1. Hamiltonian principle and Lagrange’s equations (review with new notations)
If coordinates q are independent, then the above equation should be true for any variations δq, and Lagrane’s equations follow.
In the presence of constrains, coordinates q and variations δq are dependent.
Holonomic constrain:
To proceed we can use Lagrange multipliers.

2. Lagrange multipliers and constraint forces
For 1≤n ≤ N there are N-1 independent variables δq plus λ
Several constraint forces:

3. Solution of equations (1-3):
Example: x ϕ θ r
(1)
(2)
(3)
Solution of equations (1-3):