Chronology | Current Month | Current Thread | Current Date |
[Year List] [Month List (current year)] | [Date Index] [Thread Index] | [Thread Prev] [Thread Next] | [Date Prev] [Date Next] |
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.
When we know the answer before we start "deriving" it, it's hard to
tell just how much our prior knowledge has influenced the result we
get. And I think that's what's going on here. As I read the
discussion on this thread, I see the Lagrangian approach as a very
general one that asserts that, if we find the right Lagrangian, it
will give us the right equations of motion.
But what is the right
Lagrangian? Evidently, it is whatever will give us EOMs that conform
with what we observe.
How we get there is pretty much irrelevant.
We
can guess,
or we can impose some abstract symmetries,
.... Neither Lagrange nor Newton are fundamental to nature.
What is fundamental is the symmetries. If we go one way around the
circle from the symmetries, we get Lagrange, and that leads to
Newton.
What that tells me is that the Lagrangian method is good if we are
clever enough to guess the Lagrangian (i.e., the one that leads to
equations that predict what is actually observed). It seems rather
apparent to me that when we find the correct Lagrangian, we get
Einstein, or Heisenberg/Schroedinger, or Dirac/Feynman, ....