I have seen a derivation of Newton's Laws from Lagrange's equation and
the least action principle (Landau and Lifshitz, most interesting, by
the way).
In this derivation it states that the mass of a body must be positive
because otherwise the integral of the action:
S = intg (m v^2 / 2) dt would not have a minimum, which is a
prerequisite for the principle of least action.
However, this is for a free particle, with no acting field.
If there is a field, then the integral becomes:
S = intg (m v^2 / 2 - U (r, t)), where U is the potential energy of the
field.
Since U can be positive or negative, does this mean that in this case
there might not be a minimum? Does it mean that in some cases the mass
might be negative?
Thanks for any help.
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