Lagrangian mechanics

Lagrangian mechanics is a way to do classical mechanics by choosing useful coordinates and comparing possible paths. Instead of starting with forces, start with a scalar function $L$ and the Action.

For many systems,

$$L(q,\dot{q},t)=T-V,$$

where $q_i$ are generalized coordinates, ${\dot{q}}_i$ are their velocities, $T$ is kinetic energy, and $V$ is Potential energy. The physical path obeys the Euler-Lagrange equation:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{q}}_i}\right)-\frac{\partial L}{\partial q_i}=0.$$

This equation is the variational version of Newton’s Laws of Motion. For a particle in one dimension with $L=\frac{1}{2}m{\dot{x}}^2-V(x)$, it becomes

$$m\ddot{x}=-\frac{dV}{dx},$$

which is Newton’s second law for a conservative force.

Why use it

• Constraints can be built into the coordinates.
• Symmetries are easier to see than in component force equations.
• It prepares the move to Hamiltonian mechanics, Phase space, and quantum path ideas.

Common pitfalls

• $L$ is not usually the total energy; for simple conservative systems $L=T-V$, while energy is $T+V$.
• Generalized coordinates do not have to be Cartesian distances.
• Adding a total time derivative to $L$ can leave the equations of motion unchanged.

Source trail: Susskind The Theoretical Minimum index; reference book: Classical Mechanics: The Theoretical Minimum by Leonard Susskind and George Hrabovsky.