equations:geodesic_equation
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equations:geodesic_equation [2018/05/05 21:59] jakobadmin [Concrete] |
equations:geodesic_equation [2018/05/05 22:00] jakobadmin [Concrete] |
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| The Lagrangian for a free point particle in a spacetime $Q$ is | The Lagrangian for a free point particle in a spacetime $Q$ is | ||
| - | \begin{align} | + | \begin{align*} |
| L(q,\dot{q}) &= m\sqrt{g(q)(\dot{q},\dot{q})} \\ | L(q,\dot{q}) &= m\sqrt{g(q)(\dot{q},\dot{q})} \\ | ||
| &= m\sqrt{g_{ij}\dot{q}^i\dot{q}^j} | &= m\sqrt{g_{ij}\dot{q}^i\dot{q}^j} | ||
| - | \end{align} | + | \end{align*} |
| just like in [[theories:special_relativity|special relativity]] but instead of the [[advanced_tools:minkowski_metric|Minkowski metric]] $\eta_{ij}$, we now have a general metric $g_{ij}$. Alternatively we can use | just like in [[theories:special_relativity|special relativity]] but instead of the [[advanced_tools:minkowski_metric|Minkowski metric]] $\eta_{ij}$, we now have a general metric $g_{ij}$. Alternatively we can use | ||
| \begin{align*} | \begin{align*} | ||
equations/geodesic_equation.txt · Last modified: 2018/12/19 11:01 by jakobadmin
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