Equilibrium in Newtonian Mechanics: What is the Connection to Lagrangians?

AI Thread Summary
The discussion centers on the relationship between the Lagrangian of a mechanical system and the conditions for equilibrium. It questions whether the condition ∂L/∂qi = 0 indicates equilibrium configurations, drawing parallels to Newtonian mechanics where the equilibrium condition is ∂V/∂x = 0. Participants express confusion about how to connect these concepts, highlighting the need for clarity on the role of the Lagrangian in determining equilibrium. The inquiry reflects a deeper exploration of the principles governing mechanical systems and their stability. Understanding this connection is crucial for applying Lagrangian mechanics effectively.
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So let's say we have a mechanical system described by some Lagrangian L=L(q_i,\dot{q}_i), where the qi's are the generalized coordinates of the system. Does the condition
\frac{\partial L}{\partial q_i}=0
give the equilibrium configurations of the system? Intuitively it seems so, but I can't prove it.
 
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What is the condition for equilibrium in Newtonian mechanics?
 
Jorriss said:
What is the condition for equilibrium in Newtonian mechanics?
\frac{\partial V}{\partial x}=0./tex]. I'm having trouble connecting this to Lagrangians though...
 
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