Action in terms of Kinetic Energy

Definition
Let $A$ be the action for a given segment from point $A$ to point $B$ on the trajectory of a system is defined by the line integral:


 * $\ds \sum_i \int_A^B p_i \rd q_i$

Let $A$ act over a time interval of length $t$.

Then:
 * $A = 2 K t$

where $K$ denotes the mean kinetic energy of the system over $t$.