Hamilton's Method with Lagrange Equation and Constraint

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SUMMARY

This discussion focuses on applying Hamilton's Method using the Lagrange Equation with constraints, specifically for a closed loop in a robot linkage system. The user seeks a straightforward example of incorporating equality constraints into Hamilton's Principle, particularly in the context of a particle moving in a vertical plane under gravity. The constraint mentioned is formulated as ax + by = 0. The discussion highlights the user's background as a retiring mechanical engineer with limited mathematical skills, emphasizing the need for simplicity in examples.

PREREQUISITES
  • Understanding of Hamilton's Principle and Lagrangian mechanics
  • Familiarity with the calculus of variations
  • Basic knowledge of Lagrange multipliers
  • Concept of constraints in mechanical systems
NEXT STEPS
  • Research "Hamilton's Principle examples with constraints" for practical applications
  • Study "Lagrange multipliers in mechanical systems" for deeper insights
  • Explore "Euler Lagrange Equations" to understand their derivation and application
  • Investigate "robot linkage systems" to see real-world implementations of these concepts
USEFUL FOR

This discussion is beneficial for mechanical engineers, physics students, and anyone interested in applying advanced mechanics to robotic systems, particularly those with a focus on constraints and Hamiltonian dynamics.

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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.
 
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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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