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 [itex]\lambda[/itex]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 [itex]\lambda[/itex]=|x-dot|² simply by the fact that it minimizes I* = int [|x-dot|² - [itex]\lambda[/itex](|x|²-1)dt] which is [itex]\geq0[/itex], so the [itex]\lambda[/itex] 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:
[tex]
\int |\dot{x}|^{2}dt
[/tex]
One way to think about this to say that [itex]\dot{x}=0[/itex] which integrates up to [itex]x=\textrm{constant}[/itex] which implies that [itex]|x|^{2}=\textrm{constant}[/itex]. 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 [itex]\lambda[/itex]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 [itex]\lambda[/itex]=|x-dot|² simply by the fact that it minimizes I* = int [|x-dot|² - [itex]\lambda[/itex](|x|²-1)dt] which is [itex]\geq0[/itex], so the [itex]\lambda[/itex] 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 [itex] \min \int |\dot{x}(t)|^2 dt [/itex], subject to [itex] |x(t)|^2 = 1 \; \forall t [/itex]. 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
[tex]
\int |\dot{x}(t)|^{2}dt\geqslant 0
[/tex]
for all t and so it must be smallest when the integrand is identically zero.

 

[c299792458]

Because
[tex]
\int |\dot{x}(t)|^{2}dt\geqslant 0
[/tex]
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.