We show the existence of connecting orbits for a class of singular second-order Hamiltonian systems

where, as opposed to most of the existing literature, we assume that the potential V has not one but any finite number of maxima of equal value. We use variational methods under the assumption that V (t, u) satisfies the so-called ‘strong-force’ condition at the singularity.

Jeffrey and Kirwan gave expressions for intersection pairings on the reduced space ${{M}_{0\,}}=\,{{\mu }^{-1}}(0)/G$ of a Hamiltonian $G$-space $M$ in terms of multiple residues. In this paper we prove a residue formula for symplectic volumes of reduced spaces of a quasi-Hamiltonian $\text{SU(2)}$-space. The definition of quasi-Hamiltonian $G$-spaces was introduced by Alekseev, Malkin and Meinrenken.

Soit ${{H}_{0}}\,=\,\frac{{{x}^{2}}+{{y}^{2}}}{2}$ un hamiltonien isochrone du plan ${{\mathbb{R}}^{2}}$. On met en évidence une classe d’hamiltoniens isochrones qui sont des perturbations polynomiales de ${{H}_{0}}$. On obtient alors une condition nécessaire d’isochronisme, et un critère de choix pour les hamiltoniens isochrones. On voit ce résultat comme étant une généralisation du caractère isochrone des perturbations hamiltoniennes homogènes considérées dans $\left[ \text{L} \right],\,\left[ \text{P} \right],\,\left[ \text{S} \right]$.

In this paper the germ of neighborhood of a compact leaf in a Lagrangian foliation is symplectically classified when the compact leaf is ${{\mathbb{T}}^{2}}$, the affine structure induced by the Lagrangian foliation on the leaf is complete, and the holonomy of ${{\mathbb{T}}^{2}}$ in the foliation linearizes. The germ of neighborhood is classified by a function, depending on one transverse coordinate, this function is related to the affine structure of the nearly compact leaves.

There is a well-known correspondence (due to Mehta and Seshadri in the unitary case, and extended by Bhosle and Ramanathan to other groups), between the symplectic variety ${{M}_{h}}$ of representations of the fundamental group of a punctured Riemann surface into a compact connected Lie group $G$, with fixed conjugacy classes $h$ at the punctures, and a complex variety ${{\mathcal{M}}_{h}}$ of holomorphic bundles on the unpunctured surface with a parabolic structure at the puncture points. For $G\,=\,\text{SU}\left( 2 \right)$, we build a symplectic variety $P$ of pairs (representations of the fundamental group into $G$, “weighted frame” at the puncture points), and a corresponding complex variety $\mathcal{P}$ of moduli of “framed parabolic bundles”, which encompass respectively all of the spaces ${{M}_{h}}$, ${{\mathcal{M}}_{h}}$, in the sense that one can obtain ${{M}_{h}}$ from $P$ by symplectic reduction, and ${{\mathcal{M}}_{h}}$ from $\mathcal{P}$ by a complex quotient. This allows us to explain certain features of the toric geometry of the $\text{SU(2)}$ moduli spaces discussed by Jeffrey and Weitsman, by giving the actual toric variety associated with their integrable system.

This paper treats the moduli space ${{\mathcal{M}}_{g,1}}(\Lambda)$ of representations of the fundamental group of a Riemann surface of genus $g$ with one boundary component which send the loop around the boundary to an element conjugate to exp $\Lambda$, where $\Lambda$ is in the fundamental alcove of a Lie algebra. We construct natural line bundles over ${{\mathcal{M}}_{g,1}}(\Lambda)$ and exhibit natural homology cycles representing the Poincaré dual of the first Chern class. We use these cycles to prove differential equations satisfied by the symplectic volumes of these spaces. Finally we give a bound on the degree of a nonvanishing element of a particular subring of the cohomology of the moduli space of stable bundles of coprime rank $k$ and degree $d$.