What Is the Force of Constraint Using Lagrange Multiplier?

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SUMMARY

The discussion centers on calculating the force of constraint using the Lagrange multiplier method in a system of masses along the z-axis, where the potential energy is defined as V = (1/2)kx². The user derived the Lagrangian L = T - V + λf and found the equation of motion mẋ - λ = 0, concluding that since z is not moving (ẋ = 0), λ must equal 0. This raises a question about the validity of the problem, as three equations of motion should be expected for the coordinates (x, y, z).

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Homework Statement


A combination of masses along the z-axis is separated by a distance 'a' with middle mass at origin. The potential is
[tex]V = \frac{1}{2}kx^2[/tex].
What is the force of constraint using Lagrange multiplier?

Homework Equations


[tex]L = T - V + \lambda f[/tex]


The Attempt at a Solution


I found L and calculated the lagrange eqn of motions but still I am getting
[tex]m \ddot{z} - \lambda = 0[/tex]

z is not moving, so [tex]\ddot{z} = 0[/tex].
Based on this [tex]\lambda = 0[/tex]
Is the question wrong?
 
Last edited:
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You should get 3 equations of motion, one for each coordinate (x,y,z)
 

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