Saturday, June 14, 2008

Hamilton, Lagrange, or Newton?

Matt offers a nice toy problem solved with Lagrangian methods. (Luckily, he got it right despite posting on Friday the 13th.)

So which approach is inherently cooler, Hamilton's, Lagrange's, or Newton's?

Is it Hamilton's mechanics, because it introduces the symbol p for the generalized momentum conjugate to the generalized coordinate q, because it provides the neatest formal connection to non-relativistic quantum mechanics, or because you get to say "symplectic" once in awhile? Is it Lagrange's mechanics, because it is based on the very important ideas of the calculus of variations, because it is used in relativistic quantum mechanics, or because it knows where the "action" is? Or is it Newton's mechanics, because things should be pushed around by forces, or because his work started it all?

According to Hamilton, it is Lagrange.


Matt said...

Hmm, tough one. Newton's is the most intuitive, Lagrange's is the most beautiful, and Hamilton's is the most useful. But that's only personal opinion!

Coolest? If I had to make a subjective judgment... probably Lagrange, by a hair. Just because the principle behind it feels like magic.

Anonymous said...

Lagrange. Definitely the most aesthetically pleasing.

Doctor Pion said...

I suppose I must add my dos centavos:
My pick is also Lagrange.

Probably the mathematician in me, but the principle has so many other applications and I also like the way a problem almost sets itself up when you choose the right coordinates.