Lagrangian method for Christoffel symbols and Geodesics equations calculations
Basic Concepts and principles
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
we will use
.
Principle of least action from Euler-Lagrange
Let
})
and let C be the set of functions
)
such us
y(a)=c, e y(b)=d
if
, y'(x), x) dx)
reaches its minimum at some
then,
is a solution of differential equation
 - \frac{{\partial L}}{{\partial y}} = 0)
Extended theory
All is based on the following Theorem
Theorem
Let M be a m-dimensional variety and lets g
ij the metric tensor components. Then given a geodesic in M
 = (x^1(\lambda), ..., x^m(\lambda)))
Let L be the Lagrangian associated to the metric
Then Euler-Lagrange equations are satisfied
 - \frac{{\partial L}}{{\partial x^i}} = 0)
Example
Lets the usual induced metric on a sphere S
2 (Fig 1.), ie
|
|
Fig. 1 The sphere S2
|
We write the Lagrangian associated with this metric, ie
And then, we calculate
 = 2\ddot{\theta})
,
 = 2\ddot{\phi} sin^2 \theta %2B 4 \dot{\theta}\dot{\phi} sin \theta cos \theta)
,
This allows us and write the equations of the geodesics, we obtain
Remembering that the the geodesic equation are
We obtain the Chistoffel symbols too, are
The rest of Christoffel symbols are zero.
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Post from other users
marciliofds:
2015-03-23 13:38:56
Where can i meet the theorem? Do you know a book?
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