i know i can write the geodesic equation for a massive particle as:

${\dot{x}}^{\nu}{\mathrm{\nabla}}_{\nu}{\dot{x}}^{\mu}=0$

and then we can express this using the 4 momentum, ${p}^{\mu}=m{u}^{\mu}=m{\dot{x}}^{\mu}$

${p}^{\nu}{\mathrm{\nabla}}_{\nu}{p}^{\mu}=0$

I want to show that this can be written as

$m\frac{d{p}_{\mu}}{d\tau}=\frac{1}{2}{p}^{\sigma}{p}^{\rho}{\mathrm{\partial}}_{\mu}{g}_{\rho \sigma}$

i expanded the covariant derivative out and lowered some of the indices using the metric to obtain that

${g}^{\mu \rho}{p}^{\nu}[{\mathrm{\partial}}_{\nu}{p}_{\rho}-{\mathrm{\Gamma}}_{\nu \rho}^{\alpha}{p}_{\alpha}]=0$

the first term in that expression can be written as

${g}^{\mu \rho}m\frac{\mathrm{\partial}{p}_{\rho}}{\mathrm{\partial}\tau}$

and i know i can get a 1/2 out of the christofell symbol but everything i try ends in a mess of metrics derivatives and indices.

${\dot{x}}^{\nu}{\mathrm{\nabla}}_{\nu}{\dot{x}}^{\mu}=0$

and then we can express this using the 4 momentum, ${p}^{\mu}=m{u}^{\mu}=m{\dot{x}}^{\mu}$

${p}^{\nu}{\mathrm{\nabla}}_{\nu}{p}^{\mu}=0$

I want to show that this can be written as

$m\frac{d{p}_{\mu}}{d\tau}=\frac{1}{2}{p}^{\sigma}{p}^{\rho}{\mathrm{\partial}}_{\mu}{g}_{\rho \sigma}$

i expanded the covariant derivative out and lowered some of the indices using the metric to obtain that

${g}^{\mu \rho}{p}^{\nu}[{\mathrm{\partial}}_{\nu}{p}_{\rho}-{\mathrm{\Gamma}}_{\nu \rho}^{\alpha}{p}_{\alpha}]=0$

the first term in that expression can be written as

${g}^{\mu \rho}m\frac{\mathrm{\partial}{p}_{\rho}}{\mathrm{\partial}\tau}$

and i know i can get a 1/2 out of the christofell symbol but everything i try ends in a mess of metrics derivatives and indices.