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
First of all, a notation matter: from now on we will use the dot character "." to denote the derivate, this is a notation very used in physics, for example to denote
\( \frac{dx}{d\lambda} \) we will use \(\dot{x} \).
Principle of least action from Euler-Lagrange
Let \( L\in{C^2(\mathbb{R}^3)} \) and let C be the set of functions
\( y \in C^1([a,b]) \) such us
y(a)=c, e y(b)=d
if
\( \int_a^b L(y(x), y'(x), x) dx \)
reaches its minimum at some \( y_0\in{C} \) then, \( y_0 \) is a solution of differential equation
\( \frac{d}{dx}(\frac{{\partial L}}{{\partial y^{\prime}}}) - \frac{{\partial L}}{{\partial y}} = 0 \)
Lets the usual induced metric on a sphere S
2 (Fig 1.), ie
\( ds^2 = d \theta^2 + sin^2 \theta d\phi^2 \)
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Fig. 1 The sphere S2
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We write the Lagrangian associated with this metric, ie
\( L = \dot{\theta}^2 + sin^2 \theta \dot{\phi}^2 \)
And then, we calculate
\( \frac{d}{d\lambda}(\frac{{\partial L}}{{\partial \dot{\theta}}}) = 2\ddot{\theta} \),
\( \frac{{\partial L}}{{\partial \theta}} = 2\dot{\phi}^2 sin \theta cos \theta \)
\( \frac{d}{d\lambda}(\frac{{\partial L}}{{\partial \dot{\phi}}}) = 2\ddot{\phi} sin^2 \theta + 4 \dot{\theta}\dot{\phi} sin \theta cos \theta \),
\( \frac{{\partial L}}{{\partial \ \phi}} = 0 \)
This allows us and write the equations of the geodesics, we obtain
\( \left\{\begin{matrix} \ddot{\theta}- sin \theta cos \theta \dot{\phi}^2 = 0 \\ \ddot{\phi} + 2 \frac{cos \theta}{sin \theta} \dot{\theta} \dot{\phi} = 0 \end{matrix}\right. \)
Remember that geodesic equations are
\( \frac{d^2x^k}{d\lambda^2} + \Gamma_{ij}^k \frac{dx^i}{d\lambda}\frac{dx^j}{d\lambda} = 0 \)
We obtain the Chistoffel symbols too, are
\( \Gamma_{22}^{1}=-sin\theta cos\theta \),
\( \Gamma_{12}^{2}=\Gamma_{21}^{2}=\frac{cos\theta}{sin\theta}\)
The rest of Christoffel symbols are zero.
