Lagrangian Formulation

Generalized coordinates, Lagrangian function, and Euler-Lagrange equations for constrained systems.

General idea of the topic

The Lagrangian formulation summarizes the dynamics of a system in a single function, L = T − U, the difference between the kinetic and potential energy written in generalized coordinates. Lagrange's equations give one equation per coordinate, without ideal constraint forces.

The topic writes T and U in suitable coordinates, obtains the equations of motion with d/dt(∂L/∂q̇) − ∂L/∂q = 0 and presents, as an introduction, Noether's theorem: every continuous symmetry of the Lagrangian implies a conserved quantity. Before the equations the variational principle of the action is justified, and at the end the Routhian is shown to reduce the problem by eliminating the cyclic coordinates.

What you will truly understand

By the end you will be able to write the Lagrangian of a system, derive its equations of motion and recognize cyclic coordinates, whose conjugate momentum is conserved.

You will also be able to relate translational, rotational and time symmetries to linear momentum, angular momentum and energy.

Recommended learning route

1. Kinetic and Potential Energy: expression of T and U in generalized coordinates, including constraints.
2. Variational Principle and Action: the real motion makes the action stationary, from which the Euler-Lagrange equations are deduced.
3. Lagrange's Equations: application of the Euler-Lagrange equations to concrete systems.
4. Symmetries and Conservation Laws (Noether Intro): symmetry of the Lagrangian and conserved quantity: a first idea of Noether's theorem.
5. Routhian and Cyclic Coordinates: elimination of cyclic coordinates through their conserved momenta.

Physical criterion you must acquire

The essential criterion is to write T and U before differentiating, and check the result with a known case, such as the simple pendulum, which must give θ̈ = −(g/l) sin θ. If a coordinate does not appear in L, its conjugate momentum is a constant of motion.

Connection with real problems

• articulated mechanisms and robots.
• coupled pendulums.
• motion of particles in fields.
• classical field theory.

Final goal of the topic

The final goal is to obtain the equations of motion of a system from its Lagrangian and link symmetries with conserved quantities.