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We start out formulating the resulting equations of motion based on D’Alembert’s Principle for a multiparticle system.
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Lagrange's Equations from D’Alembert’s Principle - Worked Examples
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- 1 We start out formulating the resulting equations of motion based on D’Alembert’s Principle for a multiparticle system.
- 2 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 gen…
- 3 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.
- 4 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.
- 5 we can explicitly write the equations of motion from d'Alembert's principle in terms of the kinetic energy.
- 6 are called the Lagrange's equations (or Euler-Lagrange equations) written in generalized force form.
- 7 Example 3: We write the equations of motion of a simple spring-mass system using Lagrange's equations.
- 8 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.
- 9 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.
