Least action principle remark on negative mass in Landau-Lifshitz classical mechanics In the famous Landau-Lifshitz's Classical Mechanics there is a remark I cannot fully understand at the very beginning of the book (page 7 of the second edition):

It is easy to see that mass of a particle cannot be negative. For, according to the principle of least action, the integral
\begin{equation}
S = \int_1^2 \frac{1}{2}mv^2 dt
\end{equation}
has a minimum for the actual motion of the particle in space from point 1 to point 2. If the mass were negative, the action integral would take arbitrarily large negative values for a motion in which the particle rapidly left the point 1 and rapidly approached point 2, and there would be no minimum.

Can someone explain me why a change in sign of mass implies the integral to take "arbitrarily large negative values"?
 A: I'm guessing the reasoning in Landau-Lifshitz is as follows:
There is no limit to the form of the variation. With a negative value for mass $m$ we have the following: The action $S$ evaluates to a negative value, and for any trial trajectory there are beyond that trial trajectory other trial trajectories that evaluate to a more negative value. That is, if you take a detour instead of the shortest path, then there is always a way to make an even longer detour. But the total duration is a given, so the longer the detour the higher the corresponding velocity to make it to the end point in time. So: the larger the detour the larger the $v^2$ part. Hence: if for mass $m$ a negative value is used then the larger the detour the more negative the action.


However, the reasoning presented in Landau-Lifshitz is fundamentally flawed. Hamilton's stationary action is ill suited to the purpose. Hamilton's stationary action itself is sound, of course, but here it is misused, so in the end you are none the wiser.
As a matter of principle Hamilton's action is about stationary action, not "least action". This is not a technical detail, it makes all the difference.
The derivation of the Euler-Lagrange equation to find solutions for a variational problem is agnostic as to whether the the extremum solution is a minimum or a maximum.
There are in fact classes of cases where Hamilton's action is a maximum. This is not well know, because - as I said - the Euler-Lagrange equation is agnostic as to whether the extremum is a minimum or a maximum. Whether the action is a minimum or a maximum is immaterial. Therefore any reasoning based on an assumption that there must always be a minimum action is flawed, and you are none the wiser.
A: If the mass $m<0$ is negative then OP's action functional would be manifestly $\leq 0$, and unbounded from below, i.e. it doesn't have a minimum.
A: Observe the following. When we write $S=\int dt\frac{1}{2}m v^2$ what we really mean is $S=S[x]$. That is, $S$ is a functional of the path $x(t)$ we choose to evaluate it on (and $v=\frac{d x}{d t}$). That means we get to choose what function $x(t)$ we evaluate $S$ on. The entire purpose of action minimization is to find the $x(t)$ such that $S$ takes its minimal value. If $m$ is negative, then $\frac{1}{2}mv^2$ is negative and can be made arbitrarily negative by simply selecting functions $x(t)$ with larger and larger first derivatives (make $v$ larger).
This is the argument that Landau and Lifshitz are shooting at. In reality, however, the Euler-Lagrange equations we derive only ensure we are at a local extrema in the space of paths $x(t)$. This is equivalent to the statement that the vanishing of a function's derivative can only tell you that you're at an extrema, not whether it's a minima or maxima. So typically, the solution to the equations of motion will only be a saddle point of the action, not necessarily a minima.
In so far as I'm aware, there's not actually a good classical reason why we should demand the equations of motion form a minima. The closest thing I'm aware of would have to do with boundedness of the action for the sake of Feynman path integrations in quantum mechanics, but of course that should have no bearing upon classical physics.
