Holonomic
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In mathematics, the term holonomic may occur with several different meanings.
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[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
In classical mechanics a system may be defined as holonomic if all the constraints of the system are holonomic. For a constraint to be holonomic it must be expressible as a function: f(x1,x2,x3,...xn,t) = 0 i.e. the constraint depends only on the coordinates of the system and time. It does not depend on the velocity or momentum of the system.
Examples of holonomic systems are: the simple pendulum [√(x² + y²) – L =0]; rigid bodies. See also nonholonomic system.
[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
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 is equal to the total degrees of freedom then the robot is said to be holonomic. If the controllable degrees of freedom is 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, 1 in the elbow and 3 in the wrist) 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] 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]
