arXiv subjects
Publications and source records attributed to Gianluca Gorni.
Some characteristics of the Sawada-Kotera Lagrangian system lend themselves to generalization, producing a large class of separable Lagrangian systems with two degrees of freedom.
Constants of motion are usually derived from groups of symmetry transformation of the system. Here we show that useful properties of the system can be deduced from a family of Noether-like transformations that are not inspired by any symmetry whatsoever. The system here is the Lagrangian interpretation of Poincaré's half plane, and the property is the shape of the geodesics.
We describe a recipe to generate "nonlocal" constants of motion for ODE Lagrangian systems. As a sample application, we recall a nonlocal constant of motion for dissipative mechanical systems, from which we can deduce global existence and estimates of solutions under fairly general assumptions. Then we review a generalization to Euler-Lagrange ODEs of order higher than two, leading to first integrals for the Pais-Uhlenbeck oscillator and other systems. Future developments may include adaptations of the theory to Euler-Lagrange PDEs.
A simple general theorem is used as a tool that generates nonlocal constants of motion for Lagrangian systems. We review some cases where the constants that we find are useful in the study of the systems: the homogeneous potentials of degree~$-2$, the mechanical systems with viscous fluid resistance and the conservative and dissipative Maxwell-Bloch equations of laser dynamics. We also prove a new result on explosion in the past for mechanical system with hydraulic (quadratic) fluid resistance and bounded potential.
The Maxwell-Bloch dissipative equations describe laser dynamics. Under a simple condition on the parameters there exist two time dependent first integrals, that allow a nonstandard separation of variables in the equations. That condition has a precise physical meaning. The separated differential equations lead naturally to simple conjectures on the asymptotic behavior of the physical variables.
We give a recipe to generate "nonlocal" constants of motion for ODE Lagrangian systems and we apply the method to find useful constants of motion for dissipative system, for the Lane-Emden equation, and for the Maxwell-Bloch system with RWA.
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.
If we impose infinitesimal invariance up to a boundary term of the action functional for Lagrangian ordinary differential equations, we are led to Killing-type equations, which are related to first integrals through Noether theorem. We review the "on-flow" and "strong" interpretations of the Killing-type equation, and for each we detail the complete explicit structure of the solution set in terms of the associated first integral. We give examples that reappraise the usefulness of the "on-flow" solutions. Finally, we describe an equivalent alternative approach to variational invariance.
In this paper we revisit Noether's theorem on the constants of motion for Lagrangian mechanical systems in the ODE case, with some new perspectives on both the theoretical and the applied side. We make full use of invariance up to a divergence, or, as we call it here, Bessel-Hagen (BH) invariance. By recognizing that the Bessel-Hagen (BH) function need not be a total time derivative, we can easily deduce nonlocal constants of motion. We prove that we can always trivialize either the time change or the BH-function, so that, in particular, BH-invariance turns out not to be more general than Noether's original invariance. We also propose a version of time change that simplifies some key formulas. Applications include Lane-Emden equation, dissipative systems, homogeneous potentials and superintegrable systems. Most notably, we give two derivations of the Laplace-Runge-Lenz vector for Kepler's problem that require space and time change only, without BH invariance, one with and one without use of the Lagrange equation.
We present a geometric characterization of the nonlinear smooth functions $V: R\to R$ for which the origin is a global isochronous center for the scalar equation $\ddot x=-V'(x)$. We revisit Stillinger and Dorignac isochronous potentials $V$ and show a new simple explicit family. Implicit examples are easily produced.
The main purpose of this paper is to introduce a new class of Hamiltonian scattering systems of the cone potential type that can be integrated via the asymptotic velocity. For a large subclass, the asymptotic data of the trajectories define a global canonical diffeomorphism $A$ that brings the system into the normal form $\dot P=0$, $\dot Q=P$. The integrability theory applies for example to a system of $n$ particles on the line interacting pairwise through rather general repulsive potentials. The inverse $r$-power potential for arbitrary $r>0$ is included, the reduction to normal form being carried out for the exponents $r>1$. In particular, the Calogero system is obtained for $r=2$. The treatment covers also the nonperiodic Toda lattice. The cone potentials that we allow can undergo small perturbations in any arbitrary compact set without losing the integrability and the reduction to normal form.
The problem of scattering of particles on the line with repulsive interactions, gives rise to some well-known integrable Hamiltonian systems, for example, the nonperiodic Toda lattice or Calogero's system. The aim of this note is to outline our researches which proved the integrability of a much larger class of systems, including some that had never been considered, such as the scattering with very-long-range interaction potential. The integrability of all these systems survives any small enough perturbation of the potential in an arbitrary compact set. Our framework is based on the concept of cone potentials, as defined below, which include the scattering on the line as a particular case.
In the field of the Jacobian conjecture it is well-known after Druzkowski that from a polynomial "cubic-homogeneous" mapping we can build a higher-dimensional "cubic-linear" mapping and the other way round, so that one of them is invertible if and only if the other one is. We make this point clearer through the concept of "pairing" and apply it to the related conjugability problem: one of the two maps is conjugable if and only if the other one is; moreover, we find simple formulas expressing the inverse or the conjugations of one in terms of the inverse or conjugations of the other. Two nontrivial examples of conjugable cubic-linear mappings are provided as an application.
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 $+\infty$. 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 $C^k$ ($2\le k \le+\infty$) first integrals, independent and pairwise in involution. In this way we construct a large class of completely integrable systems. We can deal with very different asymptotic behaviours of the potential and we have persistence of the integrability under any small perturbation of the potential in an arbitrary compact set.
We introduce a class of Hamiltonian scattering systems which can be reduced to the "normal form" $\dot P=0$, $\dot Q=P$, by means of a global canonical transformation $ (P,Q)=A(p,q), p,q\in R^n$, defined through asymptotic properties of the trajectories. These systems are obtained requiring certain geometrical conditions on $\dot p=-\nabla V(q)$, $\dot q=p$, where $V$ is a bounded below "cone potential", i.e., the force $-\nabla V(q)$ 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 $A$.