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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.
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
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