Determining Christoffel Symbols and Curvature Tensors for a Flat Sphere using SymPy.Diffgeom

[Imran Ali]

I just asked the following question at stackoverflow :

http://stackoverflow.com/questions/33453941/determining-christoffel-symbols-and-curvature-tensors-for-a-flat-sphere-using-sy

In essence, the question is two fold :

1) My own implementation fails for flat surfaces in 3D

2) Sympy.Diffgeom seems to give wrong results

Really appreciate any insight as to what is causing the error and why the hand-coded sympy.diffgeom produces wrong result.

[Imran Ali]

I forgot to mention in the SO post (not that I have mentioned plenty already there) that sometimes I get the following error :

Traceback (most recent call last):

  File "tensor.py", line 274, in <module>

    RC = R.find_Christoffel_tensor()

  File "tensor.py", line 210, in find_Christoffel_tensor

    R = metric_to_Riemann_components(metric)

  File "/usr/local/lib/python2.7/dist-packages/sympy-0.7.7.dev0-py2.7.egg/sympy/diffgeom/diffgeom.py", line 1575, in metric_to_Riemann_components

    ch_2nd = metric_to_Christoffel_2nd(expr)

  File "/usr/local/lib/python2.7/dist-packages/sympy-0.7.7.dev0-py2.7.egg/sympy/diffgeom/diffgeom.py", line 1525, in metric_to_Christoffel_2nd

    ch_1st = metric_to_Christoffel_1st(expr)

  File "/usr/local/lib/python2.7/dist-packages/sympy-0.7.7.dev0-py2.7.egg/sympy/diffgeom/diffgeom.py", line 1492, in metric_to_Christoffel_1st

    matrix = twoform_to_matrix(expr)

  File "/usr/local/lib/python2.7/dist-packages/sympy-0.7.7.dev0-py2.7.egg/sympy/diffgeom/diffgeom.py", line 1463, in twoform_to_matrix

    raise ValueError('The input expression concerns more than one '

ValueError: The input expression concerns more than one coordinate systems, hence there is no unambiguous way to choose a coordinate system for the matrix.

In this case I have introduced some bug in the code, I do not understand why I am getting this error.

Here is the input I give :

g = 

Matrix([[1/(u**2 + 1), 0, 0], [0, u**2, 0], [0, 0, u**2*sin(v)**2]]) 

The code produces the following two-form :

sin(v)**2*u**2*TensorProduct(dw, dw) + u**2*TensorProduct(dv, dv) + TensorProduct(du, du)/(u**2 + 1) 

For the coordinate system :

CoordSystem(nontrivial, Patch(P, Manifold(M, 3)), (u, v, w))

[Imran Ali]

On Saturday, October 31, 2015 at 6:04:14 PM UTC+1, Imran Ali wrote:

I just asked the following question at stackoverflow :

http://stackoverflow.com/questions/33453941/determining-christoffel-symbols-and-curvature-tensors-for-a-flat-sphere-using-sy

In essence, the question is two fold :

1) My own implementation fails for flat surfaces in 3D

By my own implementation,  I mean the source code that I have provided in the SO post

 

2) Sympy.Diffgeom seems to give wrong results

Here I am referring to the hand-coded example in the SO post

[Imran Ali]

If I simply hard code each metric, every thing works properly. For example, take egg carton surface,

    from sympy.diffgeom import Manifold, Patch, CoordSystem, TensorProduct

    from sympy import sin,cos

    dim = 2

    m = Manifold("M",dim)

    patch = Patch("P",m)

    system = CoordSystem('egg_carton', patch, ["u", "v"])

    u,v = system.coord_functions()

    du,dv = system.base_oneforms()

    metric = (sin(v)**2*sin(u)**2 + 1)*TensorProduct(dv, dv) +\

             (cos(v)**2*cos(u)**2 + 1)*TensorProduct(du, du) +\

             (-cos(2*v - 2*u)/8 + cos(2*v + 2*u)/8)*TensorProduct(du, dv) +\

             (-cos(2*v - 2*u)/8 + cos(2*v + 2*u)/8)*TensorProduct(dv, du)

I can use any of the metric_to_* functions without getting the ValueError. 

By the way, that metric expression, is from my code. My code produces the correct metric two form expression. But when I use it as argument for any of the metric_to_* functions, I get the ValueError. What gives ?

I posted the question on SO :

http://stackoverflow.com/questions/33545982/valueerror-using-sympy-diffgeom-when-using-metric-to-functions

[Kalevi Suominen]

On Thursday, November 5, 2015 at 3:56:09 PM UTC+2, Imran Ali wrote:

If I simply hard code each metric, every thing works properly. For example, take egg carton surface,

    from sympy.diffgeom import Manifold, Patch, CoordSystem, TensorProduct

    from sympy import sin,cos

    dim = 2

    m = Manifold("M",dim)

    patch = Patch("P",m)

    system = CoordSystem('egg_carton', patch, ["u", "v"])

    u,v = system.coord_functions()

    du,dv = system.base_oneforms()

    metric = (sin(v)**2*sin(u)**2 + 1)*TensorProduct(dv, dv) +\

             (cos(v)**2*cos(u)**2 + 1)*TensorProduct(du, du) +\

             (-cos(2*v - 2*u)/8 + cos(2*v + 2*u)/8)*TensorProduct(du, dv) +\

             (-cos(2*v - 2*u)/8 + cos(2*v + 2*u)/8)*TensorProduct(dv, du)

I can use any of the metric_to_* functions without getting the ValueError. 

By the way, that metric expression, is from my code. My code produces the correct metric two form expression. But when I use it as argument for any of the metric_to_* functions, I get the ValueError. What gives ?

I posted the question on SO :

http://stackoverflow.com/questions/33545982/valueerror-using-sympy-diffgeom-when-using-metric-to-functions

 

It is not clear how u, v, du, dv are constructed in your SO question. If you could post srepr(u) and srepr(du), it might be possible to say more.

[Aaron Meurer]

What you should do is post the complete code to stackoverflow, as you
have done here.

Aaron Meurer

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[Francesco Bonazzi]

I answered to you on StackOverflow. You should be more clear by posting questions, this looks like a problem in your calculation logic, not in SymPy.

[Imran Ali]

Hi Kalevi!

I got the following output from srepr :

Symbol('u')

Differential(BaseScalarField(CoordSystem(Symbol('egg_carton'), Patch(Symbol('P'), Manifold(Symbol('M'), Integer(2))), Tuple(Symbol('u'), Symbol('v'))), Integer(0)))

I see now what the error is! My code used u as a sympy expression, instead of it being one of the coord_functions(). 

Thanks!

[Imran Ali]

Hey Aaron!

Thanks for the tip, I posted the code on SO. But, I think that Kalevi solved the problem. I found the issue.

[Imran Ali]

I was a bit quick there. srepr(u) gives the following output :

BaseScalarField(CoordSystem(Symbol('egg_carton'), Patch(Symbol('P'), Manifold(Symbol('M'), Integer(2))), Tuple(Symbol('u'), Symbol('v'))), Integer(0))

[Kalevi Suominen]

It looks like  u  was defined differently in different contexts.