Christoffel Symbol Computation

by allenlu2007

 

什麼是 Christoffel symbol?

就是先用 tangent space 定義出 u derivative, v derivative, and normal direction coordinates.

在建立和 near-by tangent space 之間的 connection 所需要的座標變換。和 Jacobian 類似。 

 

另外一個角度是從 tensor calculus (see YouTube video on Tensor calculus).

就是 derivative of tensor (even in Euclidean space) 必須加上 Christoffel symbol 才能得到另一個 tensor (on the covariant basis or contra covariant basis). 

 

如何計算 Christoffel symbol? 

1. Direct differentiation using covariant derivative

2. Levi-Civita connection

3. Lagrangian

 

1. Direct differentiation using covariant derivative

Using Einstein summation notation

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所以什麼是 Christoffel symbol?  就是座標變換的 coefficient, 類似 Jacobian.  

可以說是 “twisting” the coordinate

 

2. Levi-Civita connection

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3. Lagrangian Method to Compute Geodesic and Christoffel Symbol

參考本文

The Christoffel symbols calculations can be quite complicated, for example for dimension 2 which is the number of symbols that has a surface, there are 2 x 2 x 2 = 8 symbols and using the symmetry would be 6. 

For dimension 4 the number of symbols is 64, and using symmetry this number is only reduced to 40. Certainly there are many calculations and this is just to find the equations of geodesics (after, we must solve or analyze it). 

We will see in this section, the Lagrangian method allows us to obtain the geodesic equations and hence obtain the Chistoffel symbols. in a simpler way. 

Ideas are the basis of the calculus of variations called principle of least action of Euler-Lagrange

先從 Euler-Lagrange Equation

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Some Examples

Example 1: Euclidean space

All Christoffel symbol = 0  (因為沒有任何 curvature)

 

Example 2: Cylindrical coordinates (direct computation, 3x3x3)

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Example 3: Spherical coordinates (using Levi-Civita connection, 3x3x3)

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 Example 4: Sphere r=1 (using Lagrangian, 2x2x2)

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