My question is simple.
- In the classical mechanics, we know the least action principle does the variation on the action $$ S = \int (T-V) dt , $$ where $T$ is the kinetic term and $V$ is the potential term, to give the classical equations of motion.
So why the least action principle extremizes (minimizes or maxmizes) this quantity
$$T-V = H- 2V = E-2V =2 T- H = 2T-E?$$ Here I also denote $H$ as Hamiltonian. And $E$ as the energy of the system. In this regard, the $E$ is conserved, but not extremized. But $$E-2V =2T-E$$ is extremized!
For example, if we write down the $(T-V)$ for a free falling object under constant gravitational acceleration $g$, we have $$T-V = \frac{1}{2} m \dot{y}^2 - m g y = E -2 m g y = m \dot{y}^2 -E$$ with $E=\frac{1}{2} m \dot{y}^2 + m g y$ is conserved.
What is the physical significance of this $T-V$?
(Provide good physical or math reasons.)
I know how to derive equations of motion. I am asking a transparent way to understand why $T-V$ is in extremization.
I suppose the extremization may minimizes or maximizes the action. (Not necessarily minimization, correct?) Do we have both examples of
The equations of motion minimize the action?
The equations of motion maximize the action?
