Constraints And Generalized Coordinates In Physics

A constraint is a restriction on the motion of a mechanical system, written as an equation that the positions of the particles must satisfy at all times. A bead threaded on a fixed circular wire cannot leave the wire; a pendulum bob tied to a string of length L must satisfy x² + y² = L²; the molecules of a gas trapped in a sealed container cannot cross the walls. Every constraint removes some freedom of motion, so instead of tracking all 3N coordinates of N particles, you first write down the k restriction equations and work with whatever freedom is left.

Generalized coordinates are any set of independent parameters q₁, q₂, …, q_s that completely specify every allowed position of the system. For the bead on the wire a single angle θ does the job; for a rigid body you might use three angles for orientation plus three coordinates for the centre of mass. The number s = 3N − k is called the number of degrees of freedom. Choosing good generalized coordinates is the gateway to Lagrangian mechanics: once every position is written as a function of the q's, the equations of motion follow almost automatically.

Key Concepts

• Constraint equation: a relation f(r₁, …, r_N, t) = 0 between particle positions and possibly time. A system with k independent constraints has 3N − k degrees of freedom.
• Holonomic constraints: can be written as equations involving only coordinates and time, such as a fixed-length string or a particle confined to a surface. They genuinely reduce the number of coordinates needed.
• Non-holonomic constraints: cannot be reduced to coordinate-only equations. A coin rolling without slipping is the classic example: the no-slip condition links velocities, and it cannot be integrated into a relation between positions alone.
• Scleronomous vs rheonomous: a constraint is scleronomous when it has no explicit time dependence (a fixed wire), and rheonomous when time appears explicitly (a bead on a wire that is itself being rotated by a motor).
• Configuration space: the abstract s-dimensional space whose points are the values (q₁, …, q_s). The motion of the entire system becomes a single curve traced by one point in this space.

Formulas

• Degrees of freedom: s = 3N − k, where N is the number of particles and k is the number of independent holonomic constraints.
• Constraint equations: fᵢ(r₁, …, r_N, t) = 0 for i = 1, …, k.
• Positions in generalized coordinates: rⱼ = rⱼ(q₁, …, q_s, t) for each particle j. Velocities follow from the chain rule: vⱼ = Σᵢ (∂rⱼ/∂qᵢ)q̇ᵢ + ∂rⱼ/∂t.
• Kinetic energy: T = ½ Σⱼ mⱼ vⱼ², which becomes a function T(q, q̇, t) after the positions are substituted.

Worked Example

Count the degrees of freedom of a double pendulum: two bobs joined by strings, swinging in a vertical plane. Without constraints there are 4 coordinates: (x₁, y₁) for bob 1 and (x₂, y₂) for bob 2. The two string-length constraints are x₁² + y₁² = L₁² and (x₂ − x₁)² + (y₂ − y₁)² = L₂², so k = 2 and s = 4 − 2 = 2 degrees of freedom. The natural generalized coordinates are the two swing angles θ₁ and θ₂, with positions x₁ = L₁ sinθ₁, y₁ = −L₁ cosθ₁, x₂ = x₁ + L₂ sinθ₂, y₂ = y₁ − L₂ cosθ₂ (y measured upward from the support). Take L₁ = L₂ = 1 m, θ₁ = 30°, θ₂ = 45°: then x₁ = 0.500 m, y₁ = −0.866 m, x₂ = 0.500 + 0.707 = 1.207 m, y₂ = −0.866 − 0.707 = −1.573 m. As a check, the second string length is √[(0.707)² + (−0.707)²] = 1.000 m, exactly L₂, confirming the coordinates satisfy the constraint.

Common Misconceptions

1. "Degrees of freedom equals the number of coordinates I started with." No — every independent constraint equation removes one degree of freedom. N free particles have 3N, but freeze them into a rigid body and only 6 remain.
2. "All constraints are holonomic." Rolling without slipping is non-holonomic: the condition v = ωR relates velocities and cannot be integrated into a position-only equation, so it does not reduce the coordinate count in the same clean way.
3. "Rheonomous means the constraint depends on velocity." It means the constraint equation contains time explicitly, like a pendulum whose support point is being shaken up and down by a motor.

Real-World Applications

• Robot arms: each joint angle is a generalized coordinate; the whole arm's motion is planned in this small coordinate set rather than in the xyz position of every link.
• Vehicle dynamics: suspension linkages are modelled as constrained multi-body systems, with wheel travel and steering angle as generalized coordinates.
• Molecular simulations: bond lengths and bond angles of molecules are treated as generalized coordinates or constraints to simulate large proteins efficiently.
• Space tethers: satellites joined by long cables are analysed with the cable-length constraint built directly into the coordinates.

Constraints turn a crowd of particles into a system with a manageable number of independent motions. Write the restriction equations, count s = 3N − k, pick the simplest generalized coordinates you can find, and the path to the Lagrangian — and to the equations of motion — is wide open.