Why Does Minimizing the Action Describe the Path of a System?

[epsilonjon]

Hi,
so today in maths we defined the Lagrangian as $$L = T-V$$ and stated Hamilton's Principle, which says that the actual path of a conservative system is the one which minimises the action

$$A(q)=\int^{t_{2}}_{t_{1}}L dt$$.

I'm a bit confused about this. What does the action represent in physical terms? Also, why on Earth would minimising the integral of $$L=T-V$$ result in the path which nature 'chooses'? How did Hamilton come up with this and why do we think it works (other than the fact it agrees with experiments!)?

Many thanks :D

 

[ansgar]

that is a postulate!

:)

just as the postulates of special relativity

the action is integral over time of the Lagrangian, nothing more nothing less.. it can be viewed as a weigthed sum over all possible configurations in energy-space, the path nature "choose" is the one that minimizes the action, i.e. the solution for the Euler-Lagrange equations

 

[phyzguy]

Try this - it's probably the best explanation you will find.

http://decision.csl.illinois.edu/~yima/psfile/ECE553/FeynmanLecturesOnPhysicsChapter2-19.pdf

 

[Dale]

why on Earth would minimising the integral of $$L=T-V$$ result in the path which nature 'chooses'? ... (other than the fact it agrees with experiments!)?

Warning: this is a non-scientific answer, but since you specifically exclude experiment I get the impression that you want a non-scientific answer.

Basically, that action is minimized by nature being lazy (anthropomorphizing). If nature can go from A to B in a variety of ways then the way it chooses is the one where it has the least kinetic energy and the most potential energy. Lazy. Think of throwing a baseball up. It spends as much time as high (most PE) and as slow (least KE) as possible.

 

[WiFO215]

If you read up on the history of this principle, you'll find that the principle of minimizing action has an interesting story. It starts way back with Fermat proposing that light, as a ray, follows a path of least time.

Then, later in the 1700s, Maupertius introduced a theorem at the time which was the earliest predecessor of the principle of least action. Based on Fermat's principle, he proposed that particles follow a path such that the product of the mass times the velocity times the distance would be minimized. This didn't always work and didn't get as much attention back then.

Much later, after Euler developed variational calculus, Lagrange applied this theory to mechanics. Hence the resulting equation of minimizing action is called the Euler-Lagrange equation. Following him, Hamilton worked on mechanics too, writing what is now known as Hamilton's Least Action principle. The rest is what you see before you in a math class. For a more detailed version of the history, look the science book discussion forum. You'll find lots of interesting links there. Have fun!

 

[patrickd]

The link given by Phyzguy is not working. Is there an alternate?

 

[Geremia]

Ch. 19 of Feynman's Lectures (vol. 2)

The link given by Phyzguy is not working. Is there an alternate?

It's just http://www.scribd.com/doc/6007778/Feynman-Lectures-on-Physics-Volume-2#outer_page_224" .