arXiv subjects

P. Morando

Publications and source records attributed to P. Morando.

10 recordsLinked to original sources

arXiv2009

Nonlocal interpretation of $λ$-variational symmetry-reduction method

In this paper we give a geometric interpretation of a reduction method based on the so called $λ$-variational symmetry (C. Muriel, J.L. Romero and P. Olver 2006 \emph{Variational $C^{\infty}$-symmetries and Euler-Lagrange equations} J. Differential equations \textbf{222} 164-184). In general this allows only a partial reduction but it is particularly suitable for the reduction of variational ODEs with a lack of computable local symmetries. We show that this method is better understood as a nonlocal symmetry-reduction.

math.DS↗

arXiv2004

On the relation between standard and $μ$-symmetries for PDEs

We give a geometrical interpretation of the notion of $μ$-prolongations of vector fields and of the related concept of $μ$-symmetry for partial differential equations (extending to PDEs the notion of $λ$-symmetry for ODEs). We give in particular a result concerning the relationship between $μ$-symmetries and standard exact symmetries. The notion is also extended to the case of conditional and partial symmetries, and we analyze the relation between local $μ$-symmetries and nonlocal standard symmetries.

math-ph↗

arXiv2004

On the geometry of lambda-symmetries, and PDEs reduction

We give a geometrical characterization of $λ$-prolongations of vector fields, and hence of $λ$-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a horizontal one-form $μ$, and we speak of $μ$-prolongations of vector fields and $μ$-symmetries of PDEs. We show that these are as good as standard symmetries in providing symmetry reduction of PDEs and systems, and explicit invariant solutions.

math-ph↗

arXiv2004

Variational principles for involutive systems of vector fields

In many relevant cases -- e.g., in hamiltonian dynamics -- a given vector field can be characterized by means of a variational principle based on a one-form. We discuss how a vector field on a manifold can also be characterized in a similar way by means of an higher order variational principle, and how this extends to involutive systems of vector fields.

math-ph↗

arXiv2003

Maximal degree variational principles

Let $M$ be smooth $n$-dimensional manifold, fibered over a $k$-dimensional submanifold $B$ as $π:M \to B$, and $\vartheta \in Λ^k (M)$; one can consider the functional on sections $ϕ$ of the bundle $π$ defined by $\int_D ϕ^* (\vartheta)$, with $D$ a domain in $B$. We show that for $k = n-2$ the variational principle based on this functional identifies a unique (up to multiplication by a smooth function) nontrivial vector field in $M$, i.e. a system of ODEs. Conversely, any vector field $X$ on $M$ satisfying $i_X ({\rm d} \vartheta) = 0$ for some $\vartheta \in Λ^{n-2} (M)$ admits such a variational characterization. We consider the general case, and also the particular case $M = P \times R$ where one of the variables (the time) has a distinguished role; in this case our results imply that any Liouville (volume-preserving) vector field on the phase space $P$ admits a variational principle of the kind considered here.

math-ph↗

arXiv2003

A variational principle for volume-preserving dynamics

We provide a variational description of any Liouville (i.e. volume preserving) autonomous vector fields on a smooth manifold. This is obtained via a ``maximal degree'' variational principle; critical sections for this are integral manifolds for the Liouville vector field. We work in coordinates and provide explicit formulae.

math-ph↗

arXiv2002

Quaternionic integrable systems

Standard (Arnold-Liouville) integrable systems are intimately related to complex rotations. One can define a generalization of these, sharing many of their properties, where complex rotations are replaced by quaternionic ones. Actually this extension is not limited to the integrable case: one can define a generalization of Hamilton dynamics based on hyperKahler structures.

math-ph↗

arXiv2002

Quaternionic Hamilton equations

The classical Hamilton equations are reinterpreted by means of complex analysis, in a non standard way. This suggests a natural extension of the Hamilton equations to the quaternionic case, extension which coincides with the one introduced in [math-ph/0204019] by a completely different approach.

math-ph↗

arXiv2002

Michel theory of symmetry breaking and gauge theories

We extend Michel's theorem on the geometry of symmetry breaking [L. Michel, {\it Comptes Rendus Acad. Sci. Paris} {\bf 272-A} (1971), 433-436] to the case of pure gauge theories, i.e. of gauge-invariant functionals defined on the space ${\cal C}$ of connections of a principal fiber bundle. Our proof follows closely the original one by Michel, using several known results on the geometry of ${\cal C}$. The result (and proof) is also extended to the case of gauge theories with matter fields.

math-ph↗

arXiv2002

Hyperhamiltonian dynamics

We introduce an extension of hamiltonian dynamics, defined on hyperkahler manifolds, which we call ``hyperhamiltonian dynamics''. We show that this has many of the attractive features of standard hamiltonian dynamics. We also discuss the prototypical integrable hyperhamiltonian systems, i.e. quaternionic oscillators.

math-ph↗