Building and Solving Lagrangians

The recipe

Every Lagrangian problem follows the same five moves. Master this loop and you can attack systems that would be a nightmare of coupled vector equations in the Newtonian picture.

Choose generalized coordinates — as many as there are degrees of freedom, matched to the constraints.
Write the kinetic energy T and potential energy V in those coordinates.
Form the Lagrangian L=T-V.
Apply the Euler-Lagrange equation once per coordinate.
Solve or interpret the resulting equations of motion.

Worked example: the simple pendulum

A bob of mass m on a massless rod of length \ell. Coordinate: the angle \theta from vertical. Its speed is \ell\dot\theta, and its height, measured from the pivot, is -\ell\cos\theta. So the kinetic and potential energies are quick to write.

T = \tfrac12 m\ell^2\dot\theta^2, \quad V = -mg\ell\cos\theta \;\Rightarrow\; L = \tfrac12 m\ell^2\dot\theta^2 + mg\ell\cos\theta

The pendulum's Lagrangian in the single coordinate \theta.

Now the Euler-Lagrange machine: \partial L/\partial\dot\theta = m\ell^2\dot\theta, whose time derivative is m\ell^2\ddot\theta; and \partial L/\partial\theta = -mg\ell\sin\theta. Setting the difference to zero gives the equation of motion.

\ddot\theta = -\frac{g}{\ell}\sin\theta \;\xrightarrow{\;\theta\ll 1\;}\; \ddot\theta \approx -\frac{g}{\ell}\,\theta,\quad \omega=\sqrt{g/\ell}

The exact pendulum equation, and its small-angle limit — simple harmonic motion.

Worked example: the Atwood machine

Two masses m_1,m_2 hang from a light string over a frictionless pulley. The string's inextensibility is one constraint, so a single coordinate suffices: let x be how far m_1 has descended, so m_2 has risen by the same x. Both move with speed \dot x.

L = \tfrac12(m_1+m_2)\dot x^2 + (m_1-m_2)g\,x

Kinetic energy of both masses, minus the (sign-tracked) gravitational potential.

One coordinate, one Euler-Lagrange equation: \tfrac{d}{dt}(m_1+m_2)\dot x = (m_1-m_2)g. The acceleration follows at once — and we never introduced the string tension that the Newtonian approach forces you to carry through two coupled equations.

a = \ddot x = \frac{(m_1-m_2)\,g}{m_1+m_2}

The Atwood acceleration, derived without ever mentioning the tension.

Generalized momentum and the energy function

For each coordinate the quantity conjugate to it is the generalized momentum p_i=\partial L/\partial\dot q_i. Sometimes it is ordinary mv; for an angular coordinate it is angular momentum; in a magnetic field it acquires an extra piece. It is this generalized p, not mv, that the theory tracks.

p_i = \frac{\partial L}{\partial \dot q_i}

The generalized (canonical) momentum conjugate to coordinate q_i.

There is also a natural energy-like quantity, the energy function (or Jacobi integral), h=\sum_i\dot q_i\,p_i-L. When L has no explicit time dependence, h is conserved; and when the coordinates are time-independent too, h equals the ordinary total energy T+V. This object is the seed of Hamiltonian mechanics.

h = \sum_i \dot q_i\,\frac{\partial L}{\partial \dot q_i} - L

The energy function — conserved when L has no explicit time dependence.