Chapter 1: Q15P (page 1)

Find the geodesics on a plane using polar coordinates.

Short Answer

Expert verified

$θ = a r c \tan (\frac{\sqrt{r^{2} - C^{2}}}{C}) + B$ , where $C$ is a constant and $B$ is the integration constant.

Step by step solution

Given Information

Geodesics on a plane is to be found out using polar coordinates by Euler equations.

Definition of Euler equation

In the calculus of variations andclassical mechanics, the Euler equations is a system of second-orderordinary differential equations whose solutions arestationary points of the givenaction functional.

Use Euler equation

Geodesics on a plane is to be found out using polar coordinates. So distance integral is to be minimized.

$\int d s = \int \sqrt{d r^{2} + r^{2} d θ} = \int \sqrt{1 + r^{2} θ'^{2}} d r$

Let $F = \sqrt{1 + r^{2} θ'^{2}}$

Euler equation for coordinates $(r, θ)$ is $\frac{d}{d r} (\frac{\partial F}{\partial θ'}) - \frac{\partial F}{\partial θ} = 0$

Calculate the required derivatives

$\frac{\partial F}{\partial θ'} = \frac{r^{2} θ'}{r \sqrt{1 + r^{2} θ'^{2}}} \frac{\partial F}{\partial θ} = 0$

Therefore,

localid="1665037647675" $\frac{d}{d r} [\frac{r^{2} θ'}{\sqrt{1 + r^{2} θ'^{2}}}] = 0 \frac{r^{2} θ'}{\sqrt{1 + r^{2} θ'^{2}}} = C θ'^{2} = \frac{C^{2}}{r^{2} (r^{2} - C^{2})} θ' = \frac{C}{r \sqrt{r^{2} - C^{2}}}$

Where $C$ is constant.

Integrate $θ' = \frac{C}{r \sqrt{r^{2} - C^{2}}}$ to get the desired result

$θ' = \int \frac{C}{r \sqrt{r^{2} - C^{2}}} d r = a r c \tan (\frac{\sqrt{r^{2} - C^{2}}}{C}) + B$

Therefore, $θ = a r c \tan (\frac{\sqrt{r^{2} - C^{2}}}{C}) + B$ , where $C$ is a constant and $B$ is the integration constant.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Find the geodesics on a plane using polar coordinates.

Short Answer

Step by step solution

Given Information

Definition of Euler equation

Use Euler equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

Solid State Physics

Translational Dynamics

Space Physics

Kinematics Physics

Fluids

Study anywhere. Anytime. Across all devices.