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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.
That was already done (more than once). That comes from the invariance of the EOM under Galilean boosts once it is granted that the resulting equations be no higher than 2nd order and >the symmetries of space & time homogeneity and spatial isotropy are imposed. Please reread the part of the argument about how only the expression v^2 obeys v^2 = v'^2 + dF/dt when v = >v' + v_0 and F = F(r',v',t). Other nonlinear functions of v^2 do not have this property.
Regarding Bob S's response:
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).
And I, for one, would argue with it. The requirements imposed come from experimental observation. The descriptive laws of nature are *observed* to be invariant under the action of the elements of the inhomogeneous Galilei group (when the boost speed is sufficiently slow compared to c). *And* they are observed to be 2nd order in time. These requirements are not given from any a priori mathematical requirements. They reflect symmetries *observed* in the real world. There *are* experimentally observed data inputted to the argument. No one denies this. It is just that the observed symmetries imposed are not an explicit version of Newton's laws. However Newton's laws certainly do respect those symmetries.
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.
That was already done (more than once). That comes from the invariance of the EOM under Galilean boosts once it is granted that the resulting equations be no higher than 2nd order and the symmetries of space & time homogeneity and spatial isotropy are imposed. Please reread the part of the argument about how only the expression v^2 obeys v^2 = v'^2 + dF/dt when v = v' + v_0 and F = F(r',v',t). Other nonlinear functions of v^2 do not have this property.
This is a statement of Newtonian dynamics and cannot be deduced
from "pure" mathematics.
Of course it can't be derived from pure mathematics. It is derived from the observed properties of nature. The requirements imposed are not *explicitly* Newtonian dynamics. Taken together they may imply Newtonian dynamics, but they are not any *explicit* form of N2. The closest the requirements come to an explicit imposition of N2 is in the explicit imposition of the 2nd order requirement on the EOM and its consequences on the form of L. Of course knowing the order of a DE does not determine it form, but it does constrain it somewhat. That constraint along with the other imposed symmetries winnows down the form of L and the resulting EOM to the point that the E-L equations produce N2.
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.
Certainly. But we all know better.
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.
Of course one could define KE at the difference between the Lagrangian of a free particle in motion and the Lagrangian of a free particle at rest. Such a definition would survive both Newtonian Dynamics and Special Relativity.
Bob Sciamanda
David Bowman
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