Raymond G. McLenaghan

Separation of variables and symmetry operators for the neutrino and Dirac equations in the space‐times admitting a two‐parameter abelian orthogonally transitive isometry group and a pair of shearfree geodesic null congruences

We show that there exist a coordinate system and null tetrad for the space‐times admitting a two‐parameter abelian orthogonally transitive isometry group and a pair of shearfree geodesic null congruences in which the neutrino equation is solvable by separation of variables if and only if the Weyl tensor is Petrov type D. The massive Dirac equation is separable if in addition the conformal factor satisfies a certain functional equation. As a corollary, we deduce that the neutrino equation is separable in a canonical system of coordinates and tetrad for the solution of Einstein’s type D vacuum or electrovac field equations with cosmological constant admitting a nonsingular aligned Maxwell field and that the Dirac equation is separable only in the subclass of Carter’s [A] solutions and the Debever–McLenaghan null orbit solution A0. We also compute the symmetry operators which arise from the above separability properties.

Invariant classification of orthogonally separable hamiltonian systems in Euclidean space

The problem of the invariant classification of the orthogonal coordinate webs defined in Euclidean space is solved within the framework of Felix Klein’s Erlangen Program. The results are applied to the problem of integrability of the Calogero-Moser model.

An extension of the classical theory of algebraic invariants to pseudo-Riemannian geometry and Hamiltonian mechanics

We develop a new approach to the study of Killing tensors defined in pseudo-Riemannian spaces of constant curvature that is ideologically close to the classical theory of invariants. The main idea, which provides the foundation of the new approach, is to treat a Killing tensor as an algebraic object determined by a set of parameters of the corresponding vector space of Killing tensors under the action of the isometry group. The spaces of group invariants and conformal group invariants of valence two Killing tensors defined in the Minkowski plane are described. The group invariants, which are the generators of the space of invariants, are applied to the problem of classification of orthogonally separable Hamiltonian systems defined in the Minkowski plane. Transformation formulas to separable coordinates expressed in terms of the parameters of the corresponding space of Killing tensors are presented. The results are applied to the problem of orthogonal separability of the Drach superintegrable potentials.

Generalized symmetries in mechanics and field theories

Generalized symmetries are introduced in a geometrical and global formalism. Such a framework applies naturally to field theories and specializes to mechanics. Generalized symmetries are characterized in a Lagrangian context by means of the transformation rules of the Poincare–Cartan form and the (generalized) Nother theorem is applied to obtain conserved quantities (first integrals in mechanics). In the particular case of mechanics it is shown how to use generalized symmetries to study the separation of variables of Hamilton–Jacobi equations recovering standard results by means of this new method. Supersymmetries (Wess–Zumino model) are considered as an intriguing example in field theory.

Group invariant classification of separable Hamiltonian systems in the Euclidean plane and the O(4)-symmetric Yang–Mills theories of Yatsun

We present a new and effective method of determining separable coordinate systems for natural Hamiltonians with two degrees of freedom in flat Riemannian space. The method is based on intrinsic properties of the associated Killing tensors and their invariants under the group of rigid motions E(2). Applications to the Hamiltonian systems derived by the late V. A. Yatsun from O(4)-symmetric Yang–Mills theories are presented. In addition, an equivalence between separability of two-dimensional Hamiltonian systems and the existence of Pfaffian quasi-bi-Hamiltonian representations is specified.

Complex recurrent space-times

It is shown that the complex recurrent space-times are of Petrov type D or N. The spaces of type D are non-empty and decomposable into the product of two 2-dimensional spaces of arbitrary curvature, the recurrence vector being necessarily real. It is established that the problem of determining the type N complex recurrent spaces is equivalent to that of determining the spaces of type N admitting a recurrent null vector field. A coordinate system is given in which the metric of such spaces is determined up to three arbitrary functions and a constant. The spaces fall into four invariant classes according to the permitted algebraic form of the Ricci tensor. The Einstein-Maxwell equations are solved, the solutions obtained providing examples for three of the classes. It seems that the remaining class cannot contain any physically reasonable solutions of Einstein’s equations. The general metric of the complex recurrent space of Petrov type N with recurrence vector proportional to the principal null vector of the Weyl tensor is also given and some of its properties are discussed. The space of plane gravitational waves admitting a singular electromagnetic field is the only physically significant space-time of this class. The conformally recurrent spaces and the conformal symmetric spaces are also determined.

Geometrical classification of Killing tensors on bidimensional flat manifolds

Valence two Killing tensors in the Euclidean and Minkowski planes are classified under the action of the group which preserves the type of the corresponding Killing web. The classification is based on an analysis of the system of determining partial differential equations for the group invariants and is entirely algebraic. The approach allows one to classify both characteristic and noncharacteristic Killing tensors.

Conformally flat solutions of the Einstein−Maxwell equations for null electromagnetic fields

The spin coefficient formalism of Newman and Penrose is employed to obtain a direct derivation of the most general conformally flat solution of the source−free Einstein−Maxwell equations for null electromagnetic fields.

A geometrical approach to the problem of integrability of Hamiltonian systems by separation of variables

Abstract We propose a geometrical approach to the problem of integrability of Hamiltonian systems of low dimensions using the Hamilton–Jacobi method of separation of variables, based on the method of moving frames. As an illustration we present a complete classification of all separable Hamiltonian systems defined in two-dimensional Riemannian manifolds of arbitrary curvature and a criterion for separability. Connections to bi-Hamiltonian theory are also found.

A new solution of the Einstein–Maxwell equations

A space–time is determined which is a solution of the Einstein–Maxwell equations for a nonsingular electromagnetic field and for which the electromagnetic field tensor is weakly parallelly propagated along its principal null directions. A coordinate system is given in which the metric depends upon one essential arbitrary constant. The space–time admits a four‐parameter simply transitive group of motions and its Weyl tensor is of Petrov type I.