• See Text section 7.7, p. 261
• Lagrange's equations give the same equations as Newton's equations
• Lagrange's equations use generalized coordinates and velocities
• There is a 'lagrangian' function L from which the equations of motion are derived
• The lagrangian L is the kinetic energy minus the potential energy ( L = T-V )
• The generalized momentum p is the partial of L with respect to the generalized velocity, holding coordinates constant.
• The equation of motion says that d/dt(p) equals the partial of the lagrangial with respect to the coordinate

Here is a simple example.

Suppose we have a particle traveling in two dimensions.

• Its KE is T = (m/2) ( x-dot^2 + y-dot^)
• If the particle is traveling subject to gravity hear the earth, the PE is V = mgy
• There are two generalized coordinates, p_x and p_y.
• p_x = partial of T wrt x-dot = m x-dot.
• p_y = m y-dot.
• The x-equation of motion is d/dt(p_x) = partial of L with respect to x =0. This says p_x is constant
• The y-equation of motion is d/dt(p_y) = partial of L with respect to y = -mg.
• Thus the y-equation of motion is d/dt( m y-dot) = -mg.
• This is what we already knew, that the y-acceleration is -g.

To test whether you understand this, try writing down the Lagrangian for a mass connected to a spring in one dimension.

• Click here to see if your lagrangian is correct.
• Click here to see if your equation of motion is correct.