Holonomic

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In mathematics, the term holonomic may occur with several different meanings.

Contents

[edit] Holonomic basis

A holonomic basis for a manifold is a set of basis vectors ek for which all Lie derivatives vanish:

[ej,ek] = 0

Some authors call a holonomic basis a coordinate basis, and a nonholonomic basis a non-coordinate basis. See also Jet bundle.

[edit] Holonomic system (Physics)

In classical mechanics a system may be defined as holonomic if all constraints of the system are holonomic. For a constraint to be holonomic it must be expressible as a function:

f(x_1,\ x_2,\ x_3,\ \dots,\ x_N,\ t)=0\,\! .

The constraint, called holonomic constraint, depends only on the coordinates x_j\,\! and time t\,\! . It does not depend on the velocities. If a constraint cannot be expressed in the form shown above, then it’s a nonholonomic constraint.

[edit] Transformation to general coordinates

The holonomic constraint equations can help us easily remove some of the dependent variables in our system. For example, if we want to remove x_d\,\! which is a parameter in the constraint equation f_i\,\! , we can rearrange the equation into the following form, assuming it can be done,

x_d=g_i(x_1,\ x_2,\ x_3,\ \dots,\ x_{d-1},\ x_{d+1},\ \dots,\ x_N,\ t) \,\! ,

and replace the x_d\,\! in every equation of the system using the above function. Thus, it is possible to remove all occurrences of the dependent variable x_d\,\! .

If, originally, a physical system has degree of freedom N\,\!. Now, h\,\! holonomic constraints are imposed on the system. Then, the degree of freedom is reduced to m=N - h\,\! . We can use m\,\! independent generalized coordinates to completely describe the motion of the system. The transformation equation can be expressed as follows:

x_i=x_i(q_1,\ q_2,\ \dots,\ q_m,\ t)\ ,\qquad\qquad\qquad i=1,\ 2,\ \dots N\,\! .

[edit] Differential form

Consider the following differential form of a constraint equation:

\sum_j\ c_{ij} dq_j+c_i dt=0\,\! ;

where c_{ij}\,\! , c_{i}\,\! are the coefficients of the differentials dq_j\,\! and dt\,\! for the ith\,\! constraint.

If the differential form is integrable, i.e., if there is a function f_i(q_1,\ q_2,\ q_3,\ \dots,\ q_N,\ t)=0\,\! satisfying the equality

df_i=\sum_j\ c_{ij} dq_j+c_i dt=0\,\! ,

then, this constraint is a holonomic constraint; otherwise, nonholonomic. Therefore, all holonomic and some nonholonomic constraints can be expressed using the differential form. Not all nonholonomic constraints can be expressed this way. Examples of nonholonomic constraints which can not be expressed this way are those that are dependent on generalized velocities. With constraint equation in differential form, whether a constraint is holonomic or nonholonomic depends on the integrability of the differential form.

[edit] Classification of physical systems

In order to study classical physics rigorously and methodically, we need to classify systems. Based on previous discussion, we can classify physical systems into holonomic systems and non-holonomic systems. One of the conditions for the applicability of many theorems and equations is that the system must be holonomic system. For example, if a physical system is a holonomic system and a monogenic system, then Hamilton's principle is the necessary and sufficient condition for the correctness of Lagrange's equation.[1]

[edit] Examples

A simple pendulum
A simple pendulum

As shown at right, a simple pendulum is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string’s length is a constant. Therefore, this system is holonomic; it obeys holonomic constraint

\sqrt{x^2+y^2} - L=0\,\! ,

where (x,\ y)\,\! is the position of the weight and L\,\! is length of the string.

The particles of a rigid body obey the holonomic constraint

(\mathbf{r}_i - \mathbf{r}_j)^2 - L_{ij}^2=0\,\! ,

where \mathbf{r}_i\,\! , \mathbf{r}_j\,\! are respectively the positions of particles P_i\,\! and P_j\,\! , andL_{ij}\,\! is the distance between them.

[edit] Holonomic system (D-modules)

In the Mikio Sato school of D-module theory, holonomic system has a further, technical meaning. Roughly speaking, with a D-module considered as a system of partial differential equations on a manifold, a holonomic system is a highly over-determined system, such that the solutions locally form a vector space of finite dimension (instead of the expected dependence on some arbitrary function). Such systems have been applied, for example, to the Riemann-Hilbert problem in higher dimensions, and to quantum field theory.

[edit] Holonomic function

A smooth function in one variable is holonomic if it satisfies a linear homogenous differential equation with polynomial coefficients. A function defined on the natural numbers is holonomic if it satisfies a linear homogenous recurrence relation (or equivalently, a linear homogenous difference equation) with polynomial coefficients. The two concepts are closely related: a function represented by a power series is holonomic if and only if the coefficients are holonomic. A holonomic function on the natural numbers is also called P-recursive.

Examples of holonomic functions are exp, ln, sin, cos, arcsin, arccos, xa, with many more. Not all elementary functions are holonomic, for example the tangent and secant are not. Holonomic functions are closed under sum, product and composition, but not division.

[edit] Robotics

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Further information might be found on the talk page or at requests for expansion. (August 2007)

In robotics holonomicity refers to the relationship between the controllable and total degrees of freedom of a given robot (or part thereof). If the controllable degrees of freedom are greater than or equal to the total degrees of freedom then the robot is said to be holonomic. If the controllable degrees of freedom are less than the total degrees of freedom it is non-holonomic. A robot is considered to be redundant if it has more controllable degrees of freedom than degrees of freedom in its task space. Holonomicity can be used to describe simple objects as well.

For example, a car is non-holonomic because although it could physically move laterally, there is no mechanism to control this movement.

A human arm, by contrast, is a holonomic, redundant system because it has 7 degrees of freedom (3 in the shoulder - rotations about each axis, 2 in the elbow - bending and rotation about the lower arm axis, and 2 in the wrist, bending up and down, and left and right) and there are only 6 physical degrees of freedom in the task of placing the hand (x, y, z, roll, pitch and yaw), while fixing the 7 degrees of freedom fixes the hand. See also sub-Riemannian geometry for a discussion of holonomic constraints in robotics.

[edit] Holonomic brain theory

Main article: Holonomic brain theory

Holonomic brain theory, developed by Karl Pribram and David Bohm, models cognitive function as being guided by a matrix of neurological wave interference patterns. This model has important implications in neurology, especially in the field of human memory.

[edit] References

  1. ^ Goldstein, Herbert (1980). Classical Mechanics, 3rd (in English), United States of America: Addison Wesley, pp. 45. ISBN 0201657023. 
  • Wolfram Koepf, The Algebra of Holonomic Equations, 20. W. Koepf: "The Algebra of Holonomic Equations", Mathematische Semesterberichte 44 (1997),

pp.173–194 [1]

  • Marko Petkovšek, Herbert S. Wilf and Doron Zeilberger, A=B, A. K. Peters, 1996 [2]
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