A nonholonomic system in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
is a
physical system
A physical system is a collection of physical objects.
In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...
whose state depends on the path taken in order to achieve it. Such a system is described by a set of
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s subject to differential constraints and non-linear constraints, such that when the system evolves along a path in its
parameter space The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for the ...
(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. Nonholonomic mechanics is autonomous division of Newtonian mechanics.
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.
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
Heinrich Rudolf Hertz ( ; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's Maxwell's equations, equations of electrom ...
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
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
has a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...
fundamentally distinct from that of a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. 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
A plotter is a machine that produces vector graphics drawings. Plotters draw lines on paper using a pen, or in some applications, use a knife to cut a material like vinyl or leather. In the latter case, they are sometimes known as a cutting pl ...
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
Turtles are an order of reptiles known as Testudines, characterized by a special shell developed mainly from their ribs. Modern turtles are divided into two major groups, the Pleurodira (side necked turtles) and Cryptodira (hidden necked tu ...
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. See this very similar
gantry crane example for a mathematical explanation of why such a system is holonomic.
History
N. M. Ferrers
Norman Macleod Ferrers D.D. (11 August 1829 – 31 January 1903) was a British mathematician and university administrator and editor of a mathematical journal.
Career and research
Ferrers was educated at Eton College before studying at Gonvill ...
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
Consider a system of
particles with positions
for
with respect to a given reference frame. In classical mechanics, any constraint that is not expressible as
is a non-
holonomic constraint
In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form:
:f(u_1, u_2, u_3,\ldots, u_n, t) = 0
where \ are the ''n'' generalized coordinates that d ...
. In other words, a nonholonomic constraint is nonintegrable
and in
Pfaffian form:
*
is the number of coordinates.
*
is the number of constraint equations.
*
are coordinates.
*
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
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calcul ...
.
For
virtual displacement
In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) \delta \gamma shows how the mechanical system's trajectory can ''hypothetically'' (hence the term ''virtual'') deviate very s ...
s only, the differential form of the constraint is
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, yet is allowed to fall off it:
*
is the distance of the particle from the centre of the sphere.
*
is the radius of the sphere.
Examples
Rolling wheel
A wheel (sometimes visualized as a unicycle or a rolling coin) is a nonholonomic system.
Layman's explanation
Consider the wheel of a bicycle that is parked in a certain place (on the ground). Initially the
inflation valve is at a certain position on the wheel. 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. Its new position depends on the path taken. If the wheel were holonomic, then the valve stem would always end up in the same position as long as the wheel were always rolled back to the same location on the Earth. Clearly, however, this is not the case, so the system is nonholonomic.
Mathematical explanation
![Unicycle drawing updat](https://upload.wikimedia.org/wikipedia/commons/5/57/Unicycle_drawing_updat.png)
It is possible to model the wheel mathematically with a system of constraint equations, and then prove that that system is nonholonomic.
First, we define the configuration space. The wheel can change its state in three ways: having a different rotation about its axle, having a different steering angle, and being at a different location. We may say that
is the rotation about the axle,
is the steering angle relative to the
-axis, and
and
define the spatial position. Thus, the configuration space is:
We must now relate these variables to each other. We notice that as the wheel changes its rotation, it changes its position. The change in rotation and position implying velocities must be present, we attempt to relate angular velocity and steering angle to linear velocities by taking simple time-derivatives of the appropriate terms:
The velocity in the
direction is equal to the angular velocity times the radius times the cosine of the steering angle, and the
velocity is similar. Now we do some algebraic manipulation to transform the equation to ''Pfaffian form'' so it is possible to test whether it is holonomic, starting with:
Then, let's separate the variables from their coefficients (left side of equation, derived from above). We also realize that we can multiply all terms by
so we end up with only the differentials (right side of equation):
The right side of the equation is now in ''Pfaffian form'':
We now use the
universal test for holonomic constraints. If this system were holonomic, we might have to do up to eight tests. However, we can use mathematical intuition to try our best to prove that the system is nonholonomic on the ''first'' test. Considering the test equation is:
we can see that if any of the terms
,
, or
were zero, that that part of the test equation would be trivial to solve and would be equal to zero. Therefore, it is often best practice to have the first test equation have as many non-zero terms as possible to maximize the chance of the sum of them not equaling zero. Therefore, we choose:
:
:
:
:
:
:
We substitute into our test equation:
and simplify:
We can easily see that this system, as described, is nonholonomic, because
is not always equal to zero.
=Additional conclusions
=
We have completed our proof that the system is nonholonomic, but our test equation gave us some insights about whether the system, if further constrained, could be holonomic. Many times test equations will return a result like
implying the system could never be constrained to be holonomic without radically altering the system, but in our result we can see that
''can'' be equal to zero, in two different ways:
*
, the radius of the wheel, can be zero. This is not helpful as the system in practice would lose all of its degrees of freedom.
*
can be zero by setting
equal to zero. This implies that if the wheel were not allowed to turn and had to move only in a straight line at all times, it would be a holonomic system.
There is one thing that we have not yet considered however, that to find all such modifications for a system, one must perform ''all'' eight test equations (four from each constraint equation) and collect ''all'' the failures to gather all requirements to make the system holonomic, if possible. In this system, out of the seven additional test equations, an additional case presents itself:
This does not pose much difficulty, however, as adding the equations and dividing by
results in:
which with some simple algebraic manipulation becomes:
which has the solution
(from
).
Refer back to the
layman's explanation above where it is said, "
he valve stem'snew position depends on the path taken. If the wheel were holonomic, then the valve stem would always end up in the same position as long as the wheel were always rolled back to the same location on the Earth. Clearly, however, this is not the case, so the system is nonholonomic." However it is easy to visualize that if the wheel were only allowed to roll in a perfectly straight line and back, the valve stem ''would'' end up in the same position! In fact, moving parallel to the given angle of
is not actually necessary in the real world as the orientation of the coordinate system itself is arbitrary. The system can become holonomic if the wheel moves only in a straight line at any fixed angle relative to a given reference. Thus, we have not only proved that the original system is nonholonomic, but we also were able to find a restriction that can be added to the system to make it holonomic.
However, there is something mathematically special about the restriction of
for the system to make it holonomic, as
in a Cartesian grid. Combining the two equations and eliminating
, we indeed see that
and therefore one of those two coordinates is completely redundant. We already know that the steering angle is a constant, so that means the holonomic system here needs to only have a configuration space of
. As discussed
here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a TV ...
, a system that is modellable by a Pfaffian constraint must be holonomic if the configuration space consists of two or fewer variables. By modifying our original system to restrict it to have only two degrees of freedom and thus requiring only two variables to be described, and assuming it can be described in Pfaffian form (which in this example we already know is true), we are assured that it is holonomic.
Rolling sphere
This example is an extension of the 'rolling wheel' problem considered above.
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 by the doubly unique
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
(''q'' and −''q'') which, when applied to the points that represent the sphere, carries points ''B'' and ''R'' to their new positions.
Foucault pendulum
An additional example of a nonholonomic system is the
Foucault pendulum
The Foucault pendulum or Foucault's pendulum is a simple device named after French physicist Léon Foucault, conceived as an experiment to demonstrate the Earth's rotation. A long and heavy pendulum suspended from the high roof above a circular a ...
. 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 plane 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 force
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
s and
Coriolis Coriolis may refer to:
* Gaspard-Gustave de Coriolis (1792–1843), French mathematician, mechanical engineer and scientist
* Coriolis force, the apparent deflection of moving objects from a straight path when viewed from a rotating frame of referen ...
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.
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
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
. 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
Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrat ...
, nonholonomic has been particularly studied in the scope of
motion planning and
feedback linearization
Feedback linearization is a common strategy employed in nonlinear control to control nonlinear systems. Feedback linearization techniques may be applied to nonlinear control systems of the form
where x(t) \in \mathbb^n is the state, u_1(t), ...
for
mobile robots.
[''Robot Motion Planning and Control'', Jean-Paul Laumond (Ed.), 1998, Lecture Notes in Control and Information Sciences, Volume 229, Springer, .]
See also
*
Holonomic constraint
In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form:
:f(u_1, u_2, u_3,\ldots, u_n, t) = 0
where \ are the ''n'' generalized coordinates that d ...
*
Bicycle and motorcycle dynamics
Bicycle and motorcycle dynamics is the science of the motion of bicycles and motorcycles and their components, due to the forces acting on them. Dynamics falls under a branch of physics known as classical mechanics. Bike motions of interest in ...
*
Falling cat problem
The falling cat problem is a problem that consists of explaining the underlying physics behind the observation of the cat righting reflex.
Although amusing and trivial to pose, the solution of the problem is not as straightforward as its state ...
*
Goryachev–Chaplygin top
In classical mechanics, the precession of a rigid body such as a spinning top under the influence of gravity is not, in general, an integrable problem. There are however three (or four) famous cases that are integrable, the Euler, the Lagrange, ...
*
Parallel parking problem
The parallel parking problem is a motion planning problem in control theory and mechanics to determine the path a car must take to parallel park into a parking space. The front wheels of a car are permitted to turn, but the rear wheels must stay ...
*
Pfaffian constraint
*
Udwadia–Kalaba equation
*
Lie group integrator
References
{{DEFAULTSORT:Nonholonomic System
Algebraic topology
Differential geometry
Differential topology
Dynamical systems