A mass of point performs an effectively 1-dimensional motion in the radial coordinate. If we use the conservation of angular momentum, the centrifugal potential should be added to the original one.
The equation of motion can be obtained also from the Lagrangian. if we substitute, however, the conserved angular momentum herein then the centrifugal potential arises with the opposite sign. So if we naively apply the Euler-Lagrange equation then the centrifugal force appears with the wrong sign in the equations of motion.
I don't know how to resolve this "paradox".
Answer
The general issue is that you cannot plug your equations of motion into the Lagrangian and naively expect to get the same equations of motion back out again. Why not? Let us look at your specific example.
For the usual story we start with L=12m(˙r2+r2˙θ2)−V(r).
Now, you want to plug ℓ back into the Lagrangian. If we do that we have L=12m(˙r2+ℓ2m2r2)−V(r).
Recall that when we call ℓ a conserved quantity we mean it is a constant in time, that is ˙ℓ=0. Explicitly writing out the Euler-Lagrange equations we have ddt[(∂L∂˙r)r,θ,˙θ]−(∂L∂r)˙r,θ,˙θ=0.
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