Ian D. Lawrie, in his "A Unified Grand Tour of Theoretical Physics", Sections 3.1 and 3.2, goes through the arguments in a much more pleasing fashion (in my opinion) than L&L. His steps are:
Consider a single, isolated particle. Its EOM can depend only on the
structure of spacetime itself: any symmetry of this structure must also
be a symmetry of the EOM. In Galilean spactime, there are three
"obvious" symmetries.
(i) The first is invariance under time-translations. L cannot depend explicitly on time: L(x, xdot, t+t0) = L(x, xdot, t), so the time argument can be omitted entirely;
(ii) Invariance under spatial translations. A translation of your coordinate origin can't chance the spatial metric tensor: L(x+x0, xdot) = L(x, xdot). Similarly to (i), the spatial dependence can be omitted from L;
(iii) Invariance under rotations. The metric tensor is unchanged by a spatial rotation, which means L cannot depend on the direction of xdot, but possibly upon its magnitude, |xdot| = (xdot.xdot)^(1/2).
Then he says:
"In order to tie down the Lagrangian completely, we have to assume a further symmetry which does not follow directly from the spacetime structure:"
(iv) Invariance under Galilean transformations. The EOM has the same form in two frames which have a constant relative velocity v. He goes from the Euler-Lagrange equations[footnote] to the EOM:
d/dt ( xdot dL/dX) = xdotdot*dL/dX + xdot*(xdotdot.xdot)*(d^2)L/dX^2 = 0,
where X = 1/2*|xdot|^2, and then makes a Galilean transformation.
To keep the EOM unchanged in form, the form of L must be dL/dX = constant, so that (d^2)L/dX^2 = 0. The constant dL/dX is the particle mass, and the Lagrangian is L = m*xdot^2 / 2, which is just the kinetic energy.
[1] my footnote: Lawrie originally argues that L is obtained as the integrand in the action integral, the stationarity of which he says "is equivalent to Newton's second law" (I have no problem with this). He then goes on to derive the Euler-Lagrange equations (fine). But his introduction to the invariance arguments is "Suppose, however, we do not assume Newton's law to be valid? Can we deduce what the Lagrangian is on a priori grounds?" To ME, o best beloved, he's used Newton's 2nd to argue for L, derived E-L's equations, and then said "what if we drop Newton's 2nd and see what forms L must take?" This is somewhat circular.
To be fair, he does say "The first assumption is that the state of the system is uniquely specified by giving the positions and velocities, and it is more of less equivalent to assuming that the equations of motion will be of second order in the time derivatives. I do not know of any justification for this beyond the fact that it works."
/************************************
Down with categorical imperative!
flutzpah@yahoo.com
************************************/
________________________________
From: Stefan Jeglinski <jeglin@4pi.com>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Wed, January 20, 2010 9:44:36 PM
Subject: [Phys-l] Landau on Lagrangian
I have a thin eclectic volume, "Mechanics," by Landau and Lifshitz
(Course of Theoretical Physics, v1).
In the discussion of the Lagrangian and equations of motion, the
concept of the inertial frame is introduced; and then:
"... a frame of reference can always be chosen in which space is
homogeneous and isotropic and time is homogeneous. This is called an
inertial frame... We can now draw some immediate inferences
concerning the form of L of a free particle in an inertial frame. The
homogeneity of space and time implies that L cannot contain
explicitly either the radius vector r of the particle or time t,
i.e., L must be a function of velocity v only. Since space is
isotropic, L must also be independent of the direction of v, and is
therefore a function only of its magnitude, i.e. v^2."
Ergo: L = L(v^2).
This is the closest I've seen to an argument from first principles
for the form of L of a free particle, that is, why KE is proportional
to v^2. I like the idea pedagogically of an argument that doesn't
resort to blatant/circular assertions about the form of KE. However,
Landau/Lifshitz do not expound further either, beyond these few
words. I think I understand the second part, essentially that the
isotropy of space cannot support a preferred direction, but it's
unclear to me how the homogeneity of space and time lead to L being a
function of v alone. I'm guessing that spatial homogeneity implies
that whatever happens in "this cubic meter" must happen identically
in "the next cubic meter," but the a priori necessity of temporal
homogeneity is lost on me - obviously, conserved quantities are
dependent on constancy over time, but his argument seems to precede
his discussion of constants of motion.