Symmetries of Hamiltonian Equations and Λ-Constants of Motion

Cicogna, Giampaolo

doi:10.1142/s1402925109000315

Cited by 13 publications

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“…The prototype of simple twisted symmetries is given by the λ-symmetries introduced by Muriel and Romero [40,41,42,43,44,45,46]; these were then generalized to µ-symmetries [16,25] (which deal also with PDEs) and to Λ-and ρ-symmetries [10,11].…”

Section: Simple Twisted Symmetries 31 Simple Twisted Symmetries: λ-Smentioning

confidence: 99%

“…The Λ i have to obey some compatibility condition (see Sect.1.5 below), but in the ODE case there is only one Λ, hence no compatibility. The special case of µ-symmetries for ODEs has been studied in detail by Cicogna [10,11,12]; to emphasize its intermediate character one speaks of Λ-symmetries, and sometimes of ρ-symmetries when dealing with the class of Λ-symmetries which effectively leads to a reduction of ODEs (in this case the ρ stands for "reducing"). The collective name for these deformed prolongations and symmetries is that of twisted prolongations and symmetries.…”

Section: Introductionmentioning

confidence: 99%

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Simple and collective twisted symmetries

Gaeta¹

2021

JNMP

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After the introduction of λ-symmetries by Muriel and Romero, several other types of so called "twisted symmetries" have been considered in the literature (their name refers to the fact they are defined through a deformation of the familiar prolongation operation); they are as useful as standard symmetries for what concerns symmetry reduction of ODEs or determination of special (invariant) solutions for PDEs and have thus attracted attention. The geometrical relation of twisted symmetries to standard ones has already been noted: for some type of twisted symmetries (in particular, λ and µ-symmetries), this amounts to a certain kind of gauge transformation.In a previous review paper [23] we have surveyed the first part of the developments of this theory; in the present paper we review recent developments. In particular, we provide a unifying geometrical description of the different types of twisted symmetries; this is based on the classical Frobenius reduction applied to distribution generated by Lie-point (local) symmetries.

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Section: Simple Twisted Symmetries 31 Simple Twisted Symmetries: λ-Smentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Simple and collective twisted symmetries

Gaeta¹

2021

JNMP

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“…It appears that the result can be extended to k λ i -symmetries with a suitable solvability condition, i.e. generating a solvable Lie module 12. A conservation law is a relation of the type D i · P i = 0 for some vector P; a µ-conservation law reads Tr ∇ i · P i = 0, with ∇ i = D i + Λ i .…”

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confidence: 98%

“…Remark 22. The situation is different in the case of ODEs (this case was studied by Cicogna (he speaks in this case of ρ-symmetries, the ρ standing for "reducing", see below) [11,12]). In this case one can proceed pretty much as in the standard reduction procedure up to a (relevant) feature: that is, the reconstruction equation, which in the standard case amounts to a quadrature, is now a proper differential equation, and its solution may very well be very hard, or turn out to be impossible.…”

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On the geometry of twisted prolongations, and dynamical systems

Gaeta¹

2020

Discrete &Amp; Continuous Dynamical Systems - S

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I give a short review of the theory of twisted symmetries of differential equations, emphasizing geometrical aspects. Some open problems are also mentioned.Dedicated to Juergen Scheurle on the occasion of his retirement 2 G. GAETA (a = 1, ..., m); partial derivatives will be denoted by u a J , where J is a multi-index J = {j 1 , ..., j n } of order |J| = j 1 + ... + j n and(here and somewhere in the following we moved the vector index of the x for typographical convenience). We denote by u (k) the set of all partial derivatives of order k, and by u [n] the set of all partial derivatives of order k ≤ n.2.1. Geometrical description of differential equations and solutions. The x are local coordinates in a manifold B, while u are local coordinates in a manifold U ; we consider the phase manifold M = B × U , which has a natural structure of bundle (M, π 0 , B) over B with fiber U . As well known [1,13,37,57,58,70] we can associate to (M, π 0 , B) its Jet bundles J k M of any order; these have a structure of fiber bundle (J k M, π k , B) over B (with projection π k ) but also of fiber bundle over those of lower order (with projectionWe also recall that the Jet bundle is equipped with a contact structure Ω [4,33,58,68,71]; this can be encoded in the contact forms 2 ω a J := du a J − u a J,i dx i . (k) f in (J k M, π k , B), which is just the set of points (x, u [k] ) with u a = f a (x) and u a J = f a J (x). Now u = f (x) is a solution to ∆ if and only if γ(k) f ⊂ S ∆ . 4 G. GAETAstart with a complete set of DIs of order zero and one, we can generate in this way a complete set of DIs of any order, basically because if U is q-dimensional, we have k · q DIs of order k (these includes those of lower orders), as follows at once from J k M being of dimension d k = (k + 1)q + 1. The situation is different for PDEs, as the dimension of J k M grows combinatorially; see e.g. the discussion in [57].2.3. Lie-point symmetries. A (Lie-point 3 ) symmetry, or more precisely a Liepoint symmetry generator, is a vector field X on M such that its prolongation X (k) to J k M is tangent to S ∆ . For a given ∆ in the form (5), this condition is expressed in local coordinates asIn these equations, also called determining equations, the F are given and one looks for ξ, ϕ (i.e. components of the vector field X) satisfying them. As the components ψ a J of X (k) along u J are given in terms of ξ, ϕ and their derivatives, all dependencies of u a J (with |J| = 0) are fully explicit, and hence (12) decouple into a system of simpler equations, one for each monomial in the u a J ; each of these is a linear PDE for the ξ and ϕ, and they can be solved algorithmically -usually with the help of a symbolic manipulation program, as the dimension of the system can be quite large. This fails in the case of first order equations.

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On the connections between symmetries and conservation rules of dynamical systems

Cicogna

2012

Math Methods in App Sciences

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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.

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Symmetries of Hamiltonian Equations and Λ-Constants of Motion

Cited by 13 publications

References 35 publications

Simple and collective twisted symmetries

Simple and collective twisted symmetries

On the geometry of twisted prolongations, and dynamical systems

On the connections between symmetries and conservation rules of dynamical systems

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