Lagrange Multipliers and Constraint Forces - Nonholonomic Constraints - Downhill Race Various Shapes

Explore Lagrange multipliers and constraint forces in nonholonomic systems through a comprehensive lecture on analytical dynamics. Delve into the derivation of generalized constraint forces using d'Alembert's principle and their connection to Newtonian forces. Examine practical examples including a two-mass rigid rod system with a wheel constraint, a roller racer with pivoted masses, and a disk rolling down a hill. Investigate how different mass distributions affect the downhill racing performance of round rigid bodies, highlighting the crucial role of moment of inertia. Gain insights into advanced concepts in dynamics, constraint equations, and their applications in various mechanical systems.

Derivation of the generalized forces of constraint using Lagrange multipliers in d'Alembert's principle.
how generalized forces are connected with the Newtonian forces and moments of constraint for bodies. .
The first example is 2 masses connected by a rigid rod, that is, a baton or dumbbell, with a 'wheel' underneath one of the masses, also called a knife-edge constraint or 'ice skate'. We solve for the Lagrange multiplier for this constraint as well as the Newtonian force of the constraint. .
We consider a pivoted-2 mass version with with wheel constraints called the roller racer (also known as a "Twistcar", "Plasma car", "Ezy Roller")..
We consider another example, of a rigid body, a disk, rolling down a hill. The constraint here is rolling without slipping, and we solve for the Lagrange multiplier, as well as the force and moment of constraint. The force is tangent to the ramp at the point of contact..
We consider different round rigid bodies with different mass distributions and attempt to .
predict which one will win a downhill race. It turns out the moment of inertia plays an important role..