One should minimize the distance between two points $p_1=(x_1,y_1)$ and $p_2=(x_2,y_2)$. The holonomic constraint states that $()=^2−2+5$ where $x_1$ is an element of this graph and $x_2$ is an element of the graph of a different function: $()=2−1$ Now the goal is to use the constraints in such a way, to eliminate two of the given coordinates $x_1,x_2,y_1,y_2$. After the elimination one should determine the minimum of the distance between these 2 points. A hint is that this process is reminiscent of the 2. Lagrangian formalism.
I just don't even know where to start off, I would be so grateful if someone could help me out somehow!
Given $p_1 = (x_1,y_1),\ \ p_2 = (x_2, y_2)$ and the restrictions $g_1(x_1,y_1) = y_1 -( x_1^2-2x_1+5)=0,\ \ g_2(x_2,y_2) = y_2 -( 2x_2-1)=0$ the problem can be formulated as the stationary points determination for the lagrangian
$$ L = \|p_1-p_2\|^2+\lambda_1 g_1(x_1,y_1)+\lambda_2 g_2(x_2,y_2) $$
The stationary points are the solutions for
$$ \nabla L = 0 = \cases{ \lambda_1(2 x_1-2)+2 (x_1-x_2) \\ 2 \lambda_2-2 (x_1-x_2) \\ 2 (y_1-y_2)-\lambda_1 \\ \lambda_2+2 (y_1-y_2) \\ x_1^2-2 x_1-y_1+5 \\ 2 x_2-y_2-1 \\ } $$