classical mechanics - What is the significance of action?

What is the physical interpretation of $$ \int_{t_1}^{t_2} (T -V) dt $$ where, $T$ is Kinetic Energy and $V$ is potential energy.
How does it give trajectory?

Answer

The quantity $$ S= \int_{t_1}^{t_2} (T -V) dt $$ is known as the classical action. There exists a physical law (called the "principle of least action") which says that the true path an object takes is that which minimizes $S$.

Check that it's true. I'll throw a ball straight up. When the ball leaves my hand its kinetic energy $T$ is high, and since nature prefers to minimize the integral $S$, the potential energy of the ball $V$ rises quickly to minimize the integrand $T-V$. The principle of least action, then, explains why balls go up when you throw them.

So why don't baseballs keep going into the stratosphere to make $T-V$ as small as possible? They would need a lot of kinetic energy to do that! So much that it would outweigh the additional negative contribution from $-V$. It turns out that the true path is somewhere in-between rising high and going fast, which is what we observe. (Balls slow down as they go up.)

Beyond this qualitative argument one may use Variational Calculus to derive Newton's laws from the principle of least action.