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] |
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