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Arnold Neumaier
#4
Feb10-06, 06:30 AM
P: n/a

Is the non-relativistic Lagrangian Galilean invariant?


Igor Khavkine wrote:
> Max wrote:
>
>>The relativistic classical/quantum Lagrangian is invariant under the
>>Lorentz boost. One would naturally assume that the classical
>>counterpart should be invariant under the Galilei boost:
>>T -> T'=T
>>X -> X'= X + uT
>>
>>However, for the non-relativistic classical Lagrangian (without extenal
>>force)
>>L = Ti(|Vi|)-Vij(|Xi-Xj|), (summation over i, j,)
>>surprise! surprise! the kinetic part is NOT invariant under the Galilei
>>boost, because |Vi| <> |Vi + u|.
>>


> A term that, when added to the Lagrangian, only change the action by a
> constant is called, in various contexts, a boundary term, a total
> derivative, or a total divergence.


The action for a translation invariant system must be translation
invariant, and then the total divergence does not change the action
(proper behavior at infinity assumed).

In general, the symmetry group of a conservative physical system is
that of the action, and not that of the Lagrangian.


Arnold Neumaier