Nonholonomic system

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A nonholonomic system in physics and mathematics is a system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints, such that when the system evolves along a path in its parameter space (the parameters varying continuously in values) but finally returns to the original set of parameter values at the start of the path, the system itself may not have returned to its original state.

Details

More precisely, a nonholonomic system, also called an anholonomic system, is one in which there is a continuous closed circuit of the governing parameters, by which the system may be transformed from any given state to any other state.[1] Because the final state of the system depends on the intermediate values of its trajectory through parameter space, the system cannot be represented by a conservative potential function as can, for example, the inverse square law of the gravitational force. This latter is an example of a holonomic system: path integrals in the system depend only upon the initial and final states of the system (positions in the potential), completely independent of the trajectory of transition between those states. The system is therefore said to be integrable, while the nonholonomic system is said to be nonintegrable. When a path integral is computed in a nonholonomic system, the value represents a deviation within some range of admissible values and this deviation is said to be an anholonomy produced by the specific path under consideration. This term was introduced by Heinrich Hertz in 1894.[2]

The general character of anholonomic systems is that of implicitly dependent parameters. If the implicit dependency can be removed, for example by raising the dimension of the space, thereby adding at least one additional parameter, the system is not truly nonholonomic, but is simply incompletely modeled by the lower-dimensional space. In contrast, if the system intrinsically cannot be represented by independent coordinates (parameters), then it is truly an anholonomic system. Some authors[citation needed] make much of this by creating a distinction between so-called internal and external states of the system, but in truth, all parameters are necessary to characterize the system, be they representative of "internal" or "external" processes, so the distinction is in fact artificial. However, there is a very real and irreconcilable difference between physical systems that obey conservation principles and those that do not. In the case of parallel transport on a sphere, the distinction is clear: a Riemannian manifold has a metric fundamentally distinct from that of a Euclidean space. For parallel transport on a sphere, the implicit dependence is intrinsic to the non-euclidean metric. The surface of a sphere is a two-dimensional space. By raising the dimension, we can more clearly see[clarification needed] the nature of the metric, but it is still fundamentally a two-dimensional space with parameters irretrievably entwined in dependency by the Riemannian metric.

History

N. M. Ferrers first suggested to extend the equations of motion with nonholonomic constraints in 1871.[3] He introduced the expressions for Cartesian velocities in terms of generalized velocities. In 1877, E. Routh wrote the equations with the Lagrange multipliers. In the third edition of his book[4] for linear non-holonomic constraints of rigid bodies, he introduced the form with multipliers, which is now called the Lagrange equations of the second kind with multipliers. The terms the holonomic and nonholonomic systems were introduced by Heinrich Hertz in 1894.[5] In 1897, S. A. Chaplygin first suggested to form the equations of motion without Lagrange multipliers.[6] Under the certain linear equations of constraints, he discriminated in the left-hand side of equations of motion the group of extra terms of the type of the Lagrange operator. The rest extra terms characterize the nonholonomicity of system and they go to zeros when the given constrains are integrable. In 1901 P. V.Voronets generalized Chaplygin's work to the cases of noncyclic holonomic coordinates and of nonstationary constraints.[7]

Constraints

Let us consider a system of ${\displaystyle N}$ particles with positions ${\displaystyle \mathbf {r} _{i}}$ for ${\displaystyle i\in \{1,\ldots ,N\}}$ with respect a given reference frame. In classical mechanics, any constraint that is not expressible as

${\displaystyle f(\mathbf {r} _{1},\mathbf {r} _{2},\mathbf {r} _{3},\ldots ,t)=0,}$

is a non-holonomic constraint. In other words, a nonholonomic constraint is nonintegrable[8]:261 and has the form

${\displaystyle \sum _{i=1}^{n}a_{s,i}\,dq_{i}+a_{s,t}\,dt=0~~~~(s=1,2,\ldots ,k)}$
${\displaystyle n}$ is the number of coordinates.
${\displaystyle k}$ is the number of constraint equations.
${\displaystyle q_{i}}$ are coordinates.
${\displaystyle a_{s,i}}$ are coefficients.

In order for the above form to be nonholonomic, it is also required that the left hand side neither be a total differential nor be able to be converted into one, perhaps via an integrating factor.[9]:2–3

For virtual displacements only, the differential form of the constraint is[8]:282

${\displaystyle \sum _{i=1}^{n}a_{s,i}\delta q_{i}=0~~~~(s=1,2,\ldots ,k).}$

It is not necessary for all non-holonomic constraints to take this form, in fact it may involve higher derivatives or inequalities.[10] A classical example of an inequality constraint is that of a particle placed on the surface of a sphere:

${\displaystyle r^{2}-a^{2}\geq 0.}$
${\displaystyle r}$ is the distance of the particle from the centre of the sphere.
${\displaystyle a}$ is the radius of the sphere.

Examples

Foucault pendulum

The classic example of a nonholonomic system is the Foucault pendulum. In the local coordinate frame the pendulum is swinging in a vertical plane with a particular orientation with respect to geographic north at the outset of the path. The implicit trajectory of the system is the line of latitude on the earth where the pendulum is located. Even though the pendulum is stationary in the earth frame, it is moving in a frame referred to the sun and rotating in synchrony with the Earth's rate of revolution, so that the only apparent motion of the pendulum is that caused by the rotation of the earth. This latter frame is considered to be an inertial reference frame, although it too is non-inertial in more subtle ways. The earth frame is well known to be non-inertial, a fact made perceivable by the apparent presence of centrifugal forces and Coriolis forces.

Motion along the line of latitude is parameterized by the passage of time, and the Foucault pendulum's plane of oscillation appears to rotate about the local vertical axis as time passes. The angle of rotation of this plane at a time t with respect to the initial orientation is the anholonomy of the system. The anholonomy induced by a complete circuit of latitude is proportional to the solid angle subtended by that circle of latitude. The path need not be constrained to latitude circles. For example, the pendulum might be mounted in an airplane. The anholonomy is still proportional to the solid angle subtended by the path, which may now be quite irregular. The Foucault pendulum is a physical example of parallel transport.

Rolling sphere

This example is very easy for the reader to demonstrate. Consider a three-dimensional orthogonal Cartesian coordinate frame, for example a level table top with a point marked on it for the origin, and the x and y axes laid out with pencil lines. Take a sphere of unit radius, for example a ping pong ball, and mark one point B in blue. Corresponding to this point is a diameter of the sphere, and the plane orthogonal to this diameter positioned at the center C of the sphere defines a great circle called the equator associated with point B. On this equator, select another point R and mark it in red. Position the sphere on the z = 0 plane such that the point B is coincident with the origin, C is located at x = 0, y = 0, z = 1, and R is located at x = 1, y = 0, and z = 1, i.e. R extends in the direction of the positive x axis. This is the initial or reference orientation of the sphere.

The sphere may now be rolled along any continuous closed path in the z = 0 plane, not necessarily a simply connected path, in such a way that it neither slips nor twists, so that C returns to x = 0, y = 0, z = 1. In general, point B is no longer coincident with the origin, and point R no longer extends along the positive x axis. In fact, by selection of a suitable path, the sphere may be re-oriented from the initial orientation to any possible orientation of the sphere with C located at x = 0, y = 0, z = 1. (reference: The Nonholonomy of the Rolling Sphere, Brody Dylan Johnson, The American Mathematical Monthly, June–July 2007, vol. 114, pp. 500–508) The system is therefore nonholonomic. The anholonomy may be represented by the doubly unique quaternion (q and −q) which, when applied to the points that represent the sphere, carries points B and R to their new positions.

Linear polarized light in an optical fiber

Take a length of optical fiber, say three meters, and lay it out in an absolutely straight line. When a vertically polarized beam is introduced at one end, it emerges from the other end, still polarized in the vertical direction. Mark the top of the fiber with a stripe, corresponding with the orientation of the vertical polarization.

Now, coil the fiber tightly around a cylinder ten centimeters in diameter. The path of the fiber now describes a helix which, like the circle, has constant curvature. The helix also has the interesting property of having constant torsion. As such the result is a gradual rotation of the fiber about the fiber's axis as the fiber's centerline progresses along the helix. Correspondingly, the stripe also twists about the axis of the helix.

When linearly polarized light is again introduced at one end, with the orientation of the polarization aligned with the stripe, it will, in general, emerge as linear polarized light aligned not with the stripe, but at some fixed angle to the stripe, dependent upon the length of the fiber, and the pitch and radius of the helix. This system is also nonholonomic, for we can easily coil the fiber down in a second helix and align the ends, returning the light to its point of origin. The anholonomy is therefore represented by the deviation of the angle of polarization with each circuit of the fiber. By suitable adjustment of the parameters, it is clear that any possible angular state can be produced.

Robotics

In robotics, nonholonomy has been particularly studied in the scope of motion planning and feedback linearization for mobile robots.[11] Refer to holonomic robotics for a more detailed description.

References

1. ^ Bryant, R.L. (2006). "Geometry of Manifolds with Special Holonomy: "100 Years of Holonomy", 150 Years of Mathematics at Washington University in St. Louis". Contemporary Mathematics. 395: 29–38. doi:10.1090/conm/395/07414.
2. ^ Berry, Michael (December 1990). "Anticipations of Geometric Phase". Physics Today. 43 (12): 34–40. Bibcode:1990PhT....43l..34B. doi:10.1063/1.881219.
3. ^ Ferrers, N.M. (1872). "Extension of Lagrange's equations". Quart. J. Pure Appl. Math. XII: 1–5.
4. ^ Routh, E. (1884). Advanced part of a Treatise on the Dynamics of a System of Rigid Bodies. London.
5. ^ Hertz, H. (1894). ie Prinzipien derMechanik in neuem Zusammenhange dargestellt.
6. ^ Chaplygin, S.A. (1897). "О движении тяжелого тела вращения по горизонтальнойплоскости" [A motion of heavy body of revolution on a horizontal plane]. антpопологии и этногpафии (in Russian). отделения физических наук общества любителей естествознания. 1 (IX): 10–16.
7. ^ Voronets, P. (1901). "Об уравнениях движения для неголономных систем" [Equations of motion of nonholonomic systems]. Мат. сб. (in Russian). 4 (22): 659–686.
8. ^ a b Torby, Bruce (1984). "Energy Methods". Advanced Dynamics for Engineers. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. ISBN 0-03-063366-4.
9. ^ Jack Sarfatti (2000-03-26). "Non Holonomic Constraints in Newtonian Mechanics" (PDF). Pedagogical Review from the Classics of Physics. stardrive.org. Archived from the original (PDF) on 2007-10-20. Retrieved 2007-09-22.
10. ^ Goldstein, Herbert (1980). Classical Mechanics (3rd ed.). United States of America: Addison Wesley. p. 16. ISBN 0-201-65702-3.
11. ^ Robot Motion Planning and Control, Jean-Paul Laumond (Ed.), 1998, Lecture Notes in Control and Information Sciences, Volume 229, Springer, doi:10.1007/BFb0036069.