Behaviour of action with respect to time

Question

I was wondering if it was possible to say something general on the behaviour of the action :

$$ S[x(\tau)]=\int_0^T L(x,\frac{dx}{d\tau},t) dt $$

(where $x(\tau)$ defines a trajectory, with certain boundary conditions at $\tau=0$ and $\tau = T$, and $L$ is the Lagrangian) at small and large values of $T$. For some systems (harmonic oscillator), we can say that the action becomes very large at small $T$ (look last formula here: http://www.oberlin.edu/physics/dstyer/FeynmanHibbs/Prob2-2.pdf). Intuitively (and quite simple "mindedly"), I see it as when the endpoints are fixed and the total time become very small, the kinetic energy must increase a lot (I cannot say anything about the potential however).

The question arised while I was reading the article "Path-Integral Derivation of Black-Hole Radiance" by Hartle and Hawking and they consider that kind of behaviour.

Answer 1

Comment to the question (v2): If the Lagrangian reads $L=\frac{1}{2}m \dot{x}-V(x)$, then then Dirichlet on-shell action reads $$\tag{1} S(x_f,t_f;x_i,t_i)~\approx~\frac{m}{2}\frac{(\Delta x)^2}{\Delta t}-V(\bar{x})\Delta t $$ for small times $\Delta t\geq 0$, where $$\tag{2} \Delta t~ :=~t_f-t_i, \qquad \Delta x~ :=~x_f-x_i, \qquad \bar{x}~ :=~ \frac{x_f+x_i}{2} .$$