# Getting geodesic from variational principle

• LCSphysicist
In summary: Then the Euler-Lagrange equation is$$L=\frac{1}{2}\left[\frac{d}{d\theta} \left( \frac{\partial L}{\partial \dot \theta}\right) - \frac{\partial L}{\partial \dot \theta} \right]$$
LCSphysicist
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?

Your approach failed because in general the Euler-Lagrange equations produced from $L_1(r, dr/d\theta)$ are not the same as those obtained from $L_2(r, dr/d\theta) = \sqrt{L_1}$: $$\begin{split} \frac{d}{d\theta} \left( \frac{\partial L_2}{\partial r'}\right) - \frac{\partial L_2}{\partial r} &= \frac{d}{d\theta} \left( \frac{1}{2L_2} \frac{\partial L_1}{\partial r'} \right) - \frac{1}{2L_2}\frac{\partial L_1}{\partial r} \\ &= \frac{1}{2L_2} \left[\frac{d}{d\theta} \left( \frac{\partial L_1}{\partial r'}\right) - \frac{\partial L_1}{\partial r}\right] + \frac{\partial L_1}{\partial r'} \frac{d}{d\theta}\left( \frac{1}{2L_2} \right) \\ &\neq \frac{1}{2L_2} \left[\frac{d}{d\theta} \left( \frac{\partial L_1}{\partial r'}\right) - \frac{\partial L_1}{\partial r}\right] \end{split}$$ If you take $s$ as the independent variable then you can work with $f(r) r'^2 + g(r) \theta'^2$ directly because by definition its total derivative with respect to $s$ is zero and you end up with the same Euler-Lagrange equation as you obtain from $\sqrt{f(r) r'^2 + g(r) \theta'^2}$.

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
$$\frac{\partial L}{\partial \dot \theta}=const$$

vanhees71

## 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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