In article <firstname.lastname@example.org>, Rob Johnson <email@example.com> wrote: >In article <firstname.lastname@example.org>, >email@example.com wrote: >>Writing E for epsilon, consider variations of the curve x(t) of the >>form >>x^E(t), defined as (x(t) + E*z(t) ) / || x(t) + E*z(t) ||, >>which lie on the surface of the unit sphere S^2, >>and require that d/dE (I [x^E]) = 0 at E = 0 for every smooth z(t), >>where I[x] is defined as int (||x'||^2) dt for x(t) in S^2. >> >>From this consideration, how can we deduce the Euler-Lagrange equation >>x'' + (||x'||^2)x = 0 for the problem of minimising I[x]? > >I worked this problem in a different way. I did note that it was >necessary to make use of the fact that ||x||^2 = 1, i.e. <x,x'> = 0. > >The equation I got for the solution was x'' = kx. From this and the >fact that ||x||^2 = 1, it can be deduced that k = -||x'||^2. This >then implies that x'' + x ||x'||^2 = 0.
I had some spare time, so I worked the problem again using Euler-Lagrange. L(u,v,w) in
|\ | L(t,x,x') dt \|
|\ 2 = | ||x'|| dt \|
is L(u,v,w) = ||w||^2. Note that L_1 = L_2 = 0. Also,
L (u,v,w) = 2w 3
Under the constraint that C(t,x,x') = ||x||^2 - 1 = 0, where C_1 = 0, C_3 = 0 and
C (u,v,w) = 2v 2
the stationary E-L equation is
d d 0 = L (t,x,x') - -- L (t,x,x') + k (C (t,x,x') - -- C (t,x,x')) 2 dt 3 2 dt 3
= -2x'' + 2kx
For some k. This is the same equation I got before.
Rob Johnson <firstname.lastname@example.org> take out the trash before replying to view any ASCII art, display article in a monospaced font