Getting geodesic from variational principle

In summary: Then the Euler-Lagrange equation is$$L=\frac{1}{2}\left[\frac{d}{d\theta} \left( \frac{\partial L}{\partial \dot \theta}\right) - \frac{\partial L}{\partial \dot \theta} \right]$$
  • #1
LCSphysicist
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Homework Statement
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Relevant Equations
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The metric is $$ds^2 = \frac{dr^2 + r^2 d\theta ^2}{r^2-a^2} - \frac{r^2 dr^2}{(r^2-a^2)^2}$$

I need to prove the geodesic is: $$a^2 (\frac{dr}{d \theta})^2 + a^2 r^2 = K r^4$$

My method was to variate the action ##\int\frac{(\frac{dr}{d\theta})^2 + r^2 }{r^2-a^2} - \frac{r^2 (\frac{dr}{d\theta})^2}{(r^2-a^2)^2} d \theta##

But the equation i am getting is $$a^2 (\frac{dr}{d \theta})^2 + r^2 = K (r^2-a^2)^2$$
Which can not be reduced to the answer.

I am a little confused, i could simpy calculate the Christoffel symbol, but i think this variation method easier, yet i am not sure how to use it.

So basically, my guess on why i have got the wrong answer is that the "lagrangean" i am variating is wrong. So what lagrangean indeed give us the right answer? Only ##g_{ab} \frac{dx^a}{ds} \frac{dx^b}{ds}= 1 ## and ##\sqrt{g_{ab} \frac{dx^a}{d\lambda} \frac{dx^b}{d\lambda}}##? Or the lagrangean i used above is right, and i have done some algebric error?
 
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  • #2
Your approach failed because in general the Euler-Lagrange equations produced from [itex]L_1(r, dr/d\theta)[/itex] are not the same as those obtained from [itex]L_2(r, dr/d\theta) = \sqrt{L_1}[/itex]: [tex]
\begin{split}
\frac{d}{d\theta} \left( \frac{\partial L_2}{\partial r'}\right) - \frac{\partial L_2}{\partial r}
&= \frac{d}{d\theta} \left( \frac{1}{2L_2} \frac{\partial L_1}{\partial r'} \right) - \frac{1}{2L_2}\frac{\partial L_1}{\partial r} \\
&= \frac{1}{2L_2} \left[\frac{d}{d\theta} \left( \frac{\partial L_1}{\partial r'}\right) - \frac{\partial L_1}{\partial r}\right] + \frac{\partial L_1}{\partial r'} \frac{d}{d\theta}\left( \frac{1}{2L_2} \right) \\
&\neq \frac{1}{2L_2} \left[\frac{d}{d\theta} \left( \frac{\partial L_1}{\partial r'}\right) - \frac{\partial L_1}{\partial r}\right]
\end{split}
[/tex] If you take [itex]s[/itex] as the independent variable then you can work with [itex]f(r) r'^2 + g(r) \theta'^2[/itex] directly because by definition its total derivative with respect to [itex]s[/itex] is zero and you end up with the same Euler-Lagrange equation as you obtain from [itex]\sqrt{f(r) r'^2 + g(r) \theta'^2}[/itex].
 
  • #3
You have a Lagrangian system
$$L=\dot r^2\Big(\frac{1}{r^2-a^2}-\frac{r^2}{(r^2-a^2)^2}\Big)+\frac{r^2}{r^2-a^2}\dot\theta^2.$$
You must consider this system at the energy level ##L=1##
perhaps the following cyclic integral will be of use
$$\frac{\partial L}{\partial \dot \theta}=const$$
 
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Related to Getting geodesic from variational principle

1. What is the variational principle?

The variational principle is a mathematical principle that states that the path taken by a system between two points is the one that minimizes or maximizes a certain quantity, such as energy or action.

2. How does the variational principle relate to geodesics?

The variational principle can be used to derive the equations of motion for a system, such as the geodesic equation for a curved space. This equation describes the path that a particle would take in a curved space, which is known as a geodesic.

3. What does it mean to "get geodesic" from the variational principle?

Getting geodesic from the variational principle means using the variational principle to derive the equations of motion for a system, specifically the geodesic equation for a curved space.

4. Why is the variational principle important in understanding geodesics?

The variational principle allows us to understand the fundamental principles that govern the behavior of systems, including the path of a particle in a curved space. It provides a powerful tool for understanding and predicting the behavior of physical systems.

5. Can the variational principle be applied to other systems besides geodesics?

Yes, the variational principle can be applied to a wide range of systems, including classical mechanics, electromagnetism, and quantum mechanics. It is a fundamental principle that has many applications in physics and other scientific fields.

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