Lagrange. These systems are described in the six-dimensional phase space and have two integrals the energy integral and the momentum integral. Of particular interest
- PDF / 622,279 Bytes
- 21 Pages / 439.37 x 666.142 pts Page_size
- 65 Downloads / 184 Views
Billiard Systems as the Models for the Rigid Body Dynamics Victoria V. Fokicheva and Anatoly T. Fomenko
Abstract Description of the rigid body dynamics is a complex problem, which goes back to Euler and Lagrange. These systems are described in the six-dimensional phase space and have two integrals the energy integral and the momentum integral. Of particular interest are the cases of rigid body dynamics, where there exists the additional integral, and where the Liouville integrability can be established. Because many of such a systems are difficult to describe, the next step in their analysis is the calculation of invariants for integrable systems, namely, the so called Fomenko–Zieschang molecules, which allow us to describe such a systems in the simple terms, and also allow us to set the Liouville equivalence between different integrable systems. Billiard systems describe the motion of the material point on a plane domain, bounded by a closed curve. The phase space is the four-dimensional manifold. Billiard systems can be integrable for a suitable choice of the boundary, for example, when the boundary consists of the arcs of the confocal ellipses, hyperbolas and parabolas. Since such a billiard systems are Liouville integrable, they are classified by the Fomenko–Zieschang invariants. In this article, we simulate many cases of motion of a rigid body in 3-space by more simple billiard systems. Namely, we set the Liouville equivalence between different systems by comparing the Fomenko–Zieschang invariants for the rigid body dynamics and for the billiard systems. For example, the Euler case can be simulated by the billiards for all values of energy integral. For many values of energy, such billard simulation is done for the systems of the Lagrange top and Kovalevskaya top, then for the Zhukovskii gyrostat, for the systems by Goryachev–Chaplygin–Sretenskii, Clebsch, Sokolov, as well as expanding the classical Kovalevskaya top Kovalevskaya–Yahia case.
V.V. Fokicheva (B) · A.T. Fomenko (B) Lomonosov Moscow State University, Leninskie Gory, 1, Moscow, Russia e-mail: [email protected] A.T. Fomenko e-mail: [email protected] © Springer International Publishing Switzerland 2016 V.A. Sadovnichiy and M.Z. Zgurovsky (eds.), Advances in Dynamical Systems and Control, Studies in Systems, Decision and Control 69, DOI 10.1007/978-3-319-40673-2_2
13
14
V.V. Fokicheva and A.T. Fomenko
2.1 Introduction Definition 2.1 A symplectic structure on a smooth manifold M is a differential 2form ω satisfying the following two properties: (1) ω is closed, i.e., dω = 0; (2) ω is non-degenerate at each point of the manifold, i.e., in local coordinates, detΩ(x) = 0, where Ω(x) = (ωij (x)) is the matrix of this form. The manifold endowed with a symplectic structure is called symplectic. Let H be a smooth function on a symplectic manifold M. We define the vector of skew-symmetric gradient sgrad H for this function by using the following identity: ω(v, sgrad H) = v(H), where v is an arbitrary tangent vector v. In local coordinates x 1 , . . . , x
Data Loading...
