Franco Cardin

Commuting Hamiltonians and Hamilton-Jacobi multi-time equations

We prove that if a sequence of pairs of smooth commuting Hamiltonians converge in the

On constrained mechanical systems: d'Alembert's and Gauss' principles

C^0

The global finite structure of generic envelope loci for Hamilton–Jacobi equations

topology to a pair of smooth Hamiltonians, these commute. This allows us define the notion of commuting continuous Hamiltonians. As an application we extend some results of Barles and Tourin on multi-time Hamilton-Jacobi equations to a more general setting.

Dynamics of a Chain of Springs with Nonconvex Potential Energy

A geometric formulation of the classical principles of D’Alembert and Gauss in analytical mechanics is given, and their equivalence for possibly non‐Riemannian mechanical systems is shown, in the case of ideal holonomic constraints. This is done by means of a Gauss’ function, which is defined in a natural way on the bundle of two‐jets on the configuration space, and which gives the ‘‘intensity’’ of the ‘‘reaction forces’’ of the constraints. It is originated by a metric structure on the bundle of semibasic forms on the phase space determined by the Finslerian kinetic energy functions of the mechanical system.

NEW ESTIMATES FOR EVANS' VARIATIONAL APPROACH TO WEAK KAM THEORY

We discuss in some detail the existence of global generating functions describing Lagrangian submanifolds connected with evolution problems for Hamilton–Jacobi (H–J) equations. First, we produce a physical application of a result by Viterbo: for generic (in a suitable sense) Hamiltonian functions and initial data, the envelopes, i.e., the wave front sets, related to Hamilton–Jacobi problems are globally finitely generated. Furthermore, we show how to compute global space–time generating functions with finite parameters for geometric solutions of a H–J equation of the evolution kind.

Exponential estimates for oscillatory integrals with degenerate phase functions

We study the dynamics of a discretized model of an elastic bar in a hard device formed by a chain of point masses connected by nonlinear springs whose total length is a controlled parameter. We compare the description of the system dynamics given by the first-order (gradient) dynamics, the second-order (Newtonian) dumped dynamics and the Relaxation Oscillation Theory. Using a technique based on Liapunovs second method, we prove a dynamic stability result concerning the above-mentioned ODEs.

Elementary symplectic topology and mechanics

We consider a recent approximate variational principle for weak KAM theory proposed by Evans. As in the case of classical integrability, for one-dimensional mechanical Hamiltonian systems all the computations can be carried out explicitly. In this setting, we illustrate the geometric content of the theory and prove new lower bounds for the estimates related to its dynamic interpretation. These estimates also extend to the case of n degrees of freedom.Abstract We consider a recent approximate variational principle for weak KAM theory proposed by Evans. As in the case of classical integrability, for one dimensional mechanical Hamiltonian systems all the computations can be carried out explicitly. In this setting, we illustrate the geometric content of the theory and prove new lower bounds for the estimates related to its dynamic interpretation. These estimates also extend to the case of n degrees of freedom.

Finite reduction and Morse index estimates for mechanical systems

In this paper we give precise asymptotic expansions and estimates of the remainder R(λ) for oscillatory integrals with non Morse phase functions, having degeneracies of any order k ≥ 2. We provide an algorithm for writing down explicitly the coefficients of the asymptotic expansion analysing precisely the combinatorial behaviour of the coefficients (Gevrey type) and deriving optimal exponential decay estimates for the remainder when λ → ∞. We recapture the fundamental asymptotic expansions by Erdelyi (1956 Asymptotic Expansions (New York: Dover)). As it concerns the remainder estimates, it seems they are novel even for the classical cases. The main application of this machinery is a derivation of uniform estimates with respect to control parameters of celebrated oscillatory integrals in optics appearing in the calculations of the intensity of the light along the caustics (umbilics), see e.g. Arnold (1988 Singularities of Differentiable Maps vol II (Boston: Birkhauser Boston Inc.)), (1974 USP. Mat. Nauk. 29 11–49) and Berry and Upstill (1980 Prog. Opt. 18 257–346). Finally, we mention that as an outcome of our abstract approach we obtain refinements for Morse phase functions provided suitable symmetry and Gevrey type regularity conditions on the phase functions and amplitudes hold. As far as we know, even this asymptotic expansion for the elliptic umbilic is a novelty.

Tonelli Principle: Finite Reduction and Fixed Energy Molecular Dynamics Trajectories

Beginning.- Notes on Differential Geometry.- Symplectic Manifolds.- Poisson brackets environment.- Cauchy Problem for H-J equations.- Calculus of Variations and Conjugate Points.- Asymptotic Theory of Oscillating Integrals.- Lusternik-Schnirelman and Morse.- Finite Exact Reductions.- Other instances.- Bibliography.

Towards a Stationary Monge--Kantorovich Dynamics: The Physarum Polycephalum Experience

A simple version of exact finite dimensional reduction for the variational setting of mechanical systems is presented. It is worked out by means of a thorough global version of the implicit function theorem for monotone operators. Moreover, the Hessian of the reduced function preserves all the relevant information of the original one, by Schur’s complement, which spontaneously appears in this context. Finally, the results are straightforwardly extended to the case of a Dirichlet problem on a bounded domain.