Nonholonomic system

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 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 the nature of the metric, but it is still fundamentally a two-dimensional space with parameters irretrievably entwined in dependency by the Riemannian metric.

By contrast, one can consider an X-Y plotter as an example of a holonomic system where the state of the system's mechanical components will have a single fixed configuration for any given position of the plotter pen. If the pen relocates between positions 0,0 and 3,3, the mechanism's gears will have the same final positions regardless of whether the relocation happens by the mechanism first incrementing 3 units on the x-axis and then 3 units on the y-axis, incrementing the Y-axis position first, or operating any other sequence of position-changes that result in a final position of 3,3. Since the final state of the machine is the same regardless of the path taken by the plotter-pen to get to its new position, the end result can be said not to be path-dependent. If we substitute a turtle plotter, the process of moving the pen from 0,0 to 3,3 can result in the gears of the robot's mechanism finishing in different positions depending on the path taken to move between the two positions.

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 certain linear constraints, he introduced on the left-hand side of the equations of motion a group of extra terms of the Lagrange-operator type. The remaining extra terms characterise the nonholonomicity of system and they become zero when the given constrains are integrable. In 1901 P. V.Voronets generalised Chaplygin's work to the cases of noncyclic holonomic coordinates and of nonstationary constraints.[7]

Consider a system of ${\displaystyle N}$ particles with positions ${\displaystyle \mathbf {r} _{i}}$ for ${\displaystyle i\in \{1,\ldots ,N\}}$ with respect to 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.

• Bicycle and motorcycle dynamics
• Falling cat problem
• Goryachev–Chaplygin top
• Holonomic (robotics)
• Parallel parking problem
• Pfaffian constraint
• Udwadia–Kalaba equation
• Lie group integrator