# Lagrange multipliers for finding geodesics on a sphere

1. Jan 29, 2013

### Leb

1. The problem statement, all variables and given/known data
Find the geodesics on a sphere $g(x,y,z)=x^{2}+y^{2}+z^{2}-1=0$
arclength element $ds=\sqrt{dx^{2}+dy^{2}+dz^{2}}$

2. Relevant equations

$f(x,y,z)=\sqrt{x'^{2}+y'^{2}+z'^{2}}$ where $x'^{2} \text{means} \frac{dx^{2}}{ds^{2}}$ and not $d^{2}x/ds^{2}$

3. The attempt at a solution

Using the fact that $x^{2}+y^{2}+z^{2}=1$ I get three equations if of the form
$d^{2}x/ds^{2}=2\lambda x$ i.e. the double derivative of x w.r.t. s
$d^{2}y/ds^{2}=2\lambda y$
$d^{2}z/ds^{2}=2\lambda z$

My lecturer now says, that we have to differentiate the constraint (g) twice w.r.t. s to get
$(dx/ds)^{2}+(dy/ds)^{2}+(dz/ds)^{2} = -2 \lambda (x^{2}+y^{2}+z^{2})$
since the LHS = 1 and the brackets in the RHS = 1 , the lecturer concludes, that $\lambda = -0.5$. Now I am most confused here, as I do not see where the double differentiation happened. Nor do I see how this helps to determine that xA+By+z=0 (A,B = const.) which defines the great circle.

Thanks.