David wrote:

> On the other hand, here is a formulation -- which is logically equivalent -- that I find more intuitively elementary:

> 1. Energy = T+V

> is conserved.

> Nature seeks paths that minimize kinetic energy T. Or, equivalently, that maximize potential energy V.

That's fine as far as it goes. Everyone who understands the principle of least action has thought about this. It's important.

But it's really wonderful how

"Nature takes paths that minimize T - V"

implies _both_ of these. Indeed, _all_ conservation laws can be derived from the principle of least action together with symmetries, thanks to Noether's theorem - and symmetry under time translation gives conservation of energy. So we wind up deciding the principle of least action is fundamental, even if it takes a while for each one of us to come around to that viewpoint.

Have you read [Feynman's little story](http://www.feynmanlectures.caltech.edu/II_19.html) about how he learned about the principle of least action in high school? Somewhere else he wrote about how he fought it, because he found it unintuitiv. He didn't like it, so he would cleverly solve all his mechanics problems using just $F = ma$, which is usually much harder. But ironically, one of his lasting claims to fame is understanding how quantum mechanics generalizes the principle of least action: in fact nature takes _all_ paths, each with amplitude $\exp(i S / \hbar)$, where $S$ is the action and $\hbar$ is Planck's constant.

> On the other hand, here is a formulation -- which is logically equivalent -- that I find more intuitively elementary:

> 1. Energy = T+V

> is conserved.

> Nature seeks paths that minimize kinetic energy T. Or, equivalently, that maximize potential energy V.

That's fine as far as it goes. Everyone who understands the principle of least action has thought about this. It's important.

But it's really wonderful how

"Nature takes paths that minimize T - V"

implies _both_ of these. Indeed, _all_ conservation laws can be derived from the principle of least action together with symmetries, thanks to Noether's theorem - and symmetry under time translation gives conservation of energy. So we wind up deciding the principle of least action is fundamental, even if it takes a while for each one of us to come around to that viewpoint.

Have you read [Feynman's little story](http://www.feynmanlectures.caltech.edu/II_19.html) about how he learned about the principle of least action in high school? Somewhere else he wrote about how he fought it, because he found it unintuitiv. He didn't like it, so he would cleverly solve all his mechanics problems using just $F = ma$, which is usually much harder. But ironically, one of his lasting claims to fame is understanding how quantum mechanics generalizes the principle of least action: in fact nature takes _all_ paths, each with amplitude $\exp(i S / \hbar)$, where $S$ is the action and $\hbar$ is Planck's constant.