Graduate Hamilton's Method with Lagrange Equation and Constraint

[Trying2Learn]

TL;DR

How do you apply constraint to calculus of variations

Good Morning

I am "comfortable" with formulating Hamilton's Principle with a Lagrangian (KE - PE), conducting the calculus of variations and obtaining the Euler Lagrange Equations. Advanced mathematical theory, is beyond me.

I also have a minimal understanding of using Lagrange multipliers.

I would like to combine both, say, for a closed loop on a robot linkage system

My issue is that I would like to see a SIMPLE example of how to apply a constraint (any constraint but formulated as an equality) to Hamilton's Principle

Could someone point me to a simple example?

I am a retiring mechanical engineer with a minimal mathematical skill set. My recent equations are now curiosity -- things I never learned.

 

[wrobel]

Perhaps the simplest example is to write formulas for the case when a particle moves in the vertical plane (x,y) in the standard gravity g and the constraint is ax+by=0