The question reads " Find the extremal of class $\displaystyle C^2$ ,if it exists,which satisfies the given final conditions for problems with lagrangian as follows:

$\displaystyle L(x,y,\dot{x},\dot{y})=\dot{x}^2+\dot{y}^2+2xy$;

$\displaystyle x(0)=y(0)=0 $

$\displaystyle x(\frac{\pi}{2})=y(\frac{\pi}{2})=1$

Please check my work:

The Euler-Lagrange Equation given by $\displaystyle \frac{d}{dt}(\frac{\partial L}{\partial \dot{x}})-\frac{\partial L}{\partial x} = 0$

I get (for x) $\displaystyle \frac{\partial L}{\partial \dot{x}} = 2\dot{x} $

$\displaystyle \frac{d}{dt}2\dot{x}=2\ddot{x}$

$\displaystyle \frac{\partial L}{\partial x}=2y$

Which yields $\displaystyle 2\ddot{x}-2y=0$

integrating wrt t: $\displaystyle \int 2\ddot{x} dt = \int 2y dt$

$\displaystyle 2\dot{x}=2yt+c$

$\displaystyle \int 2\dot{x} dt= \int (2yt+c) dt$

so $\displaystyle 2x=yt^2 +ct + d$

So that i have for x, $\displaystyle x=\frac{1}{2}[yt^2+ct+d]$

$\displaystyle x(0) => d=0; x(\frac{\pi}{2}) => c=\frac{2}{\pi}(1-\frac{\pi^2y}{8})$ didn't show calc's but i'm certain here

I find then that $\displaystyle x=\frac{1}{2}yt^2+\frac{2}{\pi}(1-\frac{\pi^2y}{8})t$

And for y, $\displaystyle y=\frac{1}{2}xt^2+\frac{2}{\pi}(1-\frac{\pi^2x}{8})t$

Am i done at this point? Have i found an extremal of the lagrangian?

I am not sure of the differentiation and integration ? I don't have much in the way of examples to work with...just started studying these...

Another question, what happens if the equation is trivially zero? i.e $\displaystyle \frac{d}{dt}(\frac{\partial L}{\partial \dot{x}}) = \frac{\partial L}{\partial x}$ ?

Does this mean that the lagrangian will have no extremal at all? Or is there some operation to be carried out, or manipulation to get an extremum? What must happen for there to be no extremum to the lagrangian?

One more thing, can anyone recommend a good textbook that deals with variational calculus comprehensively, with examples and solutions to exercises ? I only have a study guide,which is a bit of a cut and paste half story of the subject.

THANK YOU !