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I'm considering the Eddington-Robertson-Schiff line element which is given by

[tex] (ds)^2 = \left( 1 - 2 \left(\frac{\mu}{r}\right) + 2 \left(\frac{\mu^2}{r^2}\right) \right) dt^2 - \left( 1 + 2 \left( \frac{\mu}{r} \right) \right) (dr^2 + r^2 d\theta^2 + r^2 \sin^2{\theta} \;d\phi^2 ),[/tex]

where [itex]\mu = GM = \text{const.}[/itex] and [itex] r=|\mathbf{r}|.[/itex]

I'm interested in determining the equations of motion for such a line element which can be obtained from the least action principle. The classical action [itex]S[/itex] is the integral along the particle trajectory

[tex] S = \int ds, [/tex]

which can be equivalently expressed as

[tex] S = \int \left( \frac{ds}{dt} \right) dt \equiv \int L \; dt. [/tex]

We can see from the above that

[tex] L = \left[ \left( 1 - 2 \left(\frac{\mu}{r}\right) + 2 \left(\frac{\mu^2}{r^2}\right) \right) - \left( 1 + 2 \left( \frac{\mu}{r} \right) \right) (\mathbf{\dot{r}} \cdot \mathbf{\dot{r}}) \right]^{1/2}, [/tex]

where [itex]L[/itex] is the associated Lagrangian over time.

Problem and question

The associated equations of motion are given by (Eq. 20)

[tex]\frac{d^2\mathbf{r}}{dt^2} = \frac{\mu}{r^3} \left[ \left(4 \frac{\mu}{r} - v^2 \right) \mathbf{r} + 4 (\mathbf{r}\cdot \mathbf{\dot{r}} ) \mathbf{\dot{r}}\right]. [/tex]

I cannot for the life of me obtain this using the Euler-Lagrange equations.

Attempt at a solution:

The Euler-Lagrange equations are given by

[tex] \frac{d}{dt} \left( \frac{\partial L}{\partial \mathbf{\dot{r}}} \right) - \frac{\partial L}{ \partial \mathbf{r}} =0. [/tex]

I note that the equations of motion should be equivalent for either

[tex] L = \sqrt{g_{\mu\nu} \dot{x}^{\mu}\dot{x}^\mu}, [/tex]

or

[tex] L = g_{\mu\nu} \dot{x}^{\mu}\dot{x}^\mu. [/tex]

Bearing this in mind and working through the process using

[tex] L = \left[ \left( 1 - 2 \left(\frac{\mu}{r}\right) + 2 \left(\frac{\mu^2}{r^2}\right) \right) - \left( 1 + 2 \left( \frac{\mu}{r} \right) \right) (\mathbf{\dot{r}} \cdot \mathbf{\dot{r}}) \right], [/tex]

I find

[tex] \frac{d}{dt} \left( \frac{\partial L}{ \partial \mathbf{\dot{r}}} \right) = -2 \left[ \left( 1 + 2 \frac{\mu}{r} \mathbf{\ddot{r}} \right) - 2 \frac{\mu}{r^3} (\mathbf{r}\cdot \mathbf{\dot{r}}) \mathbf{\dot{r}} \right], [/tex]

and

[tex] \left( \frac{\partial L }{\partial \mathbf{r}} \right) = 2\frac{\mu}{r^3} \mathbf{r} - 4 \frac{\mu^2}{r^4} \mathbf{r} + 2 \frac{\mu}{r^3} \mathbf{r} (\mathbf{\dot{r}} \cdot \mathbf{\dot{r}} ). [/tex]

Clearly, adding these together does not give the desired result. Any suggestions?

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# A How to derive equations of motion in GR?

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