Hamilton's principle (principle of Least Action)

Bikerman

I'm currently doing quite a lot of reading to bring my woefully inadequate knowledge of math more up to date. I've been playing around with Least Action for a few hours and I'm wondering if anyone knows a piece of software or a website which does a trial and error solution to plotting world-lines ?
As always, I learn a new topic by writing a tutorial on it (teachers always make their own best students) and I'd like to include a 'brute strength' solution to a simple problem (chucking an apple up in the air) before moving on to do the Lagrangian solution.

Regards
Chris

Bikerman

To answer my own posting - I finally found what I was looking for and I must admit it fair blew my mind.

Anyone with reasonable math who has not come across the Principle of Least Action might well consider having a look at it. It sounds so simple and yet is so utterly powerful and fundamental that I'm still a bit gobsmacked from my simple tinklering with it. It tends to be taught only on grad physics courses towards the end but I think it should be taught much earlier from what I have seen of it.

In essense it states that 'nature is thrifty', or 'nature does things the easy way'. Sounds uselessly trite and cliched until you formulate it in math and then....wow.

A slightly more formal statement would be :
'Natural motions are such as to make some quantity always a minimum.'

It is now, I understand, at the heart of many areas of 'deep' science but I only looked at it in terms of motion in space. When used in this way the principle reads something like :

'A body will move through space in such a way that the difference between kinetic and potential energy is always a minimum.'

Again, I was misled and thought that was fairly innocuous and unimportant. Was I ever wrong!

The mathematical notation for that statement is written as follows:

(For any non maths readers who may be following this, the previous term reads: 'S (action) is the integral (summary over time) of Kinetic energy (the energy of motion T) minus Potential energy (the energy at rest V).
An 'integral' is a maths trick for calculating areas under curved lines and works by imagining the curve to be made up of lots of very small straight lines. It is part of calculus. If you want to read on but have no math to speak of then give it a go since there is no complex math to follow and it might interest you to try the apps I've put together).

Thanks to the kindness of Edwin F. Taylor and Slavomir Tuleja who have freely published their Java apps with no restrictions, I have put together a little demo of the principle of Least Action at work with a simple example of an apple being thrown up in the air. This will be of no interest to many I know, but if you have a bit of math and have not met this prionciple I would urge you to have a look - even if you have no math you may be able to see the implications.
If you already know this stuff then you probably think I'm raving....but to me this is amazing and I wish I'd been taught it earlier.

Here is the little demo:

http://camres.frih.net/mathslab/intermediate/leastaction/LeastActiondemo.html

redace

Principle of the Least action is really one of the most fundametal formulation od physical laws. It appears almoust everywhere in physics and with it it you can derivate such things like Maxwell equation, Newton laws and so on. But the most amazing becomes when is aplicated on quantum mechanics...there you will be shocked what it says. Try looking in some books for example Sakurai - Modern quantum mechanics. It is the topic of Feynaman path integrals.

Bikerman