Degrees of freedom & configuration space

1. How much information specifies a configuration?

In Chapter 1, the circular-wire equation involved both $x$ and $y$. Yet you cannot choose those two numbers independently. Once a point on a particular local arc is selected by one coordinate, the other follows from the circle equation. This suggests that the number of written symbols can exceed the number of independent choices.

The task of this chapter is to count those choices correctly and to understand the space formed by all possible configurations. This matters because every unnecessary coordinate carries an unnecessary constraint equation into the dynamics.

Prerequisites: Chapter 1; coordinates and the meaning of independent variables in P3.

Definition. The number of configuration degrees of freedom is the number of independent parameters required to specify the configuration locally. We denote it by $n$.

“Locally” means near an ordinary configuration where the chosen coordinates work. One angle describes a point on a circle, but a single Cartesian projection fails at some points. We will separate the physical freedom from the limitations of a particular coordinate system.

2. Counting independent holonomic constraints

Consider $N$ point particles in three dimensions subject to $k$ independent holonomic equality constraints. The constraints are assumed smooth and the configuration regular, so that each relation locally determines one positional variable from the others. For prescribed time-dependent constraints, this count is made at a fixed time.

Without restrictions, the configuration is specified by the list $(x_1,y_1,z_1,\ldots,x_N,y_N,z_N)$ containing $3N$ independent entries.

A plane constraint $z_1=0$ determines one entry. It leaves $x_1,y_1$ free. More generally, if one equation can locally determine one coordinate from the others, one independent choice is removed.

If the next relation determines another previously free coordinate, it removes another choice. Repeating this for $k$ independent relations leaves

For motion already described in a plane, the corresponding count is $n=2N-k$. Do not start with $2N$ and then subtract the planarity constraints again.

The count measures possible positions, not the number of force components or the number of integration constants. A free particle has three positional freedoms. A particle on a regular surface has two. A particle on a smooth curve has one. A particle fixed at a point has none.

3. Independence is more important than equation count

Suppose a planar particle satisfies

The second is twice the first. It removes no new choice. There is one independent constraint, and the particle has one degree of freedom, not zero.

Now suppose $z=0$ and $x^2+y^2=a^2$, with $a>0$, in three dimensions. The first restricts motion to a plane; the second restricts position within that plane to a circle. These are independent, leaving one freedom.

Remark (optional). Arrange the first derivatives of the constraint functions into rows. If no row is a linear combination of the others, the constraints are locally independent. This is a rank test on the constraint Jacobian. You do not need advanced matrix theory to see why duplicated rows cannot remove a new direction.

For $f_1=z$ and $f_2=x^2+y^2-a^2$, those rows are $(0,0,1)$ and $(2x,2y,0)$. On a circle with $a>0$, the second row is nonzero and points in a different direction, so the two restrictions are independent.

A cautionary singular example is $x^2+y^2=0$. It selects only the origin in the plane. Naively subtracting one from two gives the wrong answer because the constraint is not regular there: both first derivatives vanish. The correct configuration set is a point, with zero freedom.

4. What is configuration space?

Configuration space, denoted here by $\mathcal Q$, is the set of all admissible configurations. Each point of $\mathcal Q$ represents an entire physical arrangement of the system, not necessarily a location of one particle in ordinary space.

For one free particle, $\mathcal Q$ is ordinary three-dimensional space. For two free particles, a configuration needs six numbers, so $\mathcal Q$ is six-dimensional even though both particles live in three-dimensional physical space.

For a planar pendulum with a rigid rod and unrestricted rotation, the bob's position is specified by an angle $\theta$. Angles $\theta$ and $\theta+2\pi$ represent the same configuration. The configuration space is therefore a circle. A list such as $0\le\theta<2\pi$ is a coordinate representation with the endpoints identified; it is not an ordinary interval whose ends are unrelated.

For a double pendulum, two angles $(\theta_1,\theta_2)$ are needed. Each is periodic. The square $0\le\theta_1,\theta_2<2\pi$ represents this space with opposite edges identified. The resulting geometry is a torus. You do not need the theory of surfaces to use it: the essential point is that one point in this two-dimensional space specifies the positions of both bobs simultaneously.

For a particle on a sphere, configuration space is the spherical surface. Two angular coordinates work almost everywhere, but at the poles the azimuth becomes ambiguous. The particle still has two local directions of freedom; the coordinates, rather than the physical system, are failing there.

Configuration space contains positions/configurations. It does not contain velocities as additional axes. To specify the dynamical state of a second-order mechanical system, we also need velocities. That leads to state descriptions and later to phase space in Unit 3.

5. Time-dependent and velocity constraints

A changing pendulum length does not automatically add a freedom. If $a(t)$ is prescribed, then at any fixed time one angle still specifies the bob. The allowed circle changes with time, but the number of independent choices on that circle stays one.

If the length is an unknown dynamical variable, as in a bob on an extensible spring, we generally need both the length and angle. It then has two freedoms in a plane. Whether a quantity is known or unknown matters more than whether it appears in an equation.

For non-holonomic restrictions, distinguish the dimension of configuration space from the number of instantaneously allowed independent velocities. An upright steerable wheel has configuration variables $(x,y,\psi,\phi)$. Its two no-slip velocity relations restrict instantaneous motion, but do not reduce the configuration space to a two-dimensional surface. We will not apply $3N-k$ by blindly subtracting such velocity relations. Detailed non-holonomic dynamics lies beyond the main course.

An inequality also needs care. A particle inside a sphere has three freedoms in the interior. At the boundary its admissible velocities are one-sided unless contact is maintained as an equality. The inequality has not reduced the whole allowed region to a two-dimensional sphere.

6. Applications

Example 1 · Bead on a circular wire

For a bead confined to a fixed circle in the $xy$ plane, the plane and the fixed radius impose separate restrictions on its position.

Start with three Cartesian coordinates. The independent equations are $z=0$ and $x^2+y^2=a^2$. Thus $n=3-2=1$. Choose $\theta$ with $x=a\cos\theta$, $y=a\sin\theta$, $z=0$. Every value of $\theta$ supplies a point on the wire, and values differing by $2\pi$ describe the same point.

Example 2 · Configuration of a planar double pendulum

Consider two point bobs joined by two fixed-length rods in a plane, with the upper end attached to a fixed support. The independent motions can be counted from the Cartesian endpoint coordinates.

Work in the plane from the outset, so there are four coordinates $(x_1,y_1,x_2,y_2)$. The equations are $x_1^2+y_1^2=a^2$ and $(x_2-x_1)^2+(y_2-y_1)^2=b^2$. They constrain separate rod lengths and leave $n=4-2=2$. Two rod angles specify the configuration. Four initial numbers, two angles and two angular velocities, will normally be needed to select a motion; this does not mean four configuration degrees of freedom.

Example 3 · Translational and orientational freedom of a rigid dumbbell

Two point masses in three dimensions are joined by a rigid massless rod of length $a>0$. The freedom to translate the rod must be distinguished from the freedom to change its orientation.

The two endpoints require six coordinates. One equation fixes their separation: $|\mathbf r_2-\mathbf r_1|^2=a^2$. Hence $n=6-1=5$. Three coordinates locate the centre of mass; two angles specify the rod's direction. Rotation about the rod does not change the positions of two ideal point masses, so it adds no configuration freedom for this model. An extended non-collinear rigid body generally has six freedoms, including a physically distinct third orientation angle.

Example 4 · Prescribed support motion and a freely moving cart

A planar pendulum of fixed length $a$ has a support at horizontal position $X(t)$. The coordinate count depends on whether this motion is prescribed or the support is a cart whose motion must be determined.

With prescribed $X(t)$, only the bob's angle is unknown: one freedom. With a cart free to move horizontally, both $X$ and $\theta$ are unknown: two freedoms. The cart's vertical position is already fixed by its track. The same drawing can therefore represent different mechanical models.

7. Practice problems

See Chapter 2 hints and solutions after attempting these.

1. 2.1 How many configuration degrees of freedom does a free particle have in a plane?

2. 2.2 Does a prescribed time-dependent radius add a coordinate to a bead's circular position?

3. 2.3 Can two written constraint equations represent one independent restriction?

4. 2.4 Are $\theta=0$ and $\theta=2\pi$ different pendulum configurations?

5. 2.5 Define configuration space and distinguish it from physical space.

6. 2.6 Why do two degrees of freedom usually require four initial numbers for a second-order system?

7. 2.7 Why does azimuth fail as a unique coordinate at a sphere's pole?

8. 2.8 Explain the assumptions behind $n=3N-k$ and derive the count.

9. 2.9 Test independence of $x+y=0$ and $2x+2y=0$. Compare with $x+y=0$ and $x-y=0$.

10. 2.10 Four particles in three dimensions are subject to five independent regular holonomic constraints. Find $n$.

11. 2.11 A planar chain has three point bobs and three rigid rods, with one end fixed. Count the freedoms and name suitable coordinates.

12. 2.12 A particle lies on the intersection $z=2$ and $x^2+y^2=9$ in SI units. Give its freedom count and a parameterization.

13. 2.13 Explain configuration spaces of a free particle, planar pendulum, and double pendulum.

14. 2.14 Discuss independent, redundant, and singular constraint equations with explicit examples.

15. 2.15 Why does subtracting one equation from two coordinates fail for $x^2+y^2=0$?

16. 2.16 Explain why a steerable wheel's configuration-space dimension is not found by subtracting its two no-slip equations from its four coordinates.

8. Position, state, and initial data

Configuration space describes the positions of the system, not its velocities. Two pendulums at the same angle occupy the same point of configuration space even if one is moving left and the other right. A dynamical state therefore requires information in addition to a configuration.

For regular holonomic systems, each independent constraint removes one local positional choice. Redundant equations remove nothing further, and a singular equation may require direct examination of its solution set. The count should agree with an explicit choice of coordinates: one angle for a circular wire, two angles for a planar double pendulum, and five coordinates for a free rigid dumbbell formed from two point masses.