Is the Emmy Noether Theorem Misunderstood in Space Translation Dynamics?

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The discussion centers on the implications of the Emmy Noether theorem in the context of translation symmetries in Lagrangian mechanics. It highlights that when analyzing Lagrangians, one must consider the possibility that a change in the Lagrangian can be represented as a total time derivative of a function, which may not depend on generalized velocities. The conversation emphasizes that for a generalized translation to be a symmetry, the Lagrangian must not depend on the corresponding generalized coordinate. The participants also explore the relationship between conserved quantities and the canonical momentum derived from the Euler-Lagrange equation. Ultimately, the question remains whether the equality between the partial derivative of the Lagrangian and the total time derivative can coexist without contradiction.
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If we watch some translation in space.

L(q_i+\delta q_i,\dot{q}_i,t)=L(q_i,\dot{q}_i,t)+\frac{\partial L}{\partial q_i}\delta q_i+...

and we say then
\frac{\partial L}{\partial q_i}=0

But we know that lagrangians L and L'=L+\frac{df}{dt} are equivalent. How we know that \frac{\partial L}{\partial q_i}\delta q_i isn't time derivative of some function f?
 
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For more general symmetries than translation symmetries, you need to take into account the possibility that the Lagrangian changes by the total time derivative of a function that is only a function of the generalized coordinates (and perhaps explicitly on time) but not the generalized velocities. An example is invariance under Galileo transformations in Newtonian mechanics (Galileo boosts) or Lorentz boosts in Special Relativity, which both lead to the constant velocity of the center of mass of a closed system of point particles.
 
vanhees71 said:
For more general symmetries than translation symmetries, you need to take into account the possibility that the Lagrangian changes by the total time derivative of a function that is only a function of the generalized coordinates (and perhaps explicitly on time) but not the generalized velocities. An example is invariance under Galileo transformations in Newtonian mechanics (Galileo boosts) or Lorentz boosts in Special Relativity, which both lead to the constant velocity of the center of mass of a closed system of point particles.

I don't want more general symmetries than translation symmetries. I asked my question about invariance under translation symmetries. Can you answered the question about this problem which I asked?
 
But for this problem, you've already given the answer yourself: A generalized translation is a symmetry if the Lagrangian doesn't depend on the corresponding generalized coordinate q_1, and then from the Euler-Lagrange equation, you get

\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial L}{\partial \dot{q}_1} = \frac{\partial L}{\partial q_1}=0,

which means that the conserved quantity is given by the canonical momentum of this variable, i.e.

p_i=\frac{\partial L}{\partial \dot{q}_1}.
 
I think we don't understand each other. My question is. If two lagrangians L'=L+\frac{df}{dt} and L gives us same dynamics. Why can't be that

\frac{\partial L}{\partial q}\delta q=\frac{df}{dt}?
 
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