Constraints & constrained systems

1. Why begin with constraints?

Imagine a bead released in empty space and the same bead threaded on a rigid wire. Newton's law applies to both. Yet the first bead can move in three independent spatial directions, while the second must stay on the wire. The difference is not a new law of motion: it is a restriction on the possible configurations.

If we insist on using three Cartesian coordinates for the bead on the wire, we must also keep track of the wire's reaction force and equations describing its shape. A more economical description is possible. First understand the restriction, then choose a variable measuring position along the wire. This is the first step toward analytical mechanics.

Prerequisites: position vectors, forces, and Cartesian coordinates. Read P1 in the prerequisite guide if these need refreshing.

A constraint answers “Where may the system be, or how may it move?” An equation of motion answers “Which of those possibilities actually occurs under the given forces and initial conditions?”

2. Mechanical systems and constraint equations

A mechanical system is the collection of particles or bodies we choose to study. Its configuration specifies the positions of all its particles at an instant. An unconstrained system of $N$ point particles in three-dimensional space needs $3N$ position coordinates. “Unconstrained” does not mean force-free: gravity can act without imposing a geometrical restriction.

A constraint is a restriction on the admissible positions, velocities, or both, of a system. A rigid rod maintains a distance, a track restricts position, an inextensible string relates two displacements, and rolling without slipping restricts velocities at contact. These are idealized descriptions of physical interactions. A real rod deforms slightly; the rigid-rod model neglects that deformation.

For a particle restricted to a sphere of fixed radius $a$ centred at the origin, the distance condition is $|\mathbf r|=a$. Squaring it gives

The equation selects the spherical surface. It does not select a particular orbit on that surface. Many trajectories satisfy it, depending on forces and initial conditions.

For several particles, a position constraint can be written

The subscript $\alpha$ labels the restrictions; it is not a particle label. An explicit $t$ permits the prescribed geometry to change. For a pendulum of length $a$ with support at $(X(t),Y(t))$,

Here $X(t)$ and $Y(t)$ are specified functions, not additional unknown coordinates of the bob. If the support itself is free to move dynamically, it must instead become part of the mechanical system.

3. Holonomic and non-holonomic constraints

A constraint is holonomic if it can be expressed as an equality involving positions and possibly time, without velocities, or reduced to such an equality by integration. The sphere and fixed-length pendulum are examples.

A non-holonomic constraint cannot be reduced to an equivalent position-and-time equality. Non-integrable velocity restrictions are the most important examples for this course. Some elementary texts also group inequality restrictions under “non-holonomic”; we will state explicitly when an inequality is intended rather than relying on that broad terminology.

A velocity equation is not automatically non-holonomic

Suppose a wheel of radius $R$ rolls along a fixed straight line. Let $x$ be displacement and let $\phi$ increase in the direction of forward rolling. The no-slip relation is

Since $R$ is constant, the left side is $d(x-R\phi)/dt$. Integration gives $x-R\phi=C$, where the initial contact configuration fixes $C$. The relation is therefore integrable: within that fixed rolling branch, it is holonomic. Writing it in terms of velocities did not change this fact.

Why a steerable rolling wheel is different

For an ideal upright wheel moving on a plane, let $(x,y)$ locate its centre's horizontal projection, $\psi$ be heading, and $\phi$ be wheel rotation. A simple no-slip model gives

If heading changes, the total change in $x$ is an integral of $R\cos\psi\,d\phi$. The answer depends on how heading changes along the path, not just the final values of $\psi$ and $\phi$. For example, rolling forward, changing heading, rolling backward, and restoring heading can restore the wheel angles while changing its position. Thus no universal position-only relation of the straight-line form describes this model. These are non-holonomic restrictions.

Do not classify every instance of rolling as non-holonomic. Specify the model, write its restrictions, and test whether they integrate to position relations.

4. Time dependence and one-sided restrictions

Scleronomous constraints have no explicit time dependence in the chosen inertial-frame description. A fixed sphere or a fixed-length pendulum with fixed support is scleronomous. Rheonomous constraints explicitly depend on time: a moving track, a prescribed moving support, or a changing pendulum length are examples.

The motion of the particle itself does not make a constraint rheonomous. The equation $x(t)^2+y(t)^2=a^2$ describes a moving particle on a fixed circle. Written as a constraint function, $f(x,y)=x^2+y^2-a^2$ contains no explicit $t$.

These classifications answer different questions. A moving circular hoop remains holonomic even though it is rheonomous.

A bilateral restriction is an equality locally permitting displacement in either sense along the allowed surface. A bead threaded on a wire remains on it. A unilateral restriction permits only one side of a boundary. A particle above a floor satisfies $y\ge0$; a bob attached by a flexible string satisfies $r\le a$ because the string cannot push it outward when slack.

At contact, the floor can push upward but cannot pull downward. A taut string can pull but cannot push. If solving under the equality $r=a$ demands a negative tension, the assumed taut-string model has failed: one must allow the string to go slack. A rigid rod can support compression, so it is a different model.

Remark (optional). A change of coordinates can make a moving boundary look stationary. Its motion then reappears in the transformation equations and kinetic energy. Simply hiding time dependence in a coordinate definition does not remove the physical driving.

5. Constraint forces and constrained motion

A constraint force is the interaction needed to enforce a modeled restriction. Normal reaction from a smooth surface and tension in an ideal string are examples. Their magnitudes are usually unknown until the motion is solved.

Write Newton's law for particle $a$ as

where $\mathbf F_a$ denotes applied forces and $\mathbf R_a$ denotes constraint reactions. This division is a modeling choice, but it must be consistent. Gravity is normally applied; the reaction of a smooth track is a constraint force.

For a bead moving on a smooth circular wire, the reaction is normal to the wire. It can change the direction of velocity even if it does no work along the fixed wire. Thus “does no work” does not mean “has no dynamical effect.” It is precisely the reaction that keeps the bead from departing from the circle.

Friction needs separate attention. A rough surface exerts a tangential force that can do work. Later, we will eliminate only reactions whose total virtual work vanishes. We cannot erase all contact forces merely because they arise at a constraint.

6. Applications

Example 1 · Particle confined to a sphere

A particle is confined to the surface of a sphere of radius $2\,\mathrm m$. Its Cartesian coordinates must satisfy the fixed-radius condition.

$x^2+y^2+z^2=4\,\mathrm{m^2}$. This is a position-only equality, so it is holonomic; no explicit time appears, so it is scleronomous; the particle must remain on the surface, so the model is bilateral. Two angular coordinates locate the particle. The detailed counting is developed in Chapter 2.

The particle can move north–south and east–west locally, but it cannot move radially off the surface.

Example 2 · Pendulum with a prescribed changing length

Consider planar motion about a fixed origin with a taut string whose length is prescribed as $a(t)=a_0+ct>0$. The changing radius imposes a condition on the radial velocity as well as on the position.

The position relation is $x^2+y^2-(a_0+ct)^2=0$. It is holonomic and rheonomous when $c\ne0$. Differentiate both sides:

Divide by two to obtain $x\dot x+y\dot y=ac$. Since $r=a$, this is the radial-speed relation $\dot r=c$. A radial component of velocity is now required. When $c=0$, the velocity is tangent to the fixed circle, as expected.

Example 3 · Inextensible string connecting two masses

In an ideal Atwood machine, the pulley is fixed, massless, and frictionless, and the string is taut and inextensible. Let $x_1,x_2$ be the straight hanging lengths measured downward from fixed reference points.

The wrapped part of the string has constant length. If the remaining length is $b$, then $x_1+x_2=b$. Differentiate once: $\dot x_1+\dot x_2=0$. Differentiate again: $\ddot x_1+\ddot x_2=0$. Thus one mass moves downward exactly as far as the other moves upward. The constraint is holonomic and scleronomous.

The constraint supplies a relation, not the acceleration magnitude. The masses and gravity determine that magnitude in Chapter 6.

Example 4 · Tension and compression: string versus rod

A bob is connected to a support by a flexible string of length $a$. Although the familiar pendulum model uses $r=a$, this equality applies only while the string remains taut.

The string only limits the maximum distance, so the full geometrical condition is $r\le a$. While taut, $r=a$ and tension must satisfy $\mathcal T\ge0$. A calculation that gives $\mathcal T<0$ asks the string to push, which it cannot do. Replace the taut phase by free motion until the string becomes taut again. With a rigid rod, the equality remains valid and a compressive reaction is possible.

7. Validity of the constraint model

Do not count equations before checking independence. The equations $x^2+y^2=a^2$ and $2x^2+2y^2=2a^2$ express the same restriction. Do not call a gravitational trajectory a constraint merely because it happens to be curved: it is selected dynamically. Do not call a spring's preferred length a rigid-length constraint: an ordinary spring permits extension and supplies a force.

The distinction between prescribed and dynamical motion is equally important. If a motor prescribes the support's position, it introduces known time dependence. If the support moves because of forces, its position is another unknown and usually another degree of freedom.

8. Practice problems

Attempt these before opening Chapter 1 hints and solutions.

1. 1.1 Can a system be unconstrained while acted on by gravity?

2. 1.2 Classify a bead threaded on a fixed horizontal circular wire.

3. 1.3 Is $\dot x=c$ necessarily non-holonomic? Treat $c$ as constant.

4. 1.4 Does a rigid rod exert only a tensile force?

5. 1.5 Explain the difference between an equation of constraint and an equation of motion using a pendulum.

6. 1.6 Explain why the classifications holonomic and rheonomous can apply simultaneously.

7. 1.7 Write the full constraint for a particle that can move inside or on a sphere of radius $a$.

8. 1.8 For $x^2+y^2=a^2$, derive the relation between position and velocity. Explain it geometrically.

9. 1.9 Derive the velocity relation for $[x-X(t)]^2+[y-Y(t)]^2=a^2$.

10. 1.10 A particle is on a circle of radius $5\,\mathrm m$. At $(x,y)=(3,4)\,\mathrm m$, $\dot x=2\,\mathrm{m\,s^{-1}}$. Find $\dot y$.

11. 1.11 Two vertical string segments in an Atwood machine sum to $2.4\,\mathrm m$. If $x_1=0.9\,\mathrm m$ and $\dot x_1=0.3\,\mathrm{m\,s^{-1}}$, find $x_2$ and $\dot x_2$.

12. 1.12 A bead remains on $x=2t$ in SI units. Find its allowed $x$ velocity. Is its $y$ velocity fixed by this restriction?

13. 1.13 Define and classify constraints, giving a mathematical and physical example of each class discussed here.

14. 1.14 Discuss constraint forces, explaining normal reaction, string tension, rigidity, and friction. State which modeling assumptions matter.

15. 1.15 Why can straight-line rolling be integrable while steerable rolling is not?

16. 1.16 A bob follows a circle under a central spring force for one special initial speed. Is the circle a constraint? Explain.

9. Geometry and forces

A constraint describes the admissible configurations or velocities of a system. The forces and initial conditions determine a particular motion within those restrictions. This distinction separates, for example, a bead forced to follow a circular wire from a particle that happens to follow a circular orbit under a spring force.

Classification depends on the actual mathematical restriction. A velocity relation may be integrable; a moving boundary may still be holonomic; and an equality appropriate to a taut string may cease to hold when the string becomes slack. The next step is to determine how many independent position variables remain after the restrictions have been imposed.