Is the nonrelativistic 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 nonrelativistic classical Lagrangian (without extenal
>>force)
>>L = Ti(Vi)Vij(XiXj), (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
