[Phys-L] Re: Energy is primary and fundamental?

[jsd at AV8N.COM (John Denker)]

Stefan Jeglinski wrote:

> In the on-line book you cite, I refer to section 1.6: How to Find Lagrangians:
>
> "The key idea is to construct a Lagrangian L such that Lagrange's
> equations are Newton's equations  F = ma."

   Presumably http://mitpress.mit.edu/SICM/book-Z-H-13.html

> This would seem to be one of my points: "we have to write it this way
> to get the answer we want."

Well, I disagree with Gerry's wording and I strongly disagree with
this interpretation of that wording.  Instead I would say:
  a) In the special case where you start out knowing the F=ma law, it
suffices to work backwards from there to find the Lagrangian.
  b) In cases where you start out knowing the energy, and you can
identify part of it as potential (because it depends on x) and another
part as kinetic (because it depends on x dot), then you can write down
the Lagrangian as T-V.  You can use that to find the force laws.
  c) There exist yet other cases.  For instance, the Lagrangian for the
electromagnetic field exists, but cannot be interpreted as T-V, and
cannot be connected to an F=ma law.  (The field has neither m nor a,
hence no ma).

YMMV, but in my experience I've never seen a case where I knew the F=ma
law without also knowing the energy ... and in such cases method (b)
is far easier.  Method (a) is not "THE" key idea.