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Re: [Phys-l] Landau on Lagrangian

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)

From: "David Bowman" <>
Sent: Sunday, January 24, 2010 11:27 PM
To: "Forum for Physics Educators" <>
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.

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
Forum for Physics Educators