Constraints, Symmetries, and Conservation Laws

Constraints and d'Alembert's principle

A holonomic constraint is an equation among the coordinates, g(\mathbf{q},t)=0 — a bead confined to a wire, a rigid rod of fixed length. A nonholonomic constraint cannot be written that way (a rolling coin's velocity condition, an inequality), and needs more care. Holonomic constraints simply reduce the coordinate count.

Why can we ignore constraint forces at all? Because of d'Alembert's principle: constraint forces are perpendicular to the allowed (virtual) displacements, so they do zero virtual work. This is the foundation the whole Lagrangian edifice rests on.

\sum_i \left(\mathbf{F}_i - \dot{\mathbf{p}}_i\right)\cdot \delta \mathbf{r}_i = 0

D'Alembert's principle: the applied forces and inertial terms do no net virtual work — the constraint forces have already cancelled.

When you DO want the constraint force

Sometimes the constraint force is the answer — you want to know when the bead flies off the wire, or the tension in a cable. Then keep the redundant coordinate, impose the constraint g_k(\mathbf{q})=0 separately, and introduce a Lagrange multiplier \lambda_k for it.

\frac{d}{dt}\frac{\partial L}{\partial \dot q_j} - \frac{\partial L}{\partial q_j} = \sum_k \lambda_k\,\frac{\partial g_k}{\partial q_j}

The multipliers on the right are the generalized constraint forces.

Cyclic coordinates and conserved momenta

Here Lagrangian mechanics reveals its gift. If a coordinate q_j does not appear in L (only its velocity \dot q_j does), it is called cyclic or ignorable. Then \partial L/\partial q_j=0, and the Euler-Lagrange equation collapses to \dot p_j=0: the conjugate momentum is conserved. A conservation law, read straight off the Lagrangian.

\frac{\partial L}{\partial q_j}=0 \;\Longrightarrow\; \frac{dp_j}{dt}=0

A coordinate absent from L has a conserved conjugate momentum.

The examples are exactly the familiar conservation laws. If L does not depend on a Cartesian coordinate x (space is uniform), linear momentum p_x is conserved. If L does not depend on an angle \phi (space is isotropic), angular momentum p_\phi is conserved. The invisibility of a coordinate is a symmetry.

Noether's theorem

The cyclic-coordinate observation is a special case of one of the deepest results in physics — Noether's theorem: every continuous symmetry of the action corresponds to a conserved quantity, whether or not any coordinate happens to be cyclic in your chosen variables.

\frac{d}{dt}\!\left(\sum_i \frac{\partial L}{\partial \dot q_i}\,\frac{\partial q_i}{\partial \varepsilon}\right) = 0

For a symmetry parametrized by \varepsilon that leaves the action invariant, this Noether charge is conserved.

The three great conservation laws are now unified as three symmetries. Invariance under time translation gives conservation of energy; under space translation, momentum; under rotation, angular momentum. Conservation laws are not accidents — they are the shadows that symmetries cast.