Is it worth learning Calculus of Variations?

[torstum]

Hi everyone,

I'm already familiar with, and have used Lagrangians and Euler-Lagrange equations. I'm interested in calculus of variations, but if it all boils down to solving euler-lagrange equations (and this is probably the part where I'm mistaken), then what's the point? Please tell me if there is more to it than that. I would appreciate it.

 

[ZapperZ]

Er.. you have have this thing reversed. The calculus of variation, as far as I can tell, is the ORIGIN of Lagrangian/Hamiltonian mechanics. It is where the whole concept of a 'stationary'/maximal state and least-action principle are applied and expanded to the Lagrangian/Hamiltonian mechanics.

So no, they are not the same thing, even though they can lead to the same thing. Least action principle, for example, can account for Fermat's Least time principle. Besides, this would give you an analogous concept to the Feynman's path integral later on.

Zz.

 

[dextercioby]

To my mind, the calculus of variations is one of the most beautiful chapters in applied mathematics. There are a lot of problems in physics and engineering whose single solution requires knowledge of calculus of variations.

Think about the famous brahistochrone problem or the famous area-perimeter problems.

 

[nrqed]

Hi everyone,

I'm already familiar with, and have used Lagrangians and Euler-Lagrange equations. I'm interested in calculus of variations, but if it all boils down to solving euler-lagrange equations (and this is probably the part where I'm mistaken), then what's the point? Please tell me if there is more to it than that. I would appreciate it.

Calculus of variation is much more general than the EL equations. So it is worthwhile to understand it. The point is that th evariables with respect to which one varies the actions are not necessarily simply a generalized position and its derivative.

As one example, you could look up the derivation of Einstein's equations from a variational principle applied to the Einstein-Hilbert action. There one varies with respect to the metric so things look quite different than the usual EL equations.

 

[torstum]

Thanks for the feedback, it sure looks like a beautiful theory, and well worth getting into.