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TobyC
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The biggest problem I'm having with Noether's theorem is that I can't seem to find it stated precisely enough anywhere. The standard statement seems to be just that 'for any continuous symmetry of a system there is a corresponding conserved quantity'. I think I understand this fine when the symmetry is a symmetry of the Lagrangian itself, since then the theorem is relatively easy to prove. Where I have difficulty is when the Lagrangian changes under a transformation but the equations of motion do not, in which case I think Noether's theorem is still supposed to apply but it doesn't seem to in this example:
If you take a 1D harmonic oscillator, the Lagrangian is: [itex]L = (1/2)m\dot{q}^{2} - (1/2)kq^{2}[/itex]
Now consider the transformation of multiplying q by a constant. If you think about SHM motion it's clear that this is a symmetry, if I'm interpreting what a symmetry means correctly, because it's a transformation that maps solutions of the equation of motion to other solutions of the equation of motion. However, the Lagrangian is not invariant under this transformation.
If you apply an infinitesimal version of this transformation to a given solution (q' = (1 + ε)q), the new Lagrangian along the path as a function of time will be related to the old by:
L' = L + 2εL
If Noether's theorem applied, and I am applying it correctly, the conserved quantity associated with this symmetry would then be:
[itex]q(\partial L/\partial\dot{q}) - 2I[/itex]
Where dI/dt = L.
The trouble is L is not the total time derivative of any function, if it were then the action along any path would depend only on the end points. So there doesn't seem to be any conserved quantity associated with this symmetry.
I'd appreciate it if anyone could point out what I'm doing wrong here. Maybe under the rigorous statement of Noether's theorem it isn't supposed to apply in this case, maybe I'm misunderstanding what a symmetry is, and maybe it does apply but I'm applying it wrongly. I'd like to get this resolved anyway. Thanks in advance.
If you take a 1D harmonic oscillator, the Lagrangian is: [itex]L = (1/2)m\dot{q}^{2} - (1/2)kq^{2}[/itex]
Now consider the transformation of multiplying q by a constant. If you think about SHM motion it's clear that this is a symmetry, if I'm interpreting what a symmetry means correctly, because it's a transformation that maps solutions of the equation of motion to other solutions of the equation of motion. However, the Lagrangian is not invariant under this transformation.
If you apply an infinitesimal version of this transformation to a given solution (q' = (1 + ε)q), the new Lagrangian along the path as a function of time will be related to the old by:
L' = L + 2εL
If Noether's theorem applied, and I am applying it correctly, the conserved quantity associated with this symmetry would then be:
[itex]q(\partial L/\partial\dot{q}) - 2I[/itex]
Where dI/dt = L.
The trouble is L is not the total time derivative of any function, if it were then the action along any path would depend only on the end points. So there doesn't seem to be any conserved quantity associated with this symmetry.
I'd appreciate it if anyone could point out what I'm doing wrong here. Maybe under the rigorous statement of Noether's theorem it isn't supposed to apply in this case, maybe I'm misunderstanding what a symmetry is, and maybe it does apply but I'm applying it wrongly. I'd like to get this resolved anyway. Thanks in advance.
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