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*From*: "Bob Sciamanda" <treborsci@verizon.net>*Date*: Mon, 25 Jan 2010 08:32:13 -0500

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

_______________________________________________

Forum for Physics Educators

Phys-l@carnot.physics.buffalo.edu

https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l

**Follow-Ups**:**Re: [Phys-l] Landau on Lagrangian***From:*Hugh Haskell <haskellh@verizon.net>

**Re: [Phys-l] Landau on Lagrangian***From:*David Bowman <David_Bowman@georgetowncollege.edu>

**References**:**[Phys-l] Landau on Lagrangian***From:*Stefan Jeglinski <jeglin@4pi.com>

**Re: [Phys-l] Landau on Lagrangian***From:*"Bob Sciamanda" <treborsci@verizon.net>

**Re: [Phys-l] Landau on Lagrangian***From:*Stefan Jeglinski <jeglin@4pi.com>

**Re: [Phys-l] Landau on Lagrangian***From:*David Bowman <David_Bowman@georgetowncollege.edu>

**Re: [Phys-l] Landau on Lagrangian***From:*"Bob Sciamanda" <treborsci@verizon.net>

**Re: [Phys-l] Landau on Lagrangian***From:*David Bowman <David_Bowman@georgetowncollege.edu>

**Re: [Phys-l] Landau on Lagrangian***From:*"Bob Sciamanda" <treborsci@verizon.net>

**Re: [Phys-l] Landau on Lagrangian***From:*David Bowman <David_Bowman@georgetowncollege.edu>

**Re: [Phys-l] Landau on Lagrangian***From:*"Bob Sciamanda" <treborsci@verizon.net>

**Re: [Phys-l] Landau on Lagrangian***From:*David Bowman <David_Bowman@georgetowncollege.edu>

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