Re: [Phys-l] Landau on Lagrangian

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding Bob S's questions:

> David wrote:
> > If we have any other nonlinear function of v^2 (not proportional
> > to the above linear one or with an affine offset) then
> > substituting v' + v_0 = v into the nonlinear function of v^2
> > will not result in the >same function of v'^2 plus a total time
> > derivative of some function of the dynamical variables, and thus
> > the resulting EOM will not be frame invariant under GTs.
>
> This does not seem to exclude a linear function of v (ex., L=mv). 
> Or do I misread your argument?

You are correct that invariance under Galilean boosts *by itself* does not exclude a linear function of v.  But that was *already* excluded by the isotropy of space with the argument that L had to be a function of *square speed* v^2.  Taking the sqrt of this, namely |v| is *not* a linear function of the velocity vector.  It is a nonlinear function of v^2.  Such a nonlinear function does not preserve the EOM for Galilean boosts.  Note that there is no function f of the dynamical state (r',v',t) such that:

|v| = sqrt(v^2) = sqrt(v'^2) + d/dt(f(r',v',t)) = |v'| + df/dt

when v = v' + v_0 for constant boost velocity v_0.

> P.S.
> Does EOM mean "energy of motion"?

Not for me.  I took it to mean *equation(s)* of motion.

> Bob Sciamanda

David Bowman