Holonomic

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

Holonomic basis

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

$[e_{j},e_{k}]=0\,$

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

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 a holonomic constraint, depends only on the coordinates $x_{j}\,\!$ and time $t\,\!$ . It does not depend on the velocities. A constraint that cannot be expressed in the form shown above is a nonholonomic constraint.

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 $x_{d}\,\!$ .

Suppose that a physical system has $N\,\!$ degrees of freedom. Now, $h\,\!$ holonomic constraints are imposed on the system. Then, the number of degrees of freedom is reduced to $m=N-h\,\!$ . We can use $m\,\!$ independent generalized coordinates ($q_{j}\,\!$) to completely describe the motion of the system. The transformation equation can be expressed as follows:

$x_{i}=x_{i}(q_{1},\ q_{2},\ \dots ,\ q_{m},\ t)\ ,\qquad \qquad \qquad i=1,\ 2,\ \dots N.\,$

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.

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 a 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.

Examples

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

${x^{2}+y^{2}}-L^{2}=0,$

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 $P_{j}\,\!$ , and$L_{ij}\,\!$ is the distance between them.

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.

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, x^a, 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.

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.

References

• 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]