I now see where you are coming from.  What you are doing is basically
applying the calculus of variations to a Lagrangian.  In each
Euler-Lagrange Equation, you arrive at a geodesic equation for each
coordinate.  Of course, in this case, we are blessfully limited to only
one spatial dimension plus time.

So, let's apply the same method to the good old Newtonian Lagrangian.

        L = (dr/dt)^2 / 2 + G M / r

At (r = infinity), say (dr/dt = v).  Thus, we have

        (dr/dt)^2 = 2 (v^2 - G M / r)

Applying the operator (d/dr) to the equation above, we have

        2 (d^2r/dt^2) = 2 G M / r^2

Or

        d^2r/dt^2 = G M / r^2

This means gravitational acceleration in a Newtonian system is always
repulsive.  Of course, this is not the case.  Thus, you must be very
careful performing partial derivatives.

 See also

https://home.deds.nl/~dvdm/dirk/Physics/Fumbles/PrivateLagrangian.html