Non-holonomic Geometry Research Articles

Nonassociative Geometry of Nonholonomic Phase Spaces with Star R‐flux String Deformations and (non) Symmetric Metrics

AbstractWe elaborate on nonassociative differential geometry of phase spaces endowed with nonholonomic (non‐integrable) distributions and frames, nonlinear and linear connections, symmetric and nonsymmetric metrics, and correspondingly adapted quasi‐Hopf algebra structures. The approach is based on the concept of nonassociative star product introduced for describing closed strings moving in a constant R‐flux background. Generalized Moyal‐Weyl deformations are considered when, for nonassociative and noncommutative terms of star deformations, there are used nonholonomic frames (bases) instead of local partial derivatives. In such modified nonassociative and nonholonomic spacetimes and associated complex/real phase spaces, the coefficients of geometric and physical objects depend both on base spacetime coordinates and conventional (co) fiber velocity/momentum variables like in (non) commutative Finsler‐Lagrange‐Hamilton geometry. For nonassociative and (non) commutative phase spaces modelled as total spaces of (co) tangent bundles on Lorentz manifolds enabled with star products and nonholonomic frames, we consider associated nonlinear connection, N‐connection, structures determining conventional horizontal and (co) vertical (for instance, 4+4) splitting of dimensions and N‐adapted decompositions of fundamental geometric objects. There are defined and computed in abstract geometric and N‐adapted coefficient forms the torsion, curvature and Ricci tensors. We extend certain methods of nonholonomic geometry in order to construct R‐flux deformations of vacuum Einstein equations for the case of N‐adapted linear connections and symmetric and nonsymmetric metric structures.

A new geometrical look at Ostrogradsky’s procedure

Making use of the modern techniques of non-holonomic geometry and constrained variational calculus, a revisitation of Ostrogradsky’s Hamiltonian formulation of the evolution equations determined by a Lagrangian of order [Formula: see text] in the derivatives of the configuration variables is presented.

Nonholonomic Geometry of Viscoanelastic Media and Experimental Confirmation

A thermodynamical model for viscoanelastic media is analyzed using the nonholonomic geometry. A 27-dimensional manifold is introduced, and the differential equations for the geodetics are determined and analytically solved. It is shown that, in this manifold, the best specific entropy is a harmonic function. In the linear case the propagation of transverse acoustic waves is studied, and the theoretical results are compared with some experimental data from a polymeric material (polyisobutylene).

On the geometry of Lagrangian mechanics with non-holonomic constraints

As is well known, Lagrangian mechanics have been entirely geometrized in terms of symplectic geometry. On the other hand, the geometrization of non-holonomic mechanics has been less developed. However, due to the interest aroused by non-holonomic geometry, many papers have been devoted to this subject. In this article we generalize the construction of a connection whose geodesics are the trajectories of a system, obtained by Vershik and Feddeef in the case where the Lagrangian is quadratic and the constraints are linear on the velocities. Using the algebraic formalism of the connections theory introduced by the first author, we carry out the construction in the general case of an arbitrary mechanical system (i.e. of a manifold with a convex Lagrangian not necessarily homogeneous) with ideal non-holonomic constraints. Moreover, we prove something stronger than the result of Vershik and Feddeev: our connection has not only the above-mentioned property for the geodesics, but it preserves the Hamiltonian by parallel transport. This connection is then a generalization of the Levi-Civita connection for the Riemannian manifolds for which the metric (i.e. the kinetic energy) is preserved by parallel transport.

On the geometry of non-holonomic Lagrangian systems

We present a geometric framework for non-holonomic Lagrangian systems in terms of distributions on the configuration manifold. If the constrained system is regular, an almost product structure on the phase space of velocities is constructed such that the constrained dynamics is obtained by projecting the free dynamics. If the constrained system is singular, we develop a constraint algorithm which is very similar to that developed by Dirac and Bergmann, and later globalized by Gotay and Nester. Special attention to the case of constrained systems given by connections is paid. In particular, we extend the results of Koiller for Čaplygin systems. An application to the so-called non-holonomic geometry is given.

Bundle of nilpotent Lie algebras over a nonholonomic manifold (nilpotentization)

On the tangent bundle of a nonholonomic manifold one defines a canonical structure of bundle of homogeneous nilpotent Lie algebras. On the local ring of a nonholonomic manifold one defines a quasijet structure. With the help of these constructions one uniformizes and simplifies a number of results of nonholonomic geometry.

Geodesics in non-holonomic geometry

Geodesics in non-holonomic geometry