On Tue, May 27, 2008 at 11:42 AM, Ondrej Certik <[EMAIL PROTECTED]> wrote:
> Hi,
>
> I have a disk in the (x,y) plane rotating with the angular velocity
> omega. Now I'd like to play with differential geomtry on the disk, so
> I start with the Euclidean metrics in my laboratory system and
> transform it to the disk, I get this:
>
> In [1]: var("t omega")
> Out[1]: (t, ω)
>
> In [2]: g_dd = Matrix([(-1+omega*(x**2+y**2),2*omega*y,-2*omega*x,0),
> (2*omega*y, 1, 0, 0), (-2*omega*x, 0, 1, 0), (0, 0, 0, 1)])
>
> In [4]: g_dd
> Out[4]:
> ⎡ ⎛ 2 2⎞ ⎤
> ⎢-1 + ω*⎝x + y ⎠ 2*ω*y -2*ω*x 0⎥
> ⎢ ⎥
> ⎢ 2*ω*y 1 0 0⎥
> ⎢ ⎥
> ⎢ -2*ω*x 0 1 0⎥
> ⎢ ⎥
> ⎣ 0 0 0 1⎦
>
>
> The g_dd stands for g_\mu_\nu (i.e. both indices down). Now I want to
> calculate the Christoffel symbols from this metrics. For this, I need
> to calculate the g^\mu^\nu, which is just the matrix
> inversion:
>
> In [3]: g_uu = g_dd.inv()
>
> However, try to display this:
>
> In [6]: g_uu
> [many pages of output]
>
> So let's just try to look at [0,0]:
>
> In [9]: g_uu[0, 0]
> [still too long]
>
> In [9]: print g_uu[0, 0]
> -1/(1 - 4*omega**2*x**2/(-1 + omega*(x**2 + y**2)) -
> 16*omega**4*x**2*y**2*(-1 + omega*(x**2 + y**2))**(-2)/(1 -
> 4*omega**2*y**2/(-1 + omega*(x**2 + y**2))))*(2*omega*x/(-1 +
> omega*(x**2 + y**2)) + 8*x*omega**3*y**2*(-1 + omega*(x**2 +
> y**2))**(-2)/(1 - 4*omega**2*y**2/(-1 + omega*(x**2 +
> y**2))))*(-2*omega*x/(-1 + omega*(x**2 + y**2)) -
> 8*x*omega**3*y**2*(-1 + omega*(x**2 + y**2))**(-2)/(1 -
> 4*omega**2*y**2/(-1 + omega*(x**2 + y**2)))) + 1/(-1 + omega*(x**2 +
> y**2)) + 4*omega**2*y**2*(-1 + omega*(x**2 + y**2))**(-2)/(1 -
> 4*omega**2*y**2/(-1 + omega*(x**2 + y**2)))
>
>
> Any try to simplify this has failed (I mean simply that it didn't
> simplify much). Is there any way to get this simplified, or is it just
> so complex by nature? Maybe that's the reason to use polar
> coordinates.
Well, but try this:
In [8]: one = g_uu * g_dd
In [20]: print one[0, 0]
-1/(-1 + omega*x**2 + omega*y**2) + omega*x**2/(-1 + omega*x**2 +
omega*y**2) + omega*y**2/(-1 + omega*x**2 + omega*y**2) -
4*omega**2*x**2/(-1 + omega*x**2 + omega*y**2)/(1 -
4*omega**2*x**2/(-1 + omega*x**2 + omega*y**2) -
16*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)) -
4*omega**2*x**2*(-1 + omega*x**2 + omega*y**2)**(-2)/(1 -
4*omega**2*x**2/(-1 + omega*x**2 + omega*y**2) -
16*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)) -
4*omega**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))/(-1 + omega*x**2 + omega*y**2) + 4*omega**3*x**4*(-1 +
omega*x**2 + omega*y**2)**(-2)/(1 - 4*omega**2*x**2/(-1 + omega*x**2 +
omega*y**2) - 16*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 +
omega*x**2 + omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)) +
4*omega**3*x**2*y**2*(-1 + omega*x**2 + omega*y**2)**(-2)/(1 -
4*omega**2*x**2/(-1 + omega*x**2 + omega*y**2) -
16*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)) -
64*omega**6*x**2*y**4*(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))**(-2)*(-1 + omega*x**2 + omega*y**2)**(-4)/(1 -
4*omega**2*x**2/(-1 + omega*x**2 + omega*y**2) -
16*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)) -
64*omega**6*x**2*y**4*(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))**(-2)*(-1 + omega*x**2 + omega*y**2)**(-3)/(1 -
4*omega**2*x**2/(-1 + omega*x**2 + omega*y**2) -
16*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)) -
32*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-3)/(1 -
4*omega**2*x**2/(-1 + omega*x**2 + omega*y**2) -
16*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)) -
32*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)/(1 -
4*omega**2*x**2/(-1 + omega*x**2 + omega*y**2) -
16*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)) +
32*omega**5*x**2*y**4/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-3)/(1 -
4*omega**2*x**2/(-1 + omega*x**2 + omega*y**2) -
16*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)) +
32*omega**5*x**4*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-3)/(1 -
4*omega**2*x**2/(-1 + omega*x**2 + omega*y**2) -
16*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)) +
64*omega**7*x**2*y**6*(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))**(-2)*(-1 + omega*x**2 + omega*y**2)**(-4)/(1 -
4*omega**2*x**2/(-1 + omega*x**2 + omega*y**2) -
16*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)) +
64*omega**7*x**4*y**4*(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))**(-2)*(-1 + omega*x**2 + omega*y**2)**(-4)/(1 -
4*omega**2*x**2/(-1 + omega*x**2 + omega*y**2) -
16*omega**4*x**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)) -
4*omega**2*y**2/(1 - 4*omega**2*y**2/(-1 + omega*x**2 +
omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2) + 4*omega**3*y**4/(1
- 4*omega**2*y**2/(-1 + omega*x**2 + omega*y**2))*(-1 + omega*x**2 +
omega*y**2)**(-2) + 4*omega**3*x**2*y**2/(1 - 4*omega**2*y**2/(-1 +
omega*x**2 + omega*y**2))*(-1 + omega*x**2 + omega*y**2)**(-2)
this should simplify to a unit matrix, so one[0,0] should be 1, but it isn't.
Ondrej
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