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Re: [Phys-l] Landau on Lagrangian

David Bowman wrote:
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
2) that the general definition of the Lagrangian is L = KE - PE, or
3) something else?

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

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

--------------------------------------------------
From: "David Bowman" <David_Bowman@georgetowncollege.edu>
Sent: Saturday, January 23, 2010 3:14 PM
To: "Forum for Physics Educators" <phys-l@carnot.physics.buffalo.edu>
Subject: Re: [Phys-l] Landau on Lagrangian

Regarding Bob S's claim of circular reasoning:

David Bowman wrote:

You are correct that invariance under Galilean boosts *by itself*
does not exclude a linear function of v. But that was *already*
excluded by the isotropy of space with the argument that L had
to be a function of *square speed* v^2.

Then I need to repeat my original objection that this is a
circular argument. AFAIK the development of the Lagrange equation
uses KE(translation) = (1/2)mv^2 as an a prioi, given by Newton's
laws.

Bob Sciamanda

I do not see how you can claim the argument is completely circular.
I can see how it is a 'little circular' in that we need to assume
the allowed order of the EOM DEs. But I don't see any initial
assumption of the *form* of the resulting DEs other than their
whole argument including all the invariances.

a Lagrangian formulation of the situation.

We note that time seems to be homogeneous. Forcing our description
of nature to respect that symmetry means that the Lagrangian does
not depend explicitly on the time parameter.

We note that space seems to be homogeneous. Forcing our description
of nature to respect that symmetry means that the Lagrangian does
not depend explicitly on any of the position vector components.

We note that space seems to be isotropic. Forcing our description
of nature to respect that symmetry means that the Lagrangian does
not depend explicitly on the direction of any remaining vectorial
dynamical variables. This means that the Lagrangian can only
depend explicitly on the square magnitudes of the remaining
dynamical variables or on the mutual dot products (or other higher
order rotational scalar constructions made from more than two of
the remaining vector variables) between different ones if more
than one dynamical vector quantity is present in the Lagrangian.

We realize that if L depended explicitly on the acceleration, the
jerk or on any other higher derivatives of the position then the
resulting EOM would be of order higher than 2nd order in time. We
assume that we want our EOM to be of order no higher than 2nd order
in time since that is what nature seems to use. *This* is the only
main circular part that I can see. I think we could probably
dispense with this assumption if we imposed the symplectic geometry
of a Hamiltonian formulation as a pre-condition derived from
quantum considerations (a la a Feynmanesque functional integral
formulation of QM taken to the classical limit).

In any event, once we decide we don't want our Lagrangian EOM to be
higher than 2nd order then we see that the acceleration, jerk, etc.
are not allowable ingedients in the form of the Lagrangian. This
means that only the velocity vector is the only remaining dynamical
quantity that can be used to build the Lagrangian. With only the
velocity remaining the isotropy of space required that L depend
only on the magnitude of the speed (L = L(v^2)).

We note that Galilean boost invariance seems to be a symmetry of
nature (at least at speeds slow compared to c). Forcing our
description of nature to respect that symmetry means that the
Lagrangian's dependence on v^2 must be linear in v^2, or linear
with an affine offset (the latter having no effect on the EOM
anyway). Since an affine constant term in L has no effect we can
safely drop it. We are then left with a Lagrangian that is
linear in v^2.

We define the mass to be twice the proportionality constant
multiplying the v^2 factor in the Lagrangian. If we add to
the stationarity of Hamilton's Principle the requirement that
the physical path be a local *minimum* of the action we see that
the mass factor must be a positive constant as well.

So the upshot is that the only seriously circular part is in
assuming the order of the DEs coming out of E-L EOM can't be higher
order than second (since N2 is known to be 2nd order in the time
derivative of the position). I think if we can argue for a prior
symplectic Hamiltonian formulation, then the resulting Lagrangian
formulation won't have derivatives higher than first order in the
Lagrangian anyway, and the Lagrangian EOM will not be higher than
second order automatically. But assuming a symplectic geometry for
the dynamics is a fairly strong assumption that I can't see as
being naturally obvious, unless one invokes a prior quantum
description (which could be considered another route of circularity
if that quantum description relies on a canonical formulation for
its own derivation).

David Bowman
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