Nonholonomic Systems Revisited Within the Framework of Analytical Mechanics

1998 ◽  
Vol 51 (7) ◽  
pp. 415-433 ◽  
Author(s):  
S. Ostrovskaya ◽  
J. Angeles
Keyword(s):  
Mechanical Systems ◽  
Sufficient Conditions ◽  
Original System ◽  
Nonholonomic Systems ◽  
Review Article ◽  
Analytical Mechanics ◽  
Constraint Forces ◽  
Lagrange Equations ◽  
Constraint Force ◽  
Nonholonomic Mechanical Systems

Nonholonomic mechanical systems are revisited. This review article focuses on Lagrangian formulations leading to a system of governing equations free of constraint forces. While eliminating the constraint forces, the number of scalar Lagrange equations is reduced to a number of independent equations lower than the original system with constraint forces. In the process of constraint-force elimination and dimension-reduction, a matrix that appears to play a relevant role in the formulation of the mathematical models of mechanical systems arises naturally. We call this matrix here the holonomy matrix. It is shown that necessary and sufficient conditions for the integrability of the constraints in Pfaffian form are readily derived using the holonomy matrix. In the same vein, a class of nonholonomic systems is identified, of current engineering relevance, that is termed quasiholonomic. Examples are included to illustrate these concepts. This review article contains 40 references.

scholarly journals Alternative Lagrangians obtained by scalar deformations

2020 ◽  
Vol 17 (04) ◽  
pp. 2050050
Author(s):  
Oana A. Constantinescu ◽  
Ebtsam H. Taha
Keyword(s):  
Force Field ◽  
Differentiable Function ◽  
Mechanical Systems ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Initial System ◽  
Lagrange Equations ◽  
Necessary And Sufficient ◽  
Equations Of Motions

We study mechanical systems that can be recast into the form of a system of genuine Euler–Lagrange equations. The equations of motions of such systems are initially equivalent to the system of Lagrange equations of some Lagrangian [Formula: see text], including a covariant force field. We find necessary and sufficient conditions for the existence of a differentiable function [Formula: see text] such that the initial system is equivalent to the system of Euler–Lagrange equations of the deformed Lagrangian [Formula: see text].

scholarly journals Generalized Lagrange-D’Alembert principle

2012 ◽  
Vol 91 (105) ◽  
pp. 49-58
Author(s):  
Djordje Djukic
Keyword(s):  
Dynamic Equilibrium ◽  
Virtual Work ◽  
Analytical Mechanics ◽  
Principle Of Virtual Work ◽  
Constraint Forces ◽  
Lagrange Equations ◽  
Virtual Displacement ◽  
Generalized Coordinates ◽  
D’Alembert Principle ◽  
Applied Forces

The major issues in the analysis of the motion of a constrained dynamic system are to determine this motion and calculate constraint forces. In the analytical mechanics, only the first of the two problems is analyzed. Here, the problem is solved simultaneously using: 1) Principle of liberation of constraints; 2) Principle of generalized virtual displacement; 3) Idea of ideal constraints; 4) Concept of generalized and ?supplementary" generalized coordinates. The Lagrange-D?Alembert principle of virtual work is generalized introducing virtual displacement as vectorial sum of the classical virtual displacement and virtual displacement in the ?supplementary" directions. From such principle of virtual work we derived Lagrange equations of the second kind and equations of dynamical equilibrium in the ?supplementary" directions. Constrained forces are calculated from the equations of dynamic equilibrium. At the same time, this principle can be used for consideration of equilibrium of system of material particles. This principle simultaneously gives the connection between applied forces at equilibrium state and the constrained forces. Finally, the principle is applied to a few particular problems.

Control of nonholonomic mechanical systems using reduction and adaptation

Robotica ◽  
1999 ◽  
Vol 17 (3) ◽  
pp. 249-260 ◽  
Author(s):  
R. Colbaugh ◽  
E. Barany ◽  
M. Trabatti
Keyword(s):  
Complete System ◽  
Mechanical Systems ◽  
Original System ◽  
System Structure ◽  
System Model ◽  
Dimensional System ◽  
Effective Solution ◽  
Reduction Procedure ◽  
Nonholonomic Mechanical Systems ◽  
Lower Dimensional

This paper considers the problem of controlling the motion of nonholonomic mechanical systems in the presence of uncertainty regarding the system model and state. It is proposed that a simple and effective solution to this problem can be obtained by first using a reduction procedure to obtain a lower dimensional system which retains the mechanical system structure of the original system, and then adaptively controlling the reduced system in such a way that the complete system is driven to the goal configuration. This approach is shown to be easy to implement and to ensure accurate motion control despite measurement and model uncertainty. The efficacy of the proposed control strategy is illustrated through computer simulations and preliminary hardware experiments with nonholonomic mechanical systems arising from both explicit kinematic constraints and symmetries of the system dynamics.

What is the General Form of the Explicit Equations of Motion for Constrained Mechanical Systems?

2002 ◽  
Vol 69 (3) ◽  
pp. 335-339 ◽  
Author(s):  
F. E. Udwadia ◽  
R. E. Kalaba
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Analytical Mechanics ◽  
Constraint Forces ◽  
Constrained Mechanical Systems ◽  
Explicit Equations Of Motion ◽  
D'alembert's Principle

This paper presents the general form of the explicit equations of motion for mechanical systems. The systems may have holonomic and/or nonholonomic constraints, and the constraint forces may or may not satisfy D’Alembert’s principle at each instant of time. The explicit equations lead to new fundamental principles of analytical mechanics.

scholarly journals Variational Problems with Partial Fractional Derivative: Optimal Conditions and Noether’s Theorem

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Jun Jiang ◽  
Yuqiang Feng ◽  
Shougui Li
Keyword(s):  
Fractional Derivative ◽  
Sufficient Conditions ◽  
Variational Problems ◽  
Necessary And Sufficient Conditions ◽  
Optimal Conditions ◽  
Lagrange Equations ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Necessary And Sufficient

In this paper, the necessary and sufficient conditions of optimality for variational problems with Caputo partial fractional derivative are established. Fractional Euler-Lagrange equations are obtained. The Legendre condition and Noether’s theorem are also presented.

Reduction of Hamiltonian Mechanical Systems With Affine Constraints: A Geometric Unification

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Robin Chhabra ◽  
M. Reza Emami ◽  
Yael Karshon
Keyword(s):  
Symmetry Group ◽  
Mechanical Systems ◽  
Hamiltonian Formalism ◽  
Geometrical Approach ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Relative Configuration ◽  
D’Alembert Equation ◽  
Dynamical Reduction ◽  
Configuration Manifold

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

scholarly journals ON YOUNG TOWERS ASSOCIATED WITH INFINITE MEASURE PRESERVING TRANSFORMATIONS

2009 ◽  
Vol 09 (04) ◽  
pp. 635-655 ◽  
Author(s):  
H. BRUIN ◽  
M. NICOL ◽  
D. TERHESIU
Keyword(s):  
Dynamical System ◽  
Common Factor ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Original System ◽  
Return Time ◽  
Tail Behavior ◽  
Necessary And Sufficient ◽  
Young Towers ◽  
Measure Preserving Transformations

For a σ-finite measure preserving dynamical system (X, μ, T), we formulate necessary and sufficient conditions for a Young tower (Δ, ν, F) to be a (measure theoretic) extension of the original system. Because F is pointwise dual ergodic by construction, one immediate consequence of these conditions is that the Darling–Kac theorem carries over from F to T. One advantage of the Darling–Kac theorem in terms of Young towers is that sufficient conditions can be read off from the tail behavior and we illustrate this with relevant examples. Furthermore, any two Young towers with a common factor T, have return time distributions with tails of the same order.

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