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d-dt-dx-dt-1-2-g-g-g-g-dx-dt-dx-dt-0-




Question Number 197344 by MathedUp last updated on 14/Sep/23
((d  )/dt)∙(dx^𝛌 /dt)+(1/2)g^(𝛌𝛂) (∂_𝛍 ^  g_(𝛂𝛎) +∂_𝛎 ^  g_(𝛂𝛍) −∂_𝛂 ^  g_(𝛍𝛎) )(dx^𝛍 /dt)∙(dx^𝛎 /dt)=0
$$\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\centerdot\frac{\mathrm{d}{x}^{\boldsymbol{\lambda}} }{\mathrm{d}{t}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{g}^{\boldsymbol{\lambda\alpha}} \left(\partial_{\boldsymbol{\mu}} ^{\:} \mathrm{g}_{\boldsymbol{\alpha\nu}} +\partial_{\boldsymbol{\nu}} ^{\:} \mathrm{g}_{\boldsymbol{\alpha\mu}} −\partial_{\boldsymbol{\alpha}} ^{\:} \mathrm{g}_{\boldsymbol{\mu\nu}} \right)\frac{\mathrm{d}{x}^{\boldsymbol{\mu}} }{\mathrm{d}{t}}\centerdot\frac{\mathrm{d}{x}^{\boldsymbol{\nu}} }{\mathrm{d}{t}}=\mathrm{0} \\ $$
Commented by TheHoneyCat last updated on 20/Sep/23
I'm guessing you're looking for a paramedic expression of a geodesic, but if you don't give us the metric, we can't really do much.

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