Physic Labs

Analytical mechanics

Generalized coordinates and degrees of freedom

Illustrate generalized coordinates and holonomic constraints with a simple pendulum: describe position by the angle θ\theta alone instead of (x,y)(x, y). Verify x=lsin⁡θx = l\sin\theta, y=−lcos⁡θy = -l\cos\theta and the degree-of-freedom count f=3N−kf = 3N - k.

Undergraduate

Equipment

  • Virtual simple pendulum with rigid rod constraint
  • Sliders for angle θ and length l
  • Configuration manifold (constraint circle) display

Procedure

  1. Count the degrees of freedom

    Look at figure 1: the bob on a rigid rod of length ll rotates about the pivot. A planar position needs 2 coordinates (x,y)(x, y), but the constraint x2+y2=l2x^2 + y^2 = l^2 removes one — leaving exactly f=2−1=1f = 2 - 1 = 1 degree of freedom. Drag the "Angle θ" slider and confirm a single variable fixes the whole configuration.

  2. Relate Cartesian coordinates to θ

    With θ and l from the sliders, compute x=lsin⁡θx = l\sin\theta and y=−lcos⁡θy = -l\cos\theta, then compare with the bob position. Try θ=90°\theta = 90°: the bob is horizontal, x=lx = l, y=0y = 0. Try θ=−90°\theta = -90° for symmetry — the generalized angle is measured from the downward vertical.

  3. Explore the configuration manifold

    In figure 2, sweep θ from −170°-170° to +170°+170°: the configuration point traces a full circle — the configuration manifold is the circle S1S^1, not the plane. Change "Length l" and note the circle's radius changes but its topology stays the same. Conclusion: generalized coordinates live on the curved manifold defined by the constraints.

Simulation

Experiment history

The idea of describing motion with variables adapted to constraints grew out of analytical mechanics. In 1788 Joseph-Louis Lagrange published Mécanique analytique, rewriting Newtonian mechanics in terms of "generalized coordinates" qiq_i and the equations ddt∂L∂q˙i=∂L∂qi\frac{d}{dt}\frac{\partial L}{\partial \dot q_i} = \frac{\partial L}{\partial q_i} with L=T−UL = T - U. The book famously contains no figures — pure algebra throughout. In 1834–35 William Rowan Hamilton introduced the principle of least action and then the Hamiltonian form of the equations of motion on phase space (qi,pi)(q_i, p_i). The framework proved so general that quantum mechanics later borrowed it directly: Hamilton's canonical variables became operators, and Poisson brackets became commutators.

Related physicists

Isaac NewtonJoseph-Louis Lagrange

Related library topics

Generalized coordinates and degrees of freedom