Lagrangian mechanics

Lore & Background

Lagrangian mechanics describes a mechanical system as a pair (M, L) consisting of a configuration space M and a smooth function L within that space called a Lagrangian. For many systems, L = T − V, where T and V are the kinetic and potential energy of the system, respectively. The stationary action principle requires that the action functional of the system derived from L must remain at a stationary point throughout the time evolution of the system, allowing calculation of the equations of motion using Lagrange's equations.

Reader's Guide

Lagrangian mechanics adopts energy rather than force as its basic ingredient, leading to more abstract equations capable of tackling more complex problems. Lagrange's approach was to set up independent generalized coordinates for the position and speed of every object, allowing the writing down of a general form of the Lagrangian and summing this over all possible paths of motion to yield a formula for the 'action', which he minimized to give a generalized set of equations. This choice eliminates the need for constraint forces to enter into the resultant generalized system of equations, reducing the number of equations needed. The non-relativistic Lagrangian for a system of particles in the absence of an electromagnetic field is L = T − V, where T is total kinetic energy and V is potential energy. This formulation had crucial influence on other branches of physics, including relativity and quantum field theory.

Did You Know?

• The central quantity of Lagrangian mechanics is the Lagrangian, which for many systems equals total kinetic energy minus potential energy.
• Lagrangian mechanics eliminates the need for constraint forces to enter into the resultant generalized system of equations.
• Lagrange's approach had crucial influence on relativity and quantum field theory.