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R. Vitolo

Publications and source records attributed to R. Vitolo.

16 recordsLinked to original sources
arXiv2026

On the geometry of WDVV equations and their Hamiltonian formalism in arbitrary dimension

It is known that in low dimensions WDVV equations can be rewritten as commuting quasilinear bi-Hamiltonian systems. We extend some of these results to arbitrary dimension $N$ and arbitrary scalar product $η$. In particular, we show that suitable subsets of WDVV equations can be interpreted as a set of linear line congruences in suitable Plücker embeddings. This form leads to their representation as $N-2$ Hamiltonian systems of conservation laws. Moreover, we show that WDVV equations can be reduced to an orthonomic form, which is also passive in low dimensions $N\leq 5$. This also leads to the commutativity of the Hamiltonian systems of conservation laws ($N\leq 5$), after which we can find a solution of the WDVV equations from a joint solution of the Hamiltonian systems. Finally, we conjecture that passivity holds in all dimensions.

nlin.SI↗
arXiv2024

Bi-Hamiltonian structures of WDVV-type

We study a class of nonlinear PDEs that admit the same bi-Hamiltonian structure as WDVV equations: a Ferapontov-type first-order Hamiltonian operator and a homogeneous third-order Hamiltonian operator in a canonical Doyle--Potemin form, which are compatible. Using various equivalence groups, we classify such equations in two-component and three-component cases. In a four-component case, we add further evidence to the conjecture that there exists only one integrable system of the above type. Finally, we give an example of the six-component system with required bi-Hamiltonian structure. To streamline the symbolic computation we develop an algorithm to find the aforementioned Hamiltonian operators, which includes putting forward a conjecture on the structure of the metric parameterising the first-order Hamiltonian operator.

nlin.SI↗
arXiv2023

Bi-Hamiltonian structures of KdV type, cyclic Frobenius algebrae and Monge metrics

We study algebraic and projective geometric properties of Hamiltonian trios determined by a constant coefficient second-order operator and two first-order localizable operators of Ferapontov type. We show that first-order operators are determined by Monge metrics, and define a structure of cyclic Frobenius algebra. Examples include the AKNS system, a $2$-component generalization of Camassa-Holm equation and the Kaup--Broer system. In dimension $2$ the trio is completely determined by two conics of rank at least $2$. We provide a partial classification in dimension $4$.

math-ph↗
arXiv2023

Miura-reciprocal transformations and localizable Poisson pencils

We show that the equivalence classes of deformations of localizable semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura-reciprocal group contain a local representative and are in one-to-one correspondence with the equivalence classes of deformations of local semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura group.

math-ph↗
arXiv2021

Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds

Poisson brackets between conserved quantities are a fundamental tool in the theory of integrable systems. The subclass of weakly nonlocal Poisson brackets occurs in many significant integrable systems. Proving that a weakly nonlocal differential operator defines a Poisson bracket can be challenging. We propose a computational approach to this problem through the identification of such operators with superfunctions on supermanifolds.

math-ph↗
arXiv2019

Computing with Hamiltonian operators

Hamiltonian operators are used in the theory of integrable partial differential equations to prove the existence of infinite sequences of commuting symmetries or integrals. In this paper it is illustrated the new Reduce package \cde for computations on Hamiltonian operators. \cde can compute the Hamiltonian properties of skew-adjointness and vanishing Schouten bracket for a differential operator, as well as the compatibility property of two Hamiltonian operators and the Lie derivative of a Hamiltonian operator with respect to a vector field. It can also make computations on (variational) multivectors, or functions on supermanifolds. This can open the way to applications in other fields of Mathematical Physics.

math-ph↗
arXiv2019

Three computational approaches to weakly nonlocal Poisson brackets

We compare three different ways of checking the Jacobi identity for weakly nonlocal Poisson brackets using the theory of distributions, of pseudodifferential operators and of Poisson vertex algebras, respectively. We show that the three approaches lead to similar computations and same results.

math-ph↗
arXiv2016

Bi-Hamiltonian structures of KdV type

Combining an old idea of Olver and Rosenau with the classification of second and third order homogeneous Hamiltonian operators we classify compatible trios of two-component homogeneous Hamiltonian operators. The trios yield pairs of compatible bi-Hamiltonian operators whose structure is a direct generalization of the bi-Hamiltonian pair of the KdV equation. The bi-Hamiltonian pairs give rise to multi-parametric families of bi-Hamiltonian systems. We recover known examples and we find new integrable systems whose central invariants are non-zero; this shows that new examples are not Miura-trivial.

math-ph↗
arXiv2005

The geometry of Newton's law and rigid systems

We start by formulating geometrically the Newton's law for a classical free particle in terms of Riemannian geometry, as pattern for subsequent developments. In fact, we use this scheme for further generalisation devoted to a constrained particle, to a discrete system of several free and constrained particles. For constrained systems we have intrinsic and extrinsic viewpoints, with respect to the environmental space. In the second case, we obtain an explicit formula for the reaction force via the second fundamental form of the constrained configuration space. For multi--particle systems we describe geometrically the splitting related to the center of mass and relative velocities; in this way we emphasise the geometric source of classical formulas. Then, the above scheme is applied in detail to discrete rigid systems. We start by analysing the geometry of the rigid configuration space. In this way we recover the classical formula for the velocity of the rigid system via the parallelisation of Lie groups. Moreover, we study in detail the splitting of the tangent and cotangent environmental space into the three components of center of mass, of relative velocities and of the orthogonal subspace. This splitting yields the classical components of linear and angular momentum (which here arise from a purely geometric construction) and, moreover, a third non standard component. The third projection yields an explicit formula for the reaction force in the nodes of the rigid constraint.

math-ph↗
arXiv2005

A covariant approach to the quantisation of a rigid body

This paper concerns the quantisation of a rigid body in the framework of ``covariant quantum mechanics'' on a curved spacetime with absolute time. The basic idea is to consider the multi-configuration space, i.e. the configuration space for $n$ particles, as the $n$-fold product of the configuration space for one particle. Then we impose a rigid constraint on the multi-configuration space. The resulting space is then dealt with as a configuration space of a single abstract `particle'. The same idea is applied to all geometric and dynamical structures. We show that the above configuration space fits into the general framework of ``covariant quantum mechanics''. Hence, the methods of this theory can be applied to the rigid body. Accordingly, we find exactly two inequivalent choices of quantum structures for the rigid body. Then, we evaluate the quantum energy and momentum operators and the `rotational part' of their spectra. We provide a new mathematical interpretation of two-valued wavefunctions on SO(3) in terms of single-valued sections of a new non-trivial quantum bundle. These results have clear analogies with spin.

math-ph↗
arXiv2004

The Hessian and Jacobi Morphisms for Higher Order Calculus of Variations

We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second variation is equal (up to horizontal differentials) to the vertical differential of the Euler--Lagrange morphism which turns out to be self-adjoint along solutions of the Euler-Lagrange equations. These two objects, respectively, generalize in an invariant way the Hessian morphism and the Jacobi morphism (which is then self-adjoint along critical sections) of a given Lagrangian to the case of higher order Lagrangians. Some examples of classical Lagrangians are provided to illustrate our method.

math-ph↗
arXiv2001

Finite order formulation of Vinogradov's C-spectral sequence

The C-spectral sequence was introduced by Vinogradov in the late Seventies as a fundamental tool for the study of algebro-geometric properties of jet spaces and differential equations. A spectral sequence arise from the contact filtration of the modules of forms on jet spaces of a fibring (or on a differential equation). In order to avoid serious technical difficulties, the order of the jet space is not fixed, i. e., computations are performed on spaces containing forms on jet spaces of any order. In this paper we show that there exists a formulation of Vinogradov's C-spectral sequence in the case of finite order jet spaces of a fibred manifold. We compute all cohomology groups of the finite order C-spectral sequence. We obtain a finite order variational sequence which is shown to be naturally isomorphic with Krupka's finite order variational sequence.

math.DG↗
arXiv2001

On a class of polynomial Lagrangians

In the framework of finite order variational sequences a new class of Lagrangians arises, namely, \emph{special} Lagrangians. These Lagrangians are the horizontalization of forms on a jet space of lower order. We describe their properties together with properties of related objects, such as Poincaré--Cartan and Euler--Lagrange forms, momenta and momenta of generating forms, a new geometric object arising in variational sequences. Finally, we provide a simple but important example of special Lagrangian, namely the Hilbert--Einstein Lagrangian.

math-ph↗
arXiv2000

On symmetries in Galilei classical mechanics

In the framework of Galilei classical mechanics (i.e., general relativistic classical mechanics on a spacetime with absolute time) developed by Jadczyk and Modugno, we analyse systematically the relations between symmetries of the geometric objects. We show that the (holonomic) infinitesimal symmetries of the cosymplectic structure on spacetime and of its potentials are also symmetries of spacelike metric, gravitational and electromagnetic fields, Euler-Lagrange morphism, Lagrangians. Then, we provide a covariant momentum map associated with a group of cosymplectic symmetries by using a covariant lift of functions of phase space. In the case of an action that projects on spacetime we see that the components of this momentum map are quantisable functions in the sense of Jadczyck and Modugno. Finally, we illustrate the results in some examples.

math-ph↗
arXiv2000

On finite order variational sequences

We discuss intrinsic aspects of Krupka's approach to finite-order variational sequences. We give intrinsic isomorphisms of the quotient subsheaves of the short finite-order variational sequence with sheaves of forms on jet spaces of suitable order, obtaining a new finite-order (short exact) variational sequence which is made by sheaves of polynomial differential operators. Moreover, we present an intrinsic formulation for the Helmholtz condition of local variationality using a technique introduced by Kolar that we have adapted to our context. Finally, we provide the minimal order solution to the inverse problem of the calculus of variations and a solution of the problem of the variationally trivial Lagrangian.

math-ph↗