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*From*: David Bowman <David_Bowman@georgetowncollege.edu>*Date*: Sat, 23 Jan 2010 15:14:21 -0500

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

order. To be more clear about this let's repeat the jist of the

whole argument including all the invariances.

We start with Hamilton's principle. This essentially means we have

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

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

**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>

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