Book Description

The topic of this thesis are finite dimensional Hamiltonian integrable systems and certain aspects of the symplectic geometry of their underlying phase spaces. The main result is presented in Chapter 3. The complete integrability of a class of hamiltonian systems $T^{\ast}M,\;\omega_{can},\; H)$ is proved, where $M$ is an arbitrary compact or non-compact Riemannian symmetric space. This class contains some classical examples of integrable systems e.g. the C. Neumann's system, and the spherical pendulum. The new examples we consider are motions on projective spaces, which in turn yield integrable motions of a particle on {\bf CP}$^n$ and {\bd HP}$^n$ in a quadratic potential field with the additional presence of the magnetic and the Yang-Mills fields respectively. The connection of the systems $T^{\ast}M,\;\omega_{can},\;H)$ with The Nahm's equations is investigated. It is also indicated, how these systems fit into the context of the Hitchin's integrable systems on $T^{\ast}{\cal M}_{par}$, where ${\cal M}_{par}$ is the moduli space of stable parabolic structures on $G^{\CC}$-principal bundle over a complex curve $C$. Hitchin's systems on $T^{\ast}{\cal M}_{par}$ are studied in tha Chapter 2. In a different way, these systens were alresdy constructed by E. Markman. Our approach allows us to show how the weights of ${\cal M}_{par}$, perturb the symplectid structureof $T^{\ast}{\cal M }_{par}$ by adding a magnetic term $\gamma_{\lambda (D)}$ to the canonical structure $\omega_{can}$ on $T^{\ast}{\cal M}_{par}$. The terms are parametrised by $D\times Lie(K)$ where $D$ is the divisor of marked poins and $Lie(K)$ Cartan sub-algebra of $Lie(G^{\CC})$. The Hitchin's systems can be thought of as describing a motion of a particle on ${\cal M}_{par}$ influenced by certain forces, where the term $\gamma_{\lambda (D)}$ gives rise to a force of magnetic type. The variations of the symplectic structure on $T^{\ast}{\cal M}_{par}$, stem from an aspect of the symplectic geometry of a complex coadjoint orbit ${\cal O}^{\CC}$, which is the main topic of the Chapter 1. The orbit ${\cal O}^{\CC}$ has the natural Konstant-Kirillov symplectic structure, and the canonical cotangent structure since ${\cal O}^{\CC}=T^{\ast}{\cal O}$ for a compact orbit ${\cal O}$. The two structures differ by a magnetic term coming from a suitable mechanical connection. The construction is generalised to the case where ${\cal O}^{\CC}$ is replaced by ${\cal O}^{\CC}$-fibre bundle.