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

Explore Lagrange's equations derived from D'Alembert's Principle in this comprehensive lecture from a course on analytical dynamics. Delve into the formulation of equations of motion for multiparticle systems using generalized coordinates. Work through detailed examples including a baton sliding down a wall, a cart-pendulum system, and a spring-mass system. Discover how to express equations of motion in terms of kinetic energy and understand the structure of these equations in matrix form. Compare the D'Alembert's Principle approach with the Lagrange's equation method for solving complex dynamics problems. Gain insights into the connections between kinetic energy, generalized forces, and degrees of freedom in mechanical systems.

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..