PROBLEM:
We want to minimize the distance between a fixed point $(t_0,x_0)$ with $t_0 \leq 0$ and a line, $t = 0$.
Formalise this problem as a Lagrange problem. Solve this equation by the Euler equation.
My approach:
Obviously, $x(t_0) = x_0$ and we want to minimize the distance of the curve between $(t_0,x_0)$ and $(t_1,x_1)$ where $t_1 = 0$ and $x_1$ is free.
The formula for the length of the graph of a function between two points is the following:
$\int_{t_0}^{t_1} (1 + (x')^2)^{\frac{1}{2}}dt$
Then I have to minimise this distance but I am not sure how to do this.
My intuition says the following:
Euler equation:
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$\frac{\delta}{\delta t}L_{x'}(t) + L_x(t) = 0$
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Since $x_1$ is free we know by the transversality condition that $L_{x'}(t_1) = 0$
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$L$ is the objective function here
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So know we have to solve this and we will get an answer
Can somebody please tell me if I am correct?
Best Answer
We want to minimize the function $\;f(x,y):=(x-t_0)^2+(y-x_0)^2\;$ subject to the condition $\;x=0\;$ , which is the same as asking what's the distance between the above given point and the $\;y$-axis, and the answer is, of course, $\;|t_0|\;$, but with Lagrange we get
$$H(x,y,\lambda):=(x-t_0)^2+(y-x_0)^2-\lambda\cdot 0$$
$$\begin{align*}H'_x&=2(x-t_0)=0\iff x=t_0\\H'_y&=2(y-x_0)=0\iff y=x_0\\H'_\lambda&=0\end{align*}$$
and the answer is straightforward.