What would one study before tackling this topic?
I'm looking at this book:
http://www.amazon.com/Introduction-Calculus-Variations-Hans-Sagan/dp/0486673669
But I'm wondering what I need to be comfortable with before starting it.
In the analysis realm, I'm comfortable with differential/integral/multivariable/vector calculus.
Off the top of my head I was thinking I might need to study real analysis before I venture into Calculus of Variations, but I simply don't know enough about either to know if that's necessary.
Calculus of Variations
I would study differential equations before calculus of variations, since the basic method for solving problems in calculus of variations reduces them to differential equations.
I think the key idea from Real Analysis that's useful in studying the Calculus of Variations is just the definition of a Frechet derivative. If you spend a little time thinking about derivatives not as numbers but as linear functions, then I don't think it's too huge a leap to move to derivatives of functionals as being just linear functionals. Wiki link.
So maybe read the chapter of a real analysis text on derivatives, and then jump in? I used this one, whose main selling point is that it's very cheap, and which, though written in an older style, I thought was pretty decent. Though any calculus text with some rigor should have a decent explanation.
Personally, I always find problems to be very motivating. I think the first problem everybody wants to solve in the Calculus of Variations is to show that the curve connecting two points with minimal distance is a straight line; that might be a good place to start!
So maybe read the chapter of a real analysis text on derivatives, and then jump in? I used this one, whose main selling point is that it's very cheap, and which, though written in an older style, I thought was pretty decent. Though any calculus text with some rigor should have a decent explanation.
Personally, I always find problems to be very motivating. I think the first problem everybody wants to solve in the Calculus of Variations is to show that the curve connecting two points with minimal distance is a straight line; that might be a good place to start!
I think being really comfortable with linear algebra would be the best preparation.
Intuitively one can think of a function as being a vector with an infinite number of components (eg, the value at every point). You can add and subtract functions (pointwise), multiply them by scalars, measure their "size", etc just like vectors. So, like a n-dimensional vector is a point in an n-dimensional vector space, a function is a point in an infinite dimensional function space.
Functionals like "length of a curve" are just functions on the function space of all curves - imagine some sort of mountainous landscape where the x/y coordinates represent which curve you have, and the z coordinate represents the length of that curve. Finding the curve with the minimum length is like finding the lowest valley in your landscape. The space of curves can't be represented by just 2 coordinates (x/y) - you need an infinite number of coordinates - but other than that this picture is the right one to think of.
For example, the tangent (hyper)plane to the lowest point in the valley must have zero slope in every direction. The way to find these slopes is with the Frechet (for one slope, Gauteaux) derivative, which is basically a big word for what you would already think of: lim ||h||->0 (f(x+h)-f(x))/(||h||).
Intuitively one can think of a function as being a vector with an infinite number of components (eg, the value at every point). You can add and subtract functions (pointwise), multiply them by scalars, measure their "size", etc just like vectors. So, like a n-dimensional vector is a point in an n-dimensional vector space, a function is a point in an infinite dimensional function space.
Functionals like "length of a curve" are just functions on the function space of all curves - imagine some sort of mountainous landscape where the x/y coordinates represent which curve you have, and the z coordinate represents the length of that curve. Finding the curve with the minimum length is like finding the lowest valley in your landscape. The space of curves can't be represented by just 2 coordinates (x/y) - you need an infinite number of coordinates - but other than that this picture is the right one to think of.
For example, the tangent (hyper)plane to the lowest point in the valley must have zero slope in every direction. The way to find these slopes is with the Frechet (for one slope, Gauteaux) derivative, which is basically a big word for what you would already think of: lim ||h||->0 (f(x+h)-f(x))/(||h||).
I've studied ODE's. Do I need to study more advanced stuff or even PDE's?
I believe you are probably ready and you should go ahead and start studying calculus of variations. If you happen to find the need for something you haven't learned before, it won't be anything major, and you can just learn what you need as you go.
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