Geodesics of a Sphere: Minimizing Integral and Solving for Great Circle Equation

[c299792458]

Homework Statement

Hi, just want to get a couple of things straight regarding finding the geodesics of a sphere not using polar coordinates, but rather, Lagrange multipliers...

I want to minimize I = int (|x-dot|² dt)
subject to the constraint |x|=1 (sphere)
which gives an Euler equation of $\lambda$x - x-doubledot = 0

I have to show that the Euler equation is actually |x-dot|²x - x-doubledot = 0
Is it right to assume that $\lambda$=|x-dot|² simply by the fact that it minimizes I* = int [|x-dot|² - $\lambda$(|x|²-1)dt] which is $\geq0$, so the $\lambda$ that minimizes I* is |x-dot|²?

If I then try to integrate the Euler equation, then I get a SHM equation:

x₁= A₁ cos(|x-dot| t - C₁) where A, C are constants
and similarly for x₂, x₃

But how do I combine them to give the equation of a great circle, since I don't know the C_i's?

Thank you for any enlightenment!

Homework Equations

See above

The Attempt at a Solution

See above

 

[hunt_mat]

So you are trying to minimise the following:
$$\int |\dot{x}|^{2}dt$$
One way to think about this to say that $\dot{x}=0$ which integrates up to $x=\textrm{constant}$ which implies that $|x|^{2}=\textrm{constant}$. Does that help?

 

[c299792458]

Thanks, hunt_mat,
Why can you take x-dot = 0?

 

[Ray Vickson]

Homework Statement

Hi, just want to get a couple of things straight regarding finding the geodesics of a sphere not using polar coordinates, but rather, Lagrange multipliers...

I want to minimize I = int (|x-dot|² dt)
subject to the constraint |x|=1 (sphere)
which gives an Euler equation of $\lambda$x - x-doubledot = 0

I have to show that the Euler equation is actually |x-dot|²x - x-doubledot = 0
Is it right to assume that $\lambda$=|x-dot|² simply by the fact that it minimizes I* = int [|x-dot|² - $\lambda$(|x|²-1)dt] which is $\geq0$, so the $\lambda$ that minimizes I* is |x-dot|²?

If I then try to integrate the Euler equation, then I get a SHM equation:

x₁= A₁ cos(|x-dot| t - C₁) where A, C are constants
and similarly for x₂, x₃

But how do I combine them to give the equation of a great circle, since I don't know the C_i's?

Thank you for any enlightenment!

Homework Equations

See above

The Attempt at a Solution

See above

Your formulation is incorrect: it should be $\min \int |\dot{x}(t)|^2 dt$, subject to $|x(t)|^2 = 1 \; \forall t$. So, basically, you have infinitely many constraints, one for each t.

RGV

 

[hunt_mat]

Thanks, hunt_mat,
Why can you take x-dot = 0?

Because
$$\int |\dot{x}(t)|^{2}dt\geqslant 0$$
for all t and so it must be smallest when the integrand is identically zero.

 

[c299792458]

Because
$$\int |\dot{x}(t)|^{2}dt\geqslant 0$$
for all t and so it must be smallest when the integrand is identically zero.

I guess I meant how does one know that 0 is attained? Also, I believe I was given that |x|^2 =1 (constant)
However does your suggestion mean that I can set the augmented integrand |x-dot|2-lamda*|x|2 to 0? then I will have the desired lamda = |x-dot|2 ?

 

[c299792458]

Problem resolved! Thanks everyone.