It is known that, if a point in R^n is driven by a bounded below potential V, whose gradient is always in a closed convex cone which contains no lines, then the velocity has a finite limit as time goes to +infinity. The components of the asymptotic velocity, as functions of the initial data, are trivially constants of motion. We find sufficient conditions for these functions to be C^k (2

The main aim of the present paper is to raise the doubt that vakonomic dynamics may not be satisfactory as a model for velocity dependent constraints. 2000Academic Press

This is an exposition of some basic ideas in the realm of Global Inverse Function theorems. We address ourselves mainly to readers who are interested in the applications to Differential Equations. But we do not deal with those applications and we give a 'self-contained' elementary exposition.The first part is devoted to the celebrated Hadamard-Caccioppoli theorem on proper local homeomorphisms treated in the framework of the Hausdorff spaces. In the proof, the concept of 'ω-limit set' is used in a crucial way and this is perhaps the novelty of our approach.In the second part we deal with open sets in Banach spaces. The concept of 'attraction basin' here is the main tool of our exposition which also shows a few recent results, here extended from finite dimensional to general Banach spaces, together with the classical theorem of Hadamard-Levy which assumes that the operator norm of the inverse of the derivative does not grow too fast (roughly at most linearly).A fundamental problem in Analysis is the existence and/or uniqueness of the solutions to the equation y = f (x) in the unknown x. The function f : X → Y relates two spaces X, Y with some structure, otherwise we are impotent. From the other side, the concrete case where X, Y are subsets of the n-space R n is often too restrictive, and actually many applications arise in more general spaces. We especially think about injectivity and surjectivity problems in Differential Equations which are not discussed in this paper but constitute one of the reasons of our discussion.The books Prodi and Ambrosetti [31], and Chow and Hale [9], give the proof of global inversion theorems in general spaces and show applications to differential equations. Let us also refer to Invernizzi and Zanolin [21], Brown and Lin [6], and Radulescu and Radulescu [33] among the papers which could be mentioned for results in differential equations obtained by means of the inversion of functions in infinite dimensional Banach spaces. Finite dimensional problems are also important. The research field of the Jacobian conjectures deals with deep questions of invertibility linked to global stability problems, see Olech [27], Meisters [23], Meisters and Olech [25], [26], and the references contained therein. The inversion of functions, of course, also plays a role in the applied sciences, e.g. Economics and Network Theory.More references are listed at the end of the paper with no claim to completeness. The present paper is not a survey on the rich literature on these topics.Section 1 below is devoted to the following theorem which we call after Hadamard and Caccioppoli since Hadamard was probably the first to have the idea in finite dimension, and Caccioppoli was perhaps the most important author in the process of clarification and generalization to abstract spaces (but other mathematicians also gave a contribution). Theorem 0.1 (Hadamard-Caccioppoli) . Let f : X → Y be a local homeomorphism with X, Y path connected Hausdorff spaces and Y simply connected. Then f is a homeomorphism onto Y if and only...

We introduce a class of Hamiltonian scattering systems which can be reduced to the "normal form"Ṗ = 0,Q = P , by means of a global canonical transformation (P, Q) = A(p, q), p, q ∈ I R n , defined through asymptotic properties of the trajectories.These systems are obtained requiring certain geometrical conditions oṅ p = −∇V(q),q = p, where V is a bounded below "cone potential", i.e., the force −∇V(q) always belongs to a closed convex cone which contains no straight lines.We can deal with very different asymptotic behaviours of the potential and the potential can undergo small perturbations in any arbitrary compact set without losing the existence and the properties of A.

The aim of this paper is to introduce a class of Hamiltonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details. They have an equilibrium point which is stable for some rare elements of the class, and unstable in most cases. Anyhow, it is linearly stable (all orbits of the linearized system are bounded) and no motion is asymptotic in the past, namely no non-constant solution has the equilibrium as limit point as time goes to minus infinity. In the unstable cases, there is a sequence of initial data which converges to the equilibrium point whose corresponding solutions are unbounded and the motion is slow. So instability is quite weak and perhaps no such explicit examples of instability are known in the literature. The stable cases are also interesting since the level sets of the 2 first integrals independent and in involution keep being non-compact and stability is related to the isochronous periodicity of all orbits near the equilibrium point and the existence of a further first integral. Hopefully, these superintegrable Hamiltonian systems will deserve further research.