Lagrange's Equations from D’Alembert’s Principle - Worked Examples

We start out formulating the resulting equations of motion based on D’Alembert’s Principle for a multiparticle system..
Example 1: Baton (or dumbbell or broom, or two masses connected by a rod) sliding down a wall using d'Alembert's principle. This is an N=2 particle system with only 1 degree of freedom. For our generalized coordinate, we have some freedom, and we choose to use the angle that the rod makes with wall. .
Example 2: Cart-pendulum system (N=2 particles and 2 degrees of freedom). After setting up the problem, we use Mathematica to find the equations of motion, and solve them. .
When we put it in matrix form, we see the structure of the equations of motion, including a symmetric, invertible n x n mass matrix, which is connected to the kinetic energy..
we can explicitly write the equations of motion from d'Alembert's principle in terms of the kinetic energy. .
are called the Lagrange's equations (or Euler-Lagrange equations) written in generalized force form..
Example 3: We write the equations of motion of a simple spring-mass system using Lagrange's equations..
Example 1b: We re-visit the sliding baton problem using the Lagrange's equation approach, which gives the same answer for the equations of motion, but is even more direct..
Example 2b: We re-visit the cart-pendulum problem using the Lagrange's equation approach, which gives the same answer for the equations of motion..