How can one see that a Lagrangian in case of a free particle dictates that the mass can't be negative? Consider for example the case where the Lagrangian is given by $$ {\cal L}=\frac{1}{2} mv^2 . $$
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In fact, a free particle can certainly have negative mass. However, as soon as it gets coupled to something else with positive energy (say other particles with positive masses) you run in trouble as the system becomes unstable (e.g. the energy can be kept constant while the speed of various particle increases indefinitely) – TwoBs Dec 15 '17 at 07:36
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1Related: https://physics.stackexchange.com/q/103997/2451 and links therein. – Qmechanic Dec 15 '17 at 07:37