Jacobi fields in nonholonomic mechanics

2021 ◽  
Author(s):  
Alexandre Anahory Simoes ◽  
Juan Carlos Marrero ◽  
David Martin de Diego
Keyword(s):  
Riemannian Geometry ◽  
Nonholonomic System ◽  
Nonholonomic Mechanics ◽  
Jacobi Fields ◽  
Similar Characterization ◽  
Complete Lift ◽  
Curvature And Torsion ◽  
Nonholonomic Connection

Abstract In this paper, we define Jacobi fields for nonholonomic mechanics using a similar characterization than in Riemannian geometry. We give explicit conditions to find Jacobi fields and finally we find the nonholonomic Jacobi fields in two equivalent ways: the first one, using an appropriate complete lift of the nonholonomic system and, in the second one, using the curvature and torsion of the associated nonholonomic connection.

scholarly journals NONHOLONOMIC CONSTRAINTS IN k-SYMPLECTIC CLASSICAL FIELD THEORIES

2008 ◽  
Vol 05 (05) ◽  
pp. 799-830 ◽  
Author(s):  
M. DE LEÓN ◽  
D. MARTÍN DE DIEGO ◽  
M. SALGADO ◽  
S. VILARIÑO
Keyword(s):  
Projection Operator ◽  
Nonholonomic System ◽  
Nonholonomic Constraints ◽  
Momentum Equation ◽  
Classical Field ◽  
Nonholonomic Mechanics ◽  
Field Theories ◽  
Constrained Problem ◽  
Classical Field Theories

A k-symplectic framework for classical field theories subject to nonholonomic constraints is presented. If the constrained problem is regular one can construct a projection operator such that the solutions of the constrained problem are obtained by projecting the solutions of the free problem. Symmetries for the nonholonomic system are introduced and we show that for every such symmetry, there exist a nonholonomic momentum equation. The proposed formalism permits to introduce in a simple way many tools of nonholonomic mechanics to nonholonomic field theories.

scholarly journals A COMPLETE LIFT FOR SEMISPRAYS

2010 ◽  
Vol 07 (02) ◽  
pp. 267-287 ◽  
Author(s):  
IOAN BUCATARU ◽  
MATIAS F. DAHL
Keyword(s):  
Projective Geometry ◽  
Riemannian Metrics ◽  
Jacobi Fields ◽  
Affine Connections ◽  
Complete Lift

In this paper, we define a complete lift for semisprays. If S is a semispray on a manifold M, its complete lift is a new semispray Scon TM. The motivation for this lift is two-fold: First, geodesics for Sccorrespond to the Jacobi fields for S, and second, this complete lift generalizes and unifies previously known complete lifts for Riemannian metrics, affine connections, and regular Lagrangians. When S is a spray, we prove that the projective geometry of Scuniquely determines S. We also study how symmetries and constants of motions for S lift into symmetries and constants of motions for Sc.

A LINEAR CONNECTION FOR BOTH SUB-RIEMANNIAN GEOMETRY AND NONHOLONOMIC MECHANICS (II)

2011 ◽  
Vol 08 (05) ◽  
pp. 969-983
Author(s):  
AUREL BEJANCU
Keyword(s):  
Mechanical System ◽  
Riemannian Geometry ◽  
Classical Mechanics ◽  
Linear Connection ◽  
Nonholonomic Mechanics ◽  
Riemannian Connection ◽  
The One

We show that the sub-Riemannian connection ∇, constructed in the first part of our study [2], is the one which enables us to express the Lagrange–d'Alembert equations for a nonholonomic mechanical system in a form that is similar to the Newton's equations from classical mechanics (cf. (2.21)). Also, we define the generalized Chaplygin systems and show that the Lagrange–d'Alembert equations for these systems depend only on some horizontal variables. Two examples are given to illustrate our theory in comparison with other studies.

Geodesics and Jacobi fields in singular semi-riemannian geometry

1994 ◽  
Vol 446 (1928) ◽  
pp. 441-452 ◽  
Keyword(s):  
Singular Point ◽  
Uniqueness Theorem ◽  
Riemannian Geometry ◽  
Existence And Uniqueness ◽  
Smooth Manifold ◽  
Tensor Field ◽  
Final Section ◽  
Existence And Uniqueness Theorem ◽  
Jacobi Fields ◽  
Neutral Vector

This paper proves an existence and uniqueness theorem for geodesics tangent to a neutral vector at a stable singular point of a smooth symmetric two tensor field g on a smooth manifold M . The final section is devoted to a proof of existence and uniqueness of Jacobi fields along the above mentioned geodesics.

scholarly journals On some sub-Riemannian objects in hypersurfaces of sub-Riemannian manifolds

2004 ◽  
Vol 70 (2) ◽  
pp. 177-198 ◽  
Author(s):  
Kang-Hai Tan ◽  
Xiao-Ping Yang
Keyword(s):  
Riemannian Manifolds ◽  
Riemannian Geometry ◽  
Mean Curvature ◽  
Contact Manifold ◽  
Tangent Plane ◽  
Riemannian Metric ◽  
Horizontal Curve ◽  
Horizontal Connection ◽  
Characteristic Points ◽  
Nonholonomic Connection

We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isolated characteristic points, then there exists at least a piecewise smooth horizontal curve in this hypersurface connecting any two given points in it. In any sub-Riemannian manifold, we obtain the sub-Riemannian version of the fundamental theorem of Riemannian geometry: there exists a unique nonholonomic connection which is completely determined by the sub-Riemannian structure and is “symmetric” and compatible with the sub-Riemannian metric. We use this nonholonomic connection to study horizontal mean curvature of hypersurfaces.

scholarly journals Generalized Chebyshev problem in nonholonomic mechanics and control theory

2021 ◽  
Vol 1959 (1) ◽  
pp. 012054
Author(s):  
M P Yushkov ◽  
Sh Kh Soltakhanov ◽  
V V Dodonov
Keyword(s):  
Control Theory ◽  
Nonholonomic Mechanics ◽  
Chebyshev Problem ◽  
And Control
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