Is Calculus of Variations the Next Step After Traditional Calculus?

[sponsoredwalk]

Hi, I've seen the words "Calculus of Variations" mentioned quite a bit but never thought too much about them since it seemed too advanced.

Well, I am nearly finished the computational style calculus and am awaiting my Apostol text to get more into the theory but I also picked up a text called Calculus of Variations for like $5 (Second hand Dover book!) and I wonder will this be the kind of subject you can just jump into after calculus?

It seems to me as this subject is the basis of the Lagrangian formulation of Classical Mechanics, is this correct?

 

[Studiot]

It is also the underlying maths for some modern numerical methods and some Finite Element techniques.

 

[JSuarez]

Variational Calculus can be seen as a (vast) generalization of the methods for finding extremal points in functions, that you should have covered in Calculus; here, the "points" are functions, that belong to suitable function spaces, instead of the more common "points" that you may be used to, belonging to an euclidian space, for example.

Given this, I can tell you that the theory, as developed by mathematicians, is far too advanced for your background. As for the Physics' applications, it depends; there are some examples in Classical Mechanics that, I think, are within your reach.

As far as books are concerned, Dover re-issues are not a good bet for this topic; odds are that you spent 5$ on a book that will scare you away from VC at the first pages. There is one exception: Lanczo's "Variational principles of Mechanics"; this book is more conceptual than rigorous, but it could give you a fair view of VC, without the more spiky technicalities.

One more thing: VC is not only the foundation of lagrangian Classical Mechanics; it's also the foundation of hamiltonian Classical Mechanics (the two are related by something called a Legendre transform) and practically every major Physics theory (Electrodynamics, classical and quantum, General Relativity, etc.) admits a variational formulation.