Raffaele Vitolo

Symmetries in finite order variational sequences

We refer to Krupkas variational sequence, i.e. the quotient of the de Rham sequence on a finite order jet space with respect to a ‘variationally trivial’ subsequence. Among the morphisms of the variational sequence there are the Euler-Lagrange operator and the Helmholtz operator.In this note we show that the Lie derivative operator passes to the quotient in the variational sequence. Then we define the variational Lie derivative as an operator on the sheaves of the variational sequence. Explicit representations of this operator give us some abstract versions of Noethers theorems, which can be interpreted in terms of conserved currents for Lagrangians and Euler-Lagrange morphisms.

Symmetries in covariant 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.

On the Helmholtz operator for Euler morphisms

The variational sequence describes the Helmholtz conditions for local variationality in terms of the Helmholtz map, which is defined on a factor space. We study a tensor modification called the Helmholtz operator. For the first and second order cases we prove that, up to a multiplicative constant, the Helmholtz operator is the unique natural operator of the type in question.

On different geometric formulations of Lagrangian formalism

Abstract We consider two geometric formulations of Lagrangian formalism on fibred manifolds: Krupkas theory of finite order variational sequences, and Vinogradovs infinite order variational sequence associated with the C -spectral sequence. On one hand, we show that the direct limit of Krupkas variational bicomplex is a new infinite order variational bicomplex which yields a new infinite order variational sequence. On the other hand, by means of Vinogradovs C -spectral sequence, we provide a new finite order variational sequence whose direct limit turns out to be the Vinogradovs infinite order variational sequence. Finally, we provide an equivalence of the two finite order and infinite order variational sequences modulo the space of Euler-Lagrange morphisms.

A Unified Approach to Computation of Integrable Structures

We expose a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations.

Quantum Structures in Einstein General Relativity

We introduce the notion of a ‘quantum structure’ on an Einstein general relativistic classical spacetime M. It consists of a line bundle over M equipped with a connection fulfilling certain conditions. We give a necessary and sufficient condition for the existence of quantum structures and classify them. The existence and classification results are analogous to those of geometric quantisation (Kostant and Souriau), but they involve the topology of spacetime, rather than the topology of the configuration space. We provide physically relevant examples, such as the Dirac monopole, the Aharonov–Bohm effect and the Kerr–Newman spacetime. Our formulation is carried out by analogy with the geometric approach to quantum mechanics on a spacetime with absolute time, given by Jadczyk and Modugno.

On the Mathematical and Geometrical Structure of the Determining Equations for Shear Waves in Nonlinear Isotropic Incompressible Elastodynamics

Using the theory of

Projective-geometric aspects of homogeneous third-order Hamiltonian operators

1+1

Symmetries and conservation laws for the Karczewska–Rozmej–Rutkowski–Infeld equation

hyperbolic systems we put in perspective the mathematical and geometrical structure of the celebrated circularly polarized waves solutions for isotropic hyperelastic materials determined by Carroll in Acta Mechanica 3 (1967) 167--181. We show that a natural generalization of this class of solutions yields an infinite family of \emph{linear} solutions for the equations of isotropic elastodynamics. Moreover, we determine a huge class of hyperbolic partial differential equations having the same property of the shear wave system. Restricting the attention to the usual first order asymptotic approximation of the equations determining transverse waves we provide the complete integration of this system using generalized symmetries.

Hamiltonian Structures for General PDEs

Abstract We investigate homogeneous third-order Hamiltonian operators of differential-geometric type. Based on the correspondence with quadratic line complexes, a complete list of such operators with n ≤ 3 components is obtained.