Letter: Models of Relativistic Particle with Curvature and Torsion Revisited

Abstract

Models, describing relativistic particles, where Lagrangian densities depend linearly on both the curvature and the torsion of the trajectories, are revisited in D = 3 space forms. The moduli spaces of trajectories are completely and explicitly determined using the Lancret program. The moduli subspaces of closed solitons in the three sphere are also determined.

Integrating Hamiltonian systems defined on the Lie groups SO(4) and SO(1,3)

“…Although, in this paper we do not explicitly solve the extremal solutions as these will be problem specific, we provide a method relating the extremal solutions to the corresponding solution curves g(t) ∈ G for all systems of this form on SO (4) and SO (1,3). Applications motivating the study of Hamiltonian systems on Lie groups is the motions of relativistic particles [6] and applications of quantum control on SO(4) [5]. SO (4) and SO (1,3) are semi-simple Lie groups denoted G (see [14] for general definitions) and are the orthonormal frame bundles of the 3-dimensional non-Euclidean space forms, the sphere S 3 and Hyperboloid H 3 .…”

“…Affine control systems defined on finite-dimensional Lie groups form an important class of nonholonomic system and provide a mathematically rich setting for studying kinematic control systems [1], [2], [3], quantum control systems [4], [5] and relativistic systems [6]. The motion planning problem for such systems can be solved using optimal control theory and it follows that such problems are inseparable from problems in geometry, including the sub-Riemannian and elastic problems on the frame bundles of the planar forms [7], [8] and [9] and on the frame bundles of the space forms [10] and [11].…”

New Electromagnetic Fluids Inextensible Flows of Spacelike Particles and some Wave Solutions in Minkowski Space-time

“…The presence of magnetic fields may destroy differential rotation in nascent neutron stars, form jets and influence disk dynamics around black holes, affect collapse of massive rotating stars, etc. Many of these systems are promising sources of gravitational radiation for detection by laser interferometers, [1][2][3][4][5][6][7][8][9][10][11].…”

SURFACES IN 𝔼3MAKING CONSTANT ANGLE WITH KILLING VECTOR FIELDS

“…For example, see [5,20]. Moreover, in [2] it is shown that general helices in a 3-dimensional space form are extremal curves of a functional involving a linear combination of the curvature, the torsion (as functions) and a constant. On the other hand, in a contact 3-manifold, a curve is called a slant curve if the angle between its tangent and the Reeb vector field is constant (e.g.…”