A Hamilton's Method with Lagrange Equation and Constraint

AI Thread Summary
The discussion focuses on applying Hamilton's Principle using Lagrangian mechanics, specifically in the context of a closed loop robot linkage system. The user is comfortable with formulating Hamilton's Principle and deriving Euler Lagrange Equations but seeks a simple example of incorporating constraints using Lagrange multipliers. They request guidance on applying an equality constraint to Hamilton's Principle, particularly for a particle moving in a vertical plane under gravity with a specific constraint equation. The user expresses their background as a retiring mechanical engineer with limited mathematical skills, highlighting their desire to understand these concepts better. A straightforward example would enhance their understanding of combining these advanced mathematical theories.
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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.
 
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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
 
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