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Extremal condition calculus of variations

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Extremal condition calculus of variations

if I have a functional with a Lagrangian L(t,x(t),y(t),x’(t),y’(t)), meaning two functions x and y of one parameter t. And want to solve the minimization problem

0tLdtsuperscriptsubscript0tLdt\int_{0}^{t}Ldt

. Then I get necessary conditions to find extrema by getting the two Euler Lagrange equation

Lx-ddtLx=0LxddtLsuperscriptxnormal-′0\frac{\partial L}{\partial x}-\frac{d}{dt}\frac{\partial L}{\partial x^{{% \prime}}}=0

and

Ly-ddtLy=0LyddtLsuperscriptynormal-′0\frac{\partial L}{\partial y}-\frac{d}{dt}\frac{\partial L}{\partial y^{{% \prime}}}=0

now, if i solved these functions. how do i find out, that it is an actual minimum? are there methods to show this in general? i know, that in case of one variable it would be sufficient to show somehow that the lagrangian is convex. but is there a way to do this in this case too? or do i need to calculate a second derivative? if this is necessary, can someone give me a referece, where this is done for functionals of several functions or show me a way to do this?


Yes, a good first step is to look at the second derivative. Assuming it is not degenerate, it will tell you whether you have a local minumum, local maximum, or saddle point.

Finding a global minumum is a much harder business. In addition to looking at the value of your functional at the local minima, you need to consider its behavior at infinity; should it turn oout that the functional approaches a value less than what it attains at the local mininm as you approach infinity in some way, it will not have a global minimum.

Convexity is a useful notion here as well — if it turns out that your functional is convex, that will simplify the analysis.

As for reference, any good book on calculus of variations should answer your questions and provide examples.

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