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= Nonholonomic system =
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Introduction
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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
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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. 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.
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.
History
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N. M. Ferrers first suggested to extend the equations of motion with
nonholonomic constraints in 1871.
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 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.
In 1897, S. A. Chaplygin first suggested to form the equations of
motion without Lagrange multipliers.
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.
Constraints
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Consider a system of N particles with positions \mathbf r_i for
i\in\{1,\ldots,N\} with respect to a given reference frame. In
classical mechanics, any constraint that is not expressible as
: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 and has the form
:\sum_{i=1}^n a_{s,i} \, dq_i + a_{s,t} \, dt = 0~~~~(s = 1, 2,
\ldots, k)
::n is the number of coordinates.
::k is the number of constraint equations.
::q_i are coordinates.
::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.
For virtual displacements only, the differential form of the
constraint is
:\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. A
classical example of an inequality constraint is that of a particle
placed on the surface of a sphere:
:r^2-a^2\geq0.
::r is the distance of the particle from the centre of the sphere.
::a is the radius of the sphere.
Rolling wheel
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Consider the wheel of a bicycle that is parked in a certain place (on
the ground). Initially the inflation valve is at one position. If
the bicycle is ridden around, and then parked in 'exactly' the same
place, the valve will almost certainly not be in the same position as
before, and its new position depends on the path taken.
Rolling sphere
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This example is very easy for the reader to demonstrate. It is an
extension of the 'rolling wheel' problem considered above, with a more
mathematical treatment.
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. The
system is therefore nonholonomic. The anholonomy may be represented