Re: [Phys-l] Landau on Lagrangian

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding Bob S's response:

> David Bowman wrote:
> > We start with Hamilton's principle.  This essentially means we
> > have a Lagrangian formulation of the situation.
>
> I guess this statement illuminates the crux of my problem.  When
> you say that, at the start, you "have a Lagrangian formulation of
> the situation", are you saying that you already know
>
> 1) the details of the Lagrangian for a particular problem, or

Not at all. The details are narrowed in on *after* the rest of the
plausibility argument narrows down the possibilities.

> 2) that the general definition of the Lagrangian  is L = KE - PE,

Not at all.

> or
> 3) something else?

Nope.  It *only* means that there exists a Lagrangian that is some
function of the path between a pair of arbitrarily chosen endpoints
for which the accumulated action of the physical path between those
endpoints makes the value of the action functional stationary.  Now
since the action is the time integral of the Lagrangian between the
fixed endpoints, the Lagrangian at each moment of time is taken to
depend locally (in time) on the state of the system at each such
moment of time, so that it depends on, at most, the current
instantaneous time, the position, and a finite number of
deriviatives w.r.t. time of the position.

Other than this there is no a priori requirement on the form of
the Lagrangian.  The specific form of it is converged on later
by the *subsequent* extra requirements that the EOM are taken to
satisfy, i.e. homogeneity in time, homogeneity in space, isotropy
in space, invariance under Galilean boosts, and EOM not higher than
2nd order in derivatives w.r.t time.

> IE.; At the start, what is our a priori knowledge/definition of
> the Lagrangian?

See two paragraphs up.

> Bob Sciamanda

David Bowman