Wolfram Alpha:

```Geodesics
---------

Consider a vector field Vn.  It is contravariant because we are considering
displacement.  The rate of change of x with respect to the proper time, τ,
along a world line is given by:

dVn/dτ = (∂Vn/∂xm)(dxm/dτ)

where dxm/dτ is the TANGENT VECTOR to the world line.   It is a
contravariant 4-velocity vector.  dτ = √gμνdxμdxν

DsVn = DmVndxm/dτ  --- replace with the covariant derivative

= (∂Vn/∂xm)(dxm/dτ) + ΓnmrVrdxm/dτ

= dVn/dτ + ΓnmrVrdxm/dτ --- chain rule.

How does tangent vector vary along τ?  Compute its covariant derivative.

Vn -> dxn/dτ.  Thus,

Dsdxn/dτ = d2xn/dτ2 + Γnmr(dxm/dτ)(dxr/dτ)

The above equation describes the trajectories followed by freely falling
objects in spacetime.  If we define these as curves of zero proper
acceleration we can write:

d2xn/dτ2 + Γnmr(dxm/dτ)(dxr/dτ) = 0

Therefore,

d2xn/dτ2 = -Γnmr(dxm/dτ)(dxr/dτ)

This is the equation for a GEODESIC.  The geodesic is the equation
of motion that defines the straightest, and therefore shortest, path
through curved space-time.  By this we mean that once the
gravitational field is given (i.e. Γ is known) this equation
tells us how the object will move in such a field.

Action of a Geodesic - Particle Moving in Arbitrary Gravitational Field
-----------------------------------------------------------------------

By definition, the action of a particle is -mc2∫dτ

dτ2 = (1 + 2φ/c2)dt2 - (1/c2)dx2

A = -mc2∫√[(1 + 2φ/c2)dt2 - (1/c2)dx2]

= -mc2∫√[(1 + 2φ/c2) - (1/c2)dx2/dt2]dt
.
= -mc2∫√[(1 + (1/c2)(2φ - x2)]dt

Use the binomial theorem √(1 + x) = 1 + x/2
.
A = -mc2∫(1 + (1/2c2)(2φ - x2)dt

So the Lagrangian is:

L = -mc2[(1 + (1/2c2)(2φ - x2)]
.
= -mc2 - mφ + mx2/2```