{d^2 x^\lambda \over dT^2} = {d x^\nu \over dT} {d x^\alpha \over dT} \left is the line element. To get the geodesic equation we must vary this action. To do this lets parameterize this action with respect a parameter \lambda. Doing this we get:
S=\int\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}}d\lambda
We can now go ahead and vary this action with respect to the curve x^{\mu}. By the principle of least action we get:
0=\delta S=\int\delta\left(\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}}\right)d\lambda =\int\frac{\delta\left(g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}\right)}{2\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}}}d\lambda
For concreteness lets parameterize this action w.r.t. the proper time, \tau. Since the fourvelocity is normalized to 1 (for timelike paths) we can say that the above is equivalent to the action:
0=\int\delta\left(g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\right)d\tau
Using the product rule we get:
0=\int\left(\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\delta g_{\mu\nu}+g_{\mu\nu}\frac{d\delta x^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}+g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{d\delta x^{\nu}}{d\tau}\right)d\tau =\int\left(\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\alpha}g_{\mu\nu}\delta x^{\alpha}+2g_{\mu\nu}\frac{d\delta x^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\right)d\tau
Integrating byparts the last term and dropping the total derivative (which equals to zero at the boundaries) we get that:
0=\int d\tau\left(\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\alpha}g_{\mu\nu}\delta x^{\alpha}2\delta x^{\mu}\frac{d}{d\tau}\left(g_{\mu\nu}\frac{dx^{\nu}}{d\tau}\right)\right)=\int d\tau\left(\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\alpha}g_{\mu\nu}\delta x^{\alpha}2\delta x^{\mu}\partial_{\alpha}g_{\mu\nu}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}2\delta x^{\mu}g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}\right)
Simplifying a bit we see that:
0=\int d\tau\delta x^{\mu}\left(2g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\mu}g_{\alpha\nu}2\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\alpha}g_{\mu\nu}\right)
so,
0=\int d\tau\delta x^{\mu}\left(2g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\mu}g_{\alpha\nu}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\alpha}g_{\mu\nu}\frac{dx^{\nu}}{d\tau}\frac{dx^{\alpha}}{d\tau}\partial_{\nu}g_{\mu\alpha}\right)
multiplying this equation by \frac{1}{2} we get:
0=\int d\tau\delta x^{\mu}\left(g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\frac{1}{2}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}\left(\partial_{\alpha}g_{\mu\nu}+\partial_{\nu}g_{\mu\alpha}\partial_{\mu}g_{\alpha\nu}\right)\right)
So by Hamilton's principle we find that the Euler–Lagrange equation is
g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\frac{1}{2}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}\left(\partial_{\alpha}g_{\mu\nu}+\partial_{\nu}g_{\mu\alpha}\partial_{\mu}g_{\alpha\nu}\right)=0
Multiplying by the inverse metric tensor g^{\mu\beta} we get that
\frac{d^{2}x^{\beta}}{d\tau^{2}}+\frac{1}{2}g^{\mu\beta}\left(\partial_{\alpha}g_{\mu\nu}+\partial_{\nu}g_{\mu\alpha}\partial_{\mu}g_{\alpha\nu}\right)\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}=0
Thus we get the geodesic equation:
\frac{d^{2}x^{\beta}}{d\tau^{2}}+\Gamma^{\beta}_{\alpha\nu}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}=0
with the christoffel symbol defined in terms of the metric tensor as
\Gamma^{\beta}_{\alpha\nu}=\frac{1}{2}g^{\mu\beta}\left(\partial_{\alpha}g_{\mu\nu}+\partial_{\nu}g_{\mu\alpha}\partial_{\mu}g_{\alpha\nu}\right)
(NOTE: This derivation works for lightlike and spacelike paths too.)
Equation of motion may follow from the field equations for empty space
Albert Einstein believed that the geodesic equation of motion can be derived from the field equations for empty space, i.e. from the fact that the Ricci curvature vanishes. He wrote:^{[4]}
It has been shown that this law of motion — generalized to the case of arbitrarily large gravitating masses — can be derived from the field equations of empty space alone. According to this derivation the law of motion is implied by the condition that the field be singular nowhere outside its generating mass points.
Both physicists and philosophers have often repeated the assertion that the geodesic equation can be obtained from the field equations to describe the motion of a gravitational singularity, but this claim remains disputed.^{[5]} Less controversial is the notion that the field equations determine the motion of a fluid or dust, as distinguished from the motion of a pointsingularity.^{[6]}
Extension to the case of a charged particle
In deriving the geodesic equation from the equivalence principle, it was assumed that particles in a local inertial coordinate system are not accelerating. However, in real life, the particles may be charged, and therefore may be accelerating locally in accordance with the Lorentz force. That is:

{d^2 X^\mu \over ds^2} = {q \over m} {F^{\mu \beta}} {d X^\alpha \over ds}{\eta_{\alpha \beta}}.
with

{\eta_{\alpha \beta}}{d X^\alpha \over ds}{d X^\beta \over ds}=1.
The Minkowski tensor η is given by:

\eta = \begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}
These last three equations can be used as the starting point for the derivation of an equation of motion in General Relativity, instead of assuming that acceleration is zero in free fall.^{[2]} Because the Minkowski tensor is involved here, it becomes necessary to introduce something called the metric tensor in General Relativity. The metric tensor g is symmetric, and locally reduces to the Minkowski tensor in free fall. The resulting equation of motion is as follows:

{d^2 x^\mu \over ds^2} = \Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}\ +{q \over m} {F^{\mu \beta}} {d x^\alpha \over ds}{g_{\alpha \beta}}.
with

{g_{\alpha \beta}}{d x^\alpha \over ds}{d x^\beta \over ds}=1.
This last equation signifies that the particle is moving along a timelike geodesic; massless particles like the photon instead follow null geodesics (replace −1 with zero on the righthand side of the last equation). It is important that the last two equations are consistent with each other, when the latter is differentiated with respect to proper time, and the following formula for the Christoffel symbols ensures that consistency:

\Gamma^{\lambda}{}_{\alpha\beta}=\frac{1}{2}g^{\lambda \tau} \left(\frac{\partial g_{\tau\alpha}}{\partial x^\beta} + \frac{\partial g_{\tau\beta}}{\partial x^{\alpha}}  \frac{\partial g_{\alpha\beta}}{\partial x^{\tau}} \right)
This last equation does not involve the electromagnetic fields, and it is applicable even in the limit as the electromagnetic fields vanish. The letter g with superscripts refers to the inverse of the metric tensor. In General Relativity, indices of tensors are lowered and raised by contraction with the metric tensor or its inverse, respectively.
Geodesics as curves of stationary interval
A geodesic between two events can also be described as the curve joining those two events which has a stationary interval (4dimensional "length"). Stationary here is used in the sense in which that term is used in the calculus of variations, namely, that the interval along the curve varies minimally among curves that are nearby to the geodesic.
In Minkowski space there is only one timelike geodesic that connects any given pair of timelike separated events, and that geodesic is the curve with the longest proper time between the two events. But in curved spacetime, it's possible for a pair of widelyseparated events to have more than one timelike geodesic that connects them. In such instances, the proper times along the various geodesics will not in general be the same. And for some geodesics in such instances, it's possible for a curve that connects the two events and is nearby to the geodesic to have either a longer or a shorter proper time than the geodesic.^{[7]}
For a spacelike geodesic through two events, there are always nearby curves which go through the two events that have either a longer or a shorter proper length than the geodesic, even in Minkowski space. In Minkowski space, in an inertial frame of reference in which the two events are simultaneous, the geodesic will be the straight line between the two events at the time at which the events occur. Any curve that differs from the geodesic purely spatially in that frame of reference will have a longer proper length than the geodesic, but a curve that differs from the geodesic purely temporally in that frame of reference will have a shorter proper length.
The interval of a curve in spacetime is

l = \int \sqrt{\leftg_{\mu \nu} \dot x^\mu \dot x^\nu \right} \, ds\ .
Then, the Euler–Lagrange equation,

{d \over ds} {\partial \over \partial \dot x^\alpha} \sqrt{\left g_{\mu \nu} \dot x^\mu \dot x^\nu \right} = {\partial \over \partial x^\alpha} \sqrt{\left g_{\mu \nu} \dot x^\mu \dot x^\nu \right} \ ,
becomes, after some calculation,

2(\Gamma^\lambda {}_{\mu \nu} \dot x^\mu \dot x^\nu + \ddot x^\lambda) = U^\lambda {d \over ds} \ln U_\nu U^\nu \ ,
where U^\mu = \dot x^\mu .
Proof
The goal being to find a curve for which the value of

l = \int d\tau = \int {d\tau \over d\phi} \, d\phi = \int \sqrt \, d\phi = \int \sqrt \, d\phi = \int f \, d\phi
is stationary, where

f = \sqrt{g_{\mu \nu} \dot x^\mu \dot x^\nu}
such goal can be accomplished by calculating the Euler–Lagrange equation for f, which is

{d \over d\tau} {\partial f \over \partial \dot x^\lambda} = {\partial f \over \partial x^\lambda} .
Substituting the expression of f into the Euler–Lagrange equation (which makes the value of the integral l stationary), gives

{d \over d\tau} {\partial \sqrt{g_{\mu \nu} \dot x^\mu \dot x^\nu} \over \partial \dot x^\lambda} = {\partial \sqrt{g_{\mu \nu} \dot x^\mu \dot x^\nu} \over \partial x^\lambda}
Now calculate the derivatives: {d \over d\tau} \left( {g_{\mu \nu} {\partial \dot x^\mu \over \partial \dot x^\lambda} \dot x^\nu  g_{\mu \nu} \dot x^\mu {\partial \dot x^\nu \over \partial \dot x^\lambda} \over 2 \sqrt{g_{\mu \nu} \dot x^\mu \dot x^\nu}} \right) = {g_{\mu \nu, \lambda} \dot x^\mu \dot x^\nu \over 2 \sqrt{g_{\mu \nu} \dot x^\mu \dot x^\nu}} \qquad \qquad (1)
{d \over d\tau} \left( {g_{\mu \nu} \delta^\mu {}_\lambda \dot x^\nu + g_{\mu \nu} \dot x^\mu \delta^\nu {}_\lambda \over 2 \sqrt{g_{\mu \nu} \dot x^\mu \dot x^\nu}} \right) = {g_{\mu \nu , \lambda} \dot x^\mu \dot x^\nu \over 2 \sqrt{g_{\mu \nu} \dot x^\mu \dot x^\nu}} \qquad \qquad (2)
{d \over d\tau} \left( {g_{\lambda \nu} \dot x^\nu + g_{\mu \lambda} \dot x^\mu \over \sqrt{g_{\mu \nu} \dot x^\mu \dot x^\nu}} \right) = {g_{\mu \nu , \lambda} \dot x^\mu \dot x^\nu \over \sqrt{g_{\mu \nu} \dot x^\mu \dot x^\nu}} \qquad \qquad (3)
{\sqrt{g_{\mu \nu} \dot x^\mu \dot x^\nu} {d \over d\tau} (g_{\lambda \nu} \dot x^\nu + g_{\mu \lambda} \dot x^\mu)  (g_{\lambda \nu} \dot x^\nu + g_{\mu \lambda} \dot x^\mu) {d \over d\tau} \sqrt{g_{\mu \nu} \dot x^\mu \dot x^\nu} \over g_{\mu \nu} \dot x^\mu \dot x^\nu} = {g_{\mu \nu , \lambda} \dot x^\mu \dot x^\nu \over \sqrt{g_{\mu \nu} \dot x^\mu \dot x^\nu}} \qquad \qquad (4)
{(g_{\mu \nu} \dot x^\mu \dot x^\nu) {d \over d\tau} (g_{\lambda \nu} \dot x^\nu + g_{\mu \lambda} \dot x^\mu) + {1 \over 2} (g_{\lambda \nu} \dot x^\nu + g_{\mu \lambda} \dot x^\mu) {d \over d\tau} (g_{\mu \nu} \dot x^\mu \dot x^\nu) \over g_{\mu \nu} \dot x^\mu \dot x^\nu} = g_{\mu \nu ,\lambda} \dot x^\mu \dot x^\nu \qquad \qquad (5)
(g_{\mu \nu} \dot x^\mu \dot x^\nu) (g_{\lambda \nu ,\mu} \dot x^\nu \dot x^\mu + g_{\mu \lambda ,\nu} \dot x^\mu \dot x^\nu + g_{\lambda \nu} \ddot x^\nu + g_{\lambda \mu} \ddot x^\mu)

= (g_{\mu \nu ,\lambda} \dot x^\mu \dot x^\nu) (g_{\alpha \beta} \dot x^\alpha \dot x^\beta) + {1 \over 2} (g_{\lambda \nu} \dot x^\nu + g_{\lambda \mu} \dot x^\mu) {d \over d\tau} (g_{\mu \nu} \dot x^\mu \dot x^\nu) \qquad \qquad (6)
g_{\lambda \nu ,\mu} \dot x^\mu \dot x^\nu + g_{\lambda \mu ,\nu} \dot x^\mu \dot x^\nu  g_{\mu \nu ,\lambda} \dot x^\mu \dot x^\nu + 2 g_{\lambda \mu} \ddot x^\mu = {\dot x_\lambda {d \over d\tau} (g_{\mu \nu} \dot x^\mu \dot x^\nu) \over g_{\alpha \beta} \dot x^\alpha \dot x^\beta} \qquad \qquad (7)
2(\Gamma_{\lambda \mu \nu} \dot x^\mu \dot x^\nu + \ddot x_\lambda) = {\dot x_\lambda {d \over d\tau} (\dot x_\nu \dot x^\nu) \over \dot x_\beta \dot x^\beta} = {U_\lambda {d \over d\tau} (U_\nu U^\nu) \over U_\beta U^\beta} = U_\lambda {d \over d\tau} \ln U_\nu U^\nu \qquad \qquad (8)
This is just one step away from the geodesic equation.
If the parameter s is chosen to be affine, then the right side the above equation vanishes (because U_\nu U^\nu is constant). Finally, we have the geodesic equation

\Gamma^\lambda {}_{\mu \nu} \dot x^\mu \dot x^\nu + \ddot x^\lambda = 0\ .
See also
Bibliography

Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, (1972) John Wiley & Sons, New York ISBN 0471925675 See chapter 3.

Lev D. Landau and Evgenii M. Lifschitz, The Classical Theory of Fields, (1973) Pergammon Press, Oxford ISBN 0080181767 See section 87.

Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0716703440.

Bernard F. Schutz, A first course in general relativity, (1985; 2002) Cambridge University Press: Cambridge, UK; ISBN 0521277035. See chapter 6.

Robert M. Wald, General Relativity, (1984) The University of Chicago Press, Chicago. See Section 3.3.
References

^ Will, Clifford. Theory and Experiment in Gravitational Physics, p. 143 (Cambridge University Press 1993).

^ ^{a} ^{b} Weinberg, Steven. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Wiley 1972).

^ Plebanski, Jerzy and Krazinski, Adrzej. An Introduction to General Relativity and Cosmology, p. 34 (Cambridge University Press, 2006).

^ Einstein, Albert. The Meaning of Relativity, p. 113 (Psychology Press 2003).

^ Tamir, M. "Proving the principle: Taking geodesic dynamics too seriously in Einstein’s theory", Studies In History and Philosophy of Modern Physics 43(2), 137–154 (2012).

^ Plebanski, Jerzy and Krazinski, Adrzej. An Introduction to General Relativity and Cosmology, p. 143 (Cambridge University Press, 2006).

^
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