Holonomic constraints

In classical mechanics, holonomic constraints are relations between the position variables (and possibly time^[1]) that can be expressed in the following form:

f(q_{1},q_{2},q_{3},\ldots ,q_{n},t)=0

where $\{q_{1},q_{2},q_{3},\ldots ,q_{n}\}$ are the n generalized coordinates that describe the system. For example, the motion of a particle constrained to lie on the surface of a sphere is subject to a holonomic constraint, but if the particle is able to fall off the sphere under the influence of gravity, the constraint becomes non-holonomic. For the first case the holonomic constraint may be given by the equation:

r^{2}-a^{2}=0

where $r$ is the distance from the centre of a sphere of radius $a$ , whereas the second non-holonomic case may be given by:

r^{2}-a^{2}\geq 0

Velocity-dependent constraints such as:

f(q_{1},q_{2},\ldots ,q_{n},{\dot {q}}_{1},{\dot {q}}_{2},\ldots ,{\dot {q}}_{n},t)=0

are not usually holonomic.^{[citation needed]}

Holonomic system (physics)

In classical mechanics a system may be defined as holonomic if all constraints of the system are holonomic. For a constraint to be holonomic it must be expressible as a function:

f(x_{1},\ x_{2},\ x_{3},\ \ldots ,\ x_{N},\ t)=0,\,

i.e. a holonomic constraint depends only on the coordinates $x_{j}\,\!$ and maybe time $t\,\!$ .^[1] It does not depend on the velocities or any higher-order derivative with respect to t. A constraint that cannot be expressed in the form shown above is a nonholonomic constraint.

Introduction

As described above, a holonomic system is (simply speaking) a system in which one can deduce the state of a system by knowing only information about the change of positions of the components of the system over time, but not needing to know the velocity or in what order the components moved relative to each other. In contrast, a nonholonomic system is a system where the velocities of the components over time must be known to be able to determine the change of state of the system. Examples of holonomic systems are gantry cranes, pendulums, and robotic arms. Examples of nonholonomic systems are Segways, unicycles, and automobiles.

Terminology

The configuration space ${\overrightarrow {u}}$ lists the displacement of the components of the system, one for each degree of freedom. A system that can be described using a configuration space is called scleronomic.

{\overrightarrow {u}}={\begin{bmatrix}u_{1}&u_{2}&\ldots &u_{n}\end{bmatrix}}^{\mathrm {T} }

The event space is identical to the configuration space except for the addition of a variable $t$ to represent the change in the system over time (if needed to describe the system). A system that must be described using an event space is called rheonomic. Many systems can be described either scleronomically or rheonomically. For example, the total allowable motion of a pendulum can be described with a scleronomic constraint, but the motion over time of a pendulum must be described with a rheonomic constraint.

{\overrightarrow {u}}={\begin{bmatrix}u_{1}&u_{2}&\ldots &u_{n}&t\end{bmatrix}}^{\mathrm {T} }

The state space ${\overrightarrow {q}}$ is the configuration space, plus terms describing the velocity of each term in the configuration space.

{\overrightarrow {q}}={\begin{bmatrix}{\overrightarrow {u}}\\{\overrightarrow {\dot {u}}}\end{bmatrix}}={\begin{bmatrix}u_{1}&\ldots &u_{n}&{\dot {u}}_{1}&\ldots &{\dot {u}}_{n}\end{bmatrix}}^{\mathrm {T} }

The state-time space adds time $t$ .

{\overrightarrow {q}}={\begin{bmatrix}{\overrightarrow {u}}\\{\overrightarrow {\dot {u}}}\end{bmatrix}}={\begin{bmatrix}u_{1}&\ldots &u_{n}&t&{\dot {u}}_{1}&\ldots &{\dot {u}}_{n}\end{bmatrix}}^{\mathrm {T} }

Examples

Gantry crane

As shown on the right, a gantry crane is an overhead crane that is able to move its hook in 3 axes as indicated by the arrows. Intuitively, we can deduce that the crane should be a holonomic system as, for a given movement of its components, it doesn't matter what order or velocity the components move: as long as the total displacement of each component from a given starting condition is the same, all parts and the system as a whole will end up in the same state. Mathematically we can prove this as such:

We can define the configuration space of the system as:

{\overrightarrow {u}}={\begin{bmatrix}x\\y\\z\end{bmatrix}}

We can say that the deflection of each component of the crane from its "zero" position are $B$ , $G$ , and $O$ , for the blue, green, and orange components, respectively. The orientation and placement of the coordinate system does not matter in whether a system is holonomic, but in this example the components happen to move parallel to its axes. If the origin of the coordinate system is at the back-bottom-left of the crane, then we can write the position constraint equation as:

(x-B)+(y-G)+(z-(h-O))=0

Where $h$ is the height of the crane.

Because we have derived a holonomic constraint equation, we can see that this system must be holonomic.

Pendulum

As shown on the right, a simple pendulum is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string’s length is a constant. Therefore, this system is holonomic; it obeys the holonomic constraint

{x^{2}+y^{2}}-L^{2}=0,

where $(x,\ y)\,\!$ is the position of the weight and $L\,\!$ is length of the string.

Rigid body

The particles of a rigid body obey the holonomic constraint

(\mathbf {r} _{i}-\mathbf {r} _{j})^{2}-L_{ij}^{2}=0,\,

where $\mathbf {r} _{i}\,\!$ , $\mathbf {r} _{j}\,\!$ are respectively the positions of particles $P_{i}\,\!$ and $P_{j}\,\!$ , and $L_{ij}\,\!$ is the distance between them. If a given system is holonomic, rigidly attaching additional parts to components of the system in question cannot make it non-holonomic, assuming that the range of motion or degrees of freedom are not reduced (in other words, assuming the configuration space is unchanged).

Pfaffian form

Consider the following differential form of a constraint equation:

\sum _{j}\ c_{ij}\,du_{j}+c_{i}\,dt=0;\,

where c_ij, c_i are the coefficients of the differentials dq_j and dt for the ith constraint.

If the differential form is integrable, i.e., if there is a function $f_{i}(u_{1},\ u_{2},\ u_{3},\ \ldots ,\ u_{n},\ t)=0\,\!$ satisfying the equality

df_{i}=\sum _{j}\ c_{ij}\,du_{j}+c_{i}\,dt=0,\,

then this constraint is a holonomic constraint; otherwise, it is nonholonomic. Therefore, all holonomic and some nonholonomic constraints can be expressed using the differential form. Examples of nonholonomic constraints that cannot be expressed this way are those that are dependent on generalized velocities. With a constraint equation in differential form, whether the constraint is holonomic or nonholonomic depends on the integrability of the differential form.

Transformation to independent generalized coordinates

The holonomic constraint equations can help us easily remove some of the dependent variables in our system. For example, if we want to remove $x_{d}\,\!$ , which is a parameter in the constraint equation $f_{i}\,\!$ , we can rearrange the equation into the following form, assuming it can be done,

x_{d}=g_{i}(x_{1},\ x_{2},\ x_{3},\ \dots ,\ x_{d-1},\ x_{d+1},\ \dots ,\ x_{N},\ t),\,

and replace the $x_{d}\,\!$ in every equation of the system using the above function. This can always be done for general physical systems, provided that $f_{i}\,\!$ is $C^{1}\,\!$ , then by the implicit function theorem, the solution $g_{i}\,$ is guaranteed in some open set. Thus, it is possible to remove all occurrences of the dependent variable $x_{d}\,\!$ .

Suppose that a physical system has $N\,\!$ degrees of freedom. Now, $h\,\!$ holonomic constraints are imposed on the system. Then, the number of degrees of freedom is reduced to $m=N-h\,\!$ . We can use $m\,\!$ independent generalized coordinates ( $q_{j}\,\!$ ) to completely describe the motion of the system. The transformation equation can be expressed as follows:

x_{i}=x_{i}(q_{1},\ q_{2},\ \ldots ,\ q_{m},\ t)\ ,\qquad \qquad \qquad i=1,\ 2,\ \ldots N.\,

Classification of physical systems

In order to study classical physics rigorously and methodically, we need to classify systems. Based on previous discussion, we can classify physical systems into holonomic systems and non-holonomic systems. One of the conditions for the applicability of many theorems and equations is that the system must be a holonomic system. For example, if a physical system is a holonomic system and a monogenic system, then Hamilton's principle is the necessary and sufficient condition for the correctness of Lagrange's equation.^[2]

References

^ ^a ^b Goldstein, Herbert (2002). "1.3 Constraints". Classical mechanics (Third ed.). Pearson India: Addison-Wesley. pp. 12–13. ISBN 9788131758915. OCLC 960166650.
^ Goldstein, Herbert (1980). Classical Mechanics (3rd ed.). United States of America: Addison Wesley. p. 45. ISBN 0-201-65702-3.

[:0-1] Goldstein, Herbert (2002). "1.3 Constraints". Classical mechanics (Third ed.). Pearson India: Addison-Wesley. pp. 12–13. ISBN 9788131758915. OCLC 960166650.

[Herb1980-2] Goldstein, Herbert (1980). Classical Mechanics (3rd ed.). United States of America: Addison Wesley. p. 45. ISBN 0-201-65702-3.

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