Some Topological Aspects of Integral Rigid Body Dynamics
The main aim of this dissertation is to describe topology of isoenergy manifolds of the classical Kirchhoff case of Kirchhoff equations of rigid body motion in an ideal, incompressible fluid. The first chapter introduces the basic notions of rigid body dynamics and integrable Hamiltonian systems. To that end it also introduces the concept of symplectic and Poisson manifolds and the analytical mechanics theorems and definitions that are needed to formulate and understand the models of rigid body systems characterized by three different systems of six non-linear differential equations: the Euler-Poisson equations, the Kirchhoff equations, and the Poincar´e-Zhukovsky equations. The role of underlying Lie-Poisson algebras is stressed. In Chapter 2 we studied the Goryachev-Chaplygin top. This system is completely integrable if reduced to a level set of one first integral only. The bifurcation diagram of this completely integrable system is the region of possible motion on the plane of first integrals together with the image of the critical set. Chapter 3 gives a complete description of the topology of the iso-energy manifolds of the Kirchhoff system of the Kirchhoff equations of rigid body motion in an ideal, incompressible fluid. This is a Hamiltonian system on Lie-Poisson algebra e(3) with a Hamlitonian which is quadratic in mixed terms as well. For such general quadratic Hamiltonians on e(3) we first construct so-called reduced potential. In the special case of the Kirchhoff system we use the reduced potential to construct its Reeb graphs. Based on a theorem of Smale, we use the combinatorics of the constructed Reeb graphs to compute the topology of the isoenergy manifolds. The challenge of the presence of a large number of parameters has been compensated by a relatively simple form of the reduced potential in this case. In Chapter 4, we investigate the bifurcations of the momentum mapping for the Poincar´e model of rigid body filled with ideal incompressible vortex fluid. The equations of motion are the Poincar´e-Zhukovky equations. They can be seen as a Hamiltonian system on the Lie-Poisson algebra so(4) with a quadratic Hamiltonian. For this purpose, we find the critical points of rank zero and rank one . Finally, the bifurcations are studied for the Kirchhoff case on the Lie algebra e(3). We find critical points of rank zero and rank one.