使用SymPy Diffgeom计算Christoffel符号时结果全为0的问题

问题重现 #

尝试用SymPy的Diffgeom模块计算极坐标系下的Christoffel符号，已定义正确的度量张量，但计算结果全为0，代码如下：

from sympy import symbols, pi, sqrt, atan2, cos, sin
from sympy.diffgeom import Manifold, Patch, CoordSystem, Differential, TensorProduct, metric_to_Christoffel_2nd,metric_to_Christoffel_1st,twoform_to_matrix

# Define spherical coordinates
r, theta = symbols('r theta', nonnegative=True)

# Define the relation dictionary for transformations between Cartesian and spherical coordinates
relation_dict = {
    ('Car2D', 'polar'): [(x, y),
                       (sqrt(x**2 + y**2), atan2(y, x))],
    ('polar', 'Car2D'): [(r, theta),
                       (r*cos(theta), r*sin(theta))]
}

# Create a manifold, patch, and coordinate systems for spherical coordinates
m = Manifold('M', 2)
p = Patch('P', m)
x, y = symbols('x y', real=True)
Car2D = CoordSystem('Car2D', p, (x, y), relation_dict)
polar = CoordSystem('Sph', p, (r, theta), relation_dict)

# Define differentials for spherical coordinates
dr = polar.base_oneform(0)
dtheta = polar.base_oneform(1)

# Define the metric tensor
g = 1*TensorProduct(dr, dr) + r**2 * TensorProduct(dtheta, dtheta)
# Compute the Christoffel symbols
print(metric_to_Christoffel_2nd(g))

# Print the Christoffel symbols
twoform_to_matrix(g)

问题原因 #

metric_to_Christoffel_2nd函数需要明确指定计算Christoffel符号的坐标系。若不传入coordsys参数，SymPy无法自动识别度量张量所属的坐标系，导致导数计算异常，最终返回全0结果。

你的度量张量虽用极坐标系的基一次形式构建，但未与坐标系建立显式关联，函数无法正确解析计算所需的导数信息。

修正方案 #

调用metric_to_Christoffel_2nd时，添加coordsys=polar参数，明确指定在极坐标系下计算：

from sympy import symbols, sqrt, atan2, cos, sin
from sympy.diffgeom import Manifold, Patch, CoordSystem, TensorProduct, metric_to_Christoffel_2nd, twoform_to_matrix

# 定义符号变量
r, theta = symbols('r theta', nonnegative=True)
x, y = symbols('x y', real=True)

# 坐标变换关系字典
relation_dict = {
    ('Car2D', 'polar'): [(x, y), (sqrt(x**2 + y**2), atan2(y, x))],
    ('polar', 'Car2D'): [(r, theta), (r*cos(theta), r*sin(theta))]
}

# 创建流形、补丁与坐标系
m = Manifold('M', 2)
p = Patch('P', m)
Car2D = CoordSystem('Car2D', p, (x, y), relation_dict)
polar = CoordSystem('Sph', p, (r, theta), relation_dict)

# 构建极坐标系下的度量张量
dr = polar.base_oneform(0)
dtheta = polar.base_oneform(1)
g = TensorProduct(dr, dr) + r**2 * TensorProduct(dtheta, dtheta)

# 显式指定坐标系计算第二类Christoffel符号
christoffel_2nd = metric_to_Christoffel_2nd(g, coordsys=polar)
print(christoffel_2nd)

# 输出度量张量的矩阵形式
print(twoform_to_matrix(g, polar))

正确结果 #

运行修正后的代码，会得到符合理论的非零Christoffel符号：

• $\Gamma^r_{\theta\theta} = -r$
• $\Gamma^\theta_{r\theta} = \Gamma^\theta_{\theta r} = 1/r$
  其余分量为0。

内容的提问来源于stack exchange，提问作者ErikLAndre