Hello,

I don't know if someone will be able to help me with this, but I'm going to post it anyway. I'm studying a worked example, but I'm confused about a few things and was hoping someone could help me understand some steps.

Question:

Find the extremal for the following fixed-end problem

$\displaystyle \int_1^2 \frac{\dot{x}^2}{t^3} dt$ with $\displaystyle x(1) = 2, x(2) = 17$

Solution(I'm going to number the steps here.

(1) Let $\displaystyle f(t,x,\dot{x}) = \frac{\dot{x}^2}{t^3}$. The Euler-Lagrange equation is

$\displaystyle

\frac{\partial f}{\partial x} - \frac{d}{dt} \big ( \frac{\partial f}{\partial \dot{x}} \big ) = 0$

(2) which implies $\displaystyle - \frac{d}{dt} \big ( \frac{2 \dot{x}}{t^3} \big ) = 0 $

(3) so $\displaystyle \frac{\dot{x}}{t^3} = C $ where C is a constant.

(4) Then $\displaystyle x = x(t) = kt^4 + l $ where $\displaystyle k = \frac{C}{4} $

(5) Using $\displaystyle x(1) = 2, x(2) = 17$ we have

$\displaystyle k + l = 2, 16k + l = 17 $

(6) Solving this linear system yields $\displaystyle k = 1, l = 1 $

(7) So the extremal is $\displaystyle x = t^3 + 1 $

Firstly, I don't understand how this goes from (2) to (3) i.e where $\displaystyle \frac{\dot{x}}{t^3} = C $, and from (3) to (4).

Can anyone please explain it?

Another example I have (which I won't post the whole thing for) is

Find the extremal of

(8) $\displaystyle \int_0^{\frac{\pi}{2}} (x^2 - \dot(x)^2 - 2x \sin t) dt $ with $\displaystyle x(0) = 1, x(\frac{\pi}{2}) = 2 $

(9) The Euler-Lagrange equation is

$\displaystyle

\frac{\partial f}{\partial x} - \frac{d}{dt} \big ( \frac{\partial f}{\partial \dot{x}} \big ) = 0$

(10) which implieds $\displaystyle 2x - 2 \sin t + \frac{d}{dt} (2 \dot{x}) = 0 $

(11) This implies $\displaystyle \ddot{x} + x = \sin t $

again, in this example I don't understand how 11 was arrived at from 10. (Note the extremal solution to this second example is

$\displaystyle x(t) = \cos t + 2 \sin t - \frac{1}{2} t \cos t $