arXiv subjects

Giampaolo Cicogna

Publications and source records attributed to Giampaolo Cicogna.

13 recordsLinked to original sources

arXiv2013

On the connections between symmetries and conservation rules of dynamical systems

The strict connection between Lie point-symmetries of a dynamical system and its constants of motion is discussed and emphasized, through old and new results. It is shown in particular how the knowledge of a symmetry of a dynamical system can allow to obtain conserved quantities which are invariant under the symmetry. In the case of Hamiltonian dynamical systems it is shown that, if the system admits a symmetry of "weaker" type (specifically, a λ or a Λ-symmetry), then the generating function of the symmetry is not a conserved quantity, but the deviation from the exact conservation is "controlled" in a well defined way. Several examples illustrate the various aspects.

math-ph↗

arXiv2013

Generalized notions of symmetry of ODE's and reduction procedures

This paper describes the notion of σ-symmetry, which extends the one of λ-symmetry, and its application to reduction procedures of systems of ordinary differential equations and of dynamical systems as well. We also consider orbital symmetries, which give rise to a different form of reduction of dynamical systems. Finally, we discuss how dynamical systems can be transformed into higher-order ordinary differential equations, and how these symmetry properties of the dynamical systems can be transferred into reduction properties of the corresponding ordinary differential equations. Many examples illustrate the various situations.

math-ph↗

arXiv2013

Symmetry Classification of quasi-linear PDE's. II: an exceptional case

This short note completes the symmetry analysis of a class of quasi-linear partial differential equations considered in the previous paper (Nonlinear Dynamics, Vol. 51, 309-316 (2008)): it deals with the presence of an "exceptional" Lie point symmetry, not previously examined, which is admitted only if the involved parameters are fixed by precise relationships. The peculiarity of this symmetry is enhanced by the fact that it leads to a solution relevant in the theory of plasma physics, and also related to the presence of a nontrivial example of a conditional symmetry of weak type.

math-ph↗

arXiv2013

Dynamical systems and σ-symmetries

A deformation of the standard prolongation operation, defined on sets of vector fields in involution rather than on single ones, was recently introduced and christened "σ-prolongation"; correspondingly one has "σ-symmetries" of differential equations. These can be used to reduce the equations under study, but the general reduction procedure under σ-symmetries fails for equations of order one. In this note we discuss how σ-symmetries can be used to reduce dynamical systems, i.e. sets of first order ODEs in the form dx^a/dt = f^a (x).

math-ph↗

arXiv2012

A generalization of λ-symmetry reduction for systems of ODEs: σ-symmetries

We consider a deformation of the prolongation operation, defined on sets of vector fields and involving a mutual interaction in the definition of prolonged ones. This maintains the "invariants by differentiation" property, and can hence be used to reduce ODEs satisfying suitable invariance conditions in a fully algorithmic way, similarly to what happens for standard prolongations and symmetries.

math-ph↗

arXiv2012

Orbital reducibility and a generalization of lambda symmetries

We review the notion of reducibility and we introduce and discuss the notion of orbital reducibility for autonomous ordinary differential equations of first order. The relation between (orbital) reducibility and (orbital) symmetry is investigated and employed to construct (orbitally) reducible systems. By standard identifications, the notions extend to non-autonomous ODEs of first and higher order. Moreover we thus obtain a generalization of the lambda symmetries of Muriel and Romero. Several examples are given.

math-ph↗

arXiv2011

Lambda-symmetries of Dynamical Systems, Hamiltonian and Lagrangian equations

After a brief survey of the definition and the properties of Lambda-symmetries in the general context of dynamical systems, the notion of "Lambda-constant of motion'' for Hamiltonian equations is introduced. If the Hamiltonian problem is derived from a Lambda-invariant Lagrangian, it is shown how the Lagrangian Lambda-invariance can be transferred into the Hamiltonian context and shown that the Hamiltonian equations turn out to be Lambda-symmetric. Finally, the "partial'' (Lagrangian) reduction of the Euler-Lagrange equations is compared with the reduction obtained for the corresponding Hamiltonian equations.

math-ph↗

arXiv2010

A Discussion on the Different Notions of Symmetry of Differential Equations

A discussion is presented, within a simple unifying scheme, about different types of symmetry of PDE's, with the introduction and a precise characterization of the notions of "standard" and "weak" conditional symmetries, together with their relationship with exact and partial symmetries. An extensive use of "symmetry-adapted" variables will be made, and some clarifying examples are provided.

math-ph↗

arXiv2010

Symmetries, weak symmetries and related solutions of the Grad-Shafranov equation

We discuss a new family of solutions of the Grad--Shafranov (GS) equation that describe D-shaped toroidal plasma equilibria with sharp gradients at the plasma edge. These solutions have been derived by exploiting the continuous Lie symmetry properties of the GS equation and in particular a special type of "weak" symmetries. In addition, we review the continuous Lie symmetry properties of the GS equation and present a short but exhaustive survey of the possible choices for the arbitrary flux functions that yield GS equations admitting some continuous Lie symmetry. Particular solutions related to these symmetries are also discussed.

math-ph↗

arXiv2010

Symmetries of Hamiltonian equations and Lambda-constants of motion

We consider symmetries and perturbed symmetries of canonical Hamiltonian equations of motion. Specifically we consider the case in which the Hamiltonian equations exhibit a Lambda symmetry under some Lie point vector field. After a brief survey of the relationships between standard symmetries and the existence of first integrals, we recall the definition and the properties of Lambda symmetries. We show that in the presence of a Lambda symmetry for the Hamiltonian equations, one can introduce the notion of "Lambda-constant of motion". The presence of a Lambda symmetry leads also to a nice and useful reduction of the form of the equations. We then consider the case in which the Hamiltonian problem is deduced from a Lambda-invariant Lagrangian. We illustrate how the Lagrangian Lambda-invariance is transferred into the Hamiltonian context and show that the Hamiltonian equations are Lambda-symmetric. We also compare the "partial" (Lagrangian) reduction of the Euler-Lagrange equations with the reduction which can be obtained for the Hamiltonian equations. Several examples illustrate and clarify the various situations.

math-ph↗

arXiv2007

Symmetry classification of quasi-linear PDE's containing arbitrary functions

We consider the problem of performing the preliminary "symmetry classification'' of a class of quasi-linear PDE's containing one or more arbitrary functions: we provide an easy condition involving these functions in order that nontrivial Lie point symmetries be admitted, and a "geometrical'' characterization of the relevant system of equations determining these symmetries. Two detailed examples will elucidate the idea and the procedure: the first one concerns a nonlinear Laplace-type equation, the second a generalization of an equation (the Grad-Schlüter-Shafranov equation) which is used in magnetohydrodynamics.

math-ph↗

arXiv2006

Applications of Symmetry Methods to the Theory of Plasma Physics

The theory of plasma physics offers a number of nontrivial examples of partial differential equations, which can be successfully treated with symmetry methods. We propose three different examples which may illustrate the reciprocal advantage of this "interaction" between plasma physics and symmetry techniques. The examples include, in particular, the complete symmetry analysis of system of two PDE's, with the determination of some conditional and partial symmetries, the construction of group-invariant solutions, and the symmetry classification of a nonlinear PDE.

math-ph↗

arXiv2001

Chaos in Black holes Surrounded by Electromagnetic Fields

In this paper we prove the occurence of chaos for charged particles moving around a Schwarzshild black hole, perturbed by uniform electric and magnetic fields. The appearance of chaos is studied resorting to the Poincare'-Melnikov method.

nlin.CD↗