The Lagrangian of a dynamical system is defined as the Kinetic Energy T minus the Potential Energy U, or
To acquire equations of motion for a dynamical system, one simply computes the Lagrange Equation,
for each generalized coordinate qi. When qi = ri (i.e. the generalized coordinates are simply the Cartesian coordinates), Lagrange's equations reduce to Newton's second law.
The n generalized coordinates are the coordinates associated with the n degrees of freedom of the system. For example,
- The rotation angles θ1 and θ2 in a double pendulum system
- The 3N spatial coordinates for N particles in three-dimensional space
- The 6N spatial plus rotational coordinates for N rigid bodies in three-dimensional space
- The θ and φ of a particle confined to the surface of a sphere