User Tools

Site Tools


equations:geodesic_equation

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
equations:geodesic_equation [2018/04/14 10:48]
aresmarrero
equations:geodesic_equation [2018/12/19 11:01] (current)
jakobadmin ↷ Links adapted because of a move operation
Line 3: Line 3:
  
 <tabbox Intuitive> ​ <tabbox Intuitive> ​
 +Solutions of the geodesic equations are called geodesics. ​
  
-<note tip> +Geodesics are the "​shortest"​ paths between two points ​in a flat spacetime and the straightest path between two points in curved spacetime 
-Explanations ​in this section should contain no formulas, but instead colloquial things like you would hear them during ​coffee break or at cocktail party. +
-</​note>​+
   ​   ​
 +On a sphere the geodesics are "great circles"​. ​
 <tabbox Concrete> ​ <tabbox Concrete> ​
  
-<note tip> +**Derivation**
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas. +
-</​note>​ +
-  +
-<tabbox Abstract> ​+
  
-<note tip> +The Lagrangian for a free point particle ​in a spacetime $Q$ is  
-The motto in this section ​is: //the higher the level of abstraction, ​the better//+\begin{align*} 
-</​note>​+ ​L(q,​\dot{q}) &= m\sqrt{g(q)(\dot{q},​\dot{q})} \\  
 + &​= m\sqrt{g_{ij}\dot{q}^i\dot{q}^j} 
 +\end{align*} 
 +just like in [[models: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*} 
 + ​L(q,​\dot{q}) &= \tfrac{1}{2}m g(q)(\dot{q},​\dot{q}) \\  
 + &​= \tfrac{1}{2}m g_{ij}\dot{q}^i\dot{q}^j 
 +\end{align*}
  
-<tabbox Why is it interesting?> ​  +We now want, as usual, to find the equations of motion. ​ Using the [[equations:​euler_lagrange_equations|Euler-Lagrange equations]] we get
  
 +\begin{align*}
 + p_i = \frac{\partial L}{\partial\dot{q}^i} &= mg_{ij}\dot{q}^j \\
 + F_i = \frac{\partial L}{\partial q^i} 
 + &​= \frac{\partial}{\partial q^i}\Bigl(\tfrac{1}{2}mg_{jk}(q)\dot{q}^j\dot{q}^k\Bigr)\\
 + &​=\tfrac{1}{2}m\partial{i}g_{jk}\dot{q}^j\dot{q}^k,​
 + ​\quad(\text{where } \partial_i=\frac{\partial}{\partial q^i}).
 +\end{align*}
 +So the Euler--Lagrange equations say
 +\[
 + ​\frac{d}{dt}mg_{ij}\dot{q}^j = \tfrac{1}{2}m\partial_{i}g_{jk}\dot{q}^j\dot{q}^k.
 +\]
 +An important observation is that the mass factors away. Therefore, the motion is independent of the mass!  ​
 +
 +We can rewrite the geodesic equation as follows
 +\begin{align*}
 + ​\frac{d}{dt}g_{ij}\dot{q}^j &= \tfrac{1}{2}\partial_{i}g_{jk}\dot{q}^j\dot{q}^k \\
 + ​\hspace{-3ex}\rightarrow\quad
 + ​\partial_{k}g_{ij}\dot{q}^k\dot{q}^j + g_{ij}\ddot{q}^j ​
 + &​= \tfrac{1}{2}\partial_{i}g_{jk}\dot{q}^j\dot{q}^k \\
 + ​\hspace{-3ex}\rightarrow\quad
 + ​g_{ij}\ddot{q}^j &= \bigl(\tfrac{1}{2}\partial_{i}g_{jk}-\partial_{k}g_{ij}\bigr)\dot{q}^j\dot{q}^k \\
 + &​= \tfrac{1}{2}\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)\dot{q}^j\dot{q}^k
 +\end{align*}
 +where the last line follows since $g_{ik}=g_{ki}$.
 +Now we define the so-called [[advanced_tools:​connections:​levi_civita_connection|Christoffel symbols]]
 +\[
 + ​\Gamma_{ijk} \equiv -\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)
 +\]
 +
 +Using this definition, we can write the geodesic equation as
 +\begin{align*}
 + ​\ddot{q}_i = g_{ij}\ddot{q}^j &= -\Gamma_{ijk}\dot{q}^j\dot{q}^k \\
 + ​\hspace{-3ex}\rightarrow\quad
 + ​\ddot{q}^i &= -\Gamma^i_{jk}\dot{q}^j\dot{q}^k.
 +\end{align*}
 +
 +<tabbox Abstract> ​
 +
 +Geodesics are paths $q:​[t_0,​t_1]\rightarrow Q$ that are critical points of the action
 +\[
 + S(q) = \int_{t_0}^{t_1}\sqrt{g_{ij} \dot{q}^i\dot{q}^j}\,​dt
 +\]
 +This action is exactly the proper time when $(Q,g)$ is a Lorentzian manifold, or arclength when $(Q,g)$ is a Riemannian manifold.
 +<tabbox Why is it interesting?> ​
 Almost every problem in classical mechanics can be regarded as geodesic motion. Almost every problem in classical mechanics can be regarded as geodesic motion.
 +
 +In the geometric optics approximation light acts like particles tracing out geodesics, i.e. the shortest paths.
 +
 +Especially in [[models:​general_relativity|general relativity]] particles always follow geodesics. To be precise, a free particle in general relativity traces out a geodesic on the Lorentzian manifold, i.e. spacetime.
 +
  
 </​tabbox>​ </​tabbox>​
  
  
equations/geodesic_equation.1523695682.txt.gz · Last modified: 2018/04/14 08:48 (external edit)