Re: [Phys-l] Landau on Lagrangian

["Bob Sciamanda" <treborsci@verizon.net>]

OK
The mathematical connection between the least action principle and the 
Lagrange equations which generate the "EOM" is a purely mathematical logical 
development, with no input from Newtonian mechanics to further specify the 
Lagrangian L ( indeed it is often applied to non mechanical, purely 
mathematical problems, eg geodesics).
You have added some mathematical requirements to be imposed upon L when this 
development is to be applied to mechanical motion; ie, to generate Newtonian 
dynamics.

These requirements can be accepted as also "purely" mathematical, with no 
overt input from the conclusions of Newtonian mechanics (this may be 
arguable).
But even If I grant you these requirements and accept that they imply that 
the mechanical L = L(v^2), you still have to show that this implies: 
KE(translation) is proportional to v^2.

This is a statement of Newtonian dynamics and cannot be deduced from "pure" 
mathematics. If it could be so deduced, Newtonian dynamics would be (at 
least in part) a metaphysical necessity and not just a falsifiable 
conclusion of empirical physics.  Indeed the very concept of KE is defined 
within Newtonian dynamics and is not even a part of the language of the 
purely mathematical calculus of variations.

Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
treborsci@verizon.net
http://mysite.verizon.net/res12merh/

--------------------------------------------------
From: "David Bowman" <David_Bowman@georgetowncollege.edu>
Sent: Sunday, January 24, 2010 11:27 PM
To: "Forum for Physics Educators" <phys-l@carnot.physics.buffalo.edu>
Subject: Re: [Phys-l] Landau on Lagrangian

> 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
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