We use the properties of $Mp\left(n\right)$ to construct functions ${\mu }_{\ell }:Mp\left(n\right)\to {\mathbf{Z}}_8$ associated with the elements $\ell$ of the lagrangian grassmannian $\Lambda$ (n) which generalize the Maslov index on Mp(n) defined by J. Leray in his “Lagrangian Analysis”. We deduce from these constructions the identity between $Mp\left(n\right)$ and a subset of $Sp\left(n\right)\times {\mathbf{Z}}_8$, equipped with appropriate algebraic and topological structures.

We study the poset of Hamiltonian tori for polygon spaces. We determine some maximal elements and give examples where maximal Hamiltonian tori are not all of the same dimension.

A mixed formulation is given for elastic problems. Existence and uniqueness of the discretized problem are given for conformal continuous interpolations for the stress tensor components and for the components of the displacement vector. A counterpart of the problem is discussed in the case of an even-dimensional Euclidean space with an associated Hamiltonian vector field and the Poisson structure. For conformal interpolations of the same order the question remains open.

A multiplicative structure in the cohomological version of Conley index is described following a joint paper by the author with K. Gęba and W. Uss. In the case of equivariant flows we apply a normalization procedure known from equivariant degree theory and we propose a new continuation invariant. The theory is applied then to obtain a mountain pass type theorem. Another illustrative application is a result on multiple bifurcations for some elliptic PDE.

In this article we give an obstruction to integrability by quadratures of an ordinary differential equation on the differential Galois group of variational equations of any order along a particular solution. In Hamiltonian situation the condition on the Galois group gives Morales-Ramis-Simó theorem. The main tools used are Malgrange pseudogroup of a vector field and Artin approximation theorem.

Given a one-parameter family $\{g_{\lambda }:\lambda \in [a,b]\}$ of semi Riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials $\{V_{\lambda }:\lambda \in [a,b]\}$ and a family $\{{\sigma }_{\lambda }:\lambda \in [a,b]\}$ of trajectories connecting two points of the mechanical system defined by $(g_{\lambda },V_{\lambda })$, we show that there are trajectories bifurcating from the trivial branch ${\sigma }_{\lambda }$ if the generalized Morse indices $\mu \left({\sigma }_a\right)$ and $\mu \left({\sigma }_b\right)$ are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate...

Si dimostra l'esistenza di infinite soluzioni «multi-bump» - e conseguentemente il comportamento caotico - per una classe di sistemi Hamiltoniani del secondo ordine della forma $-\ddot{q}+q=\left(g_1\left(\omega t\right)+g_2\left(t/\omega \right)\right)V^{\prime }\left(q\right)$ per $\omega$ sufficientemente piccolo. Qui $q\in {\mathbb{R}}^n$ , $g_1$ e $g_2$ sono funzioni strettamente positive e periodiche e $V$ è un potenziale superquadratico (ad esempio $V\left(q\right)={\left|q\right|}^4$ ).