[SR] - Test Particle inside the Sun's Gravitational Field - Part 2

[Athenian]

Homework Statement

[Question Context: Consider the motion of a test particle of (constant) mass $m$ inside the gravitational field produced by the sun in the context of special relativity.

Consider the equations of motion for the test particle, which can be written as $$\frac{d(m\gamma c)}{dt} = \frac{\vec{v}}{c} \cdot \vec{F},$$

OR

$$\frac{d(m\gamma \vec{v})}{dt} = \vec{F},$$

where $\vec{v}$ is the speed of the test particle, $c$ is the (constant) speed of light, and by definition, $$\gamma \equiv \frac{1}{\sqrt{1- \frac{\vec{v}^2}{c^2}}} .$$

In addition, the gravitational force is given by $$\vec{F} \equiv -\frac{GMm}{r^2} \hat{e}_r$$

where $\hat{e}_r$ is the unit vector in the direction between the Sun (of mass M) and the test particle (of mass $m$).]
------------------------------------

Question:
Study the planar motion in the $x-y$ plane, so that for this test particle $z$ = constant = $0$. In other words, the orbit if planar. Use polar coordinates $(r, \theta)$ defined as
$$x = r cos\theta , y=r sin \theta$$

and write the equations of motions in the coordinate frame centered in the Sun and rotating by $\theta(t)$ with respect to an inertial frame, so that, in this rotating frame, the position of the test particle, at any time, is given by $\vec{x} = r \hat{e_r}$, and $\vec{v} = \dot{\vec{x}} = \dot{r} \hat{e_r} + r\dot{\theta} \hat{e_\theta}$. In other words, write down the equations of motion in the system of polar coordinates.

Relevant Equations

Refer above.

To begin with, I posted this thread ahead of time simply because I thought it may provide me some insight on how to solve for another problem that I have previously posted here: https://www.physicsforums.com/threa...inside-suns-gravitational-field.983171/unread.

With that said, my attempt for a solution is quite simply summed up in this YouTube video provided here: .

Essentially, would following the instructions of the YouTube video above provide me the solution for "the equations of motions in the system of polar coordinates"? Or, am I misunderstanding the given problem? If not, should I be mindful of any differences (or anything in general) about the question when following along the video?

Perhaps I may be coming out as rude for asking such blatant requests, however, any amount of assistance toward solving this question would be greatly appreciated as I am confused on how to properly begin. Thank you in advance!

 

[PeroK]

This video presents a derivation of the acceleration vector in polar coordinates. That's a starting point for any problem where you want to study motion using polar coordinates.

 

[Athenian]

Phew. So, here's the attempted solution. Please tell me what you all think. Thank you!

Part (a)
$$\frac{d(m\gamma c)}{dt} = \frac{\vec{v}}{c} \cdot \vec{F}$$
$$\implies \frac{1}{c} \cdot \begin{pmatrix} \dot{r} \\ r\dot{\theta} \end{pmatrix} \cdot \begin{pmatrix} -\frac{GMm}{r^2} \\ 0 \end{pmatrix}$$
$$\therefore \implies -\frac{GMm}{cr^2} \dot{r}$$

Part (b)
$$\frac{d(m\gamma \vec{v})}{dt} = \vec{F}$$
$$\implies m\gamma \frac{d}{dt} (\vec{v}) = m\gamma \frac{d}{dt} \begin{pmatrix} \dot{r} \\ r\dot{\theta} \end{pmatrix}$$
$$\implies m \gamma \frac{d}{dt} (\dot{r} \hat{e}_r + r\dot{\theta} \hat{e}_\theta) = \vec{F}$$
$$\implies m\gamma \frac{d}{dt} (\dot{r} \hat{e}_r) + m\gamma \frac{d}{dt} (r \dot{\theta} \hat{e}_\theta) = -\frac{GMm}{r^2} \hat{e}_r + 0\cdot \hat{e}_\theta$$
$$\implies \frac{d(m\gamma \dot{r})}{dt} = -\frac{GMm}{r^2}$$
and
$$\frac{d(m\gamma r \dot{\theta})}{dt} = 0$$
$$\therefore \frac{d(m\gamma \vec{v})}{dt} = \begin{pmatrix} -\frac{GMm}{r^2} \\ 0 \end{pmatrix}$$

Note: Final answers for each part have the "therefore" (i.e. $\therefore$) sign.

 

[PeroK]

You've lost me with what you are trying to do there. You've ended up back where you started.

What happened to all the maths you had in that other post?

 

[Athenian]

You've lost me with what you are trying to do there. You've ended up back where you started.

What happened to all the maths you had in that other post?

Ah, apologies for the confusion. Technically speaking, this is "question 1" of the assignment. While it appears that I have ended up back where I started, this question should be the starting point. In short, I wasn't solving the multiple questions in the assignment in the chronological order that I was supposed to be doing.

However, now with the other questions, I have already gone ahead and changed that. So, it's fine now.

 

[PeroK]

Ah, apologies for the confusion. Technically speaking, this is "question 1" of the assignment. While it appears that I have ended up back where I started, this question should be the starting point. In short, I wasn't solving the multiple questions in the assignment in the chronological order that I was supposed to be doing.

However, now with the other questions, I have already gone ahead and changed that. So, it's fine now.

What are questions a) and b)?

 

[Athenian]

What are questions a) and b)?

Questions 1 Part A and Part B are simple naming conventions. Part A refers to the first equation on the homework statement and Part B refers to the second equation on the homework statement (i.e. the equations of motion).

 

[PeroK]

Questions 1 Part A and Part B are simple naming conventions. Part A refers to the first equation on the homework statement and Part B refers to the second equation on the homework statement (i.e. the equations of motion).

I meant what are the questions asking you to do?

 

[Athenian]

Apologies for the confusion. The question, in short, is as follows: "Write down the equations of motion in the system of polar coordinates".

The equations of motion are:
$$\frac{d(m\gamma c)}{dt} = \frac{\vec{v}}{c} \cdot \vec{F}$$
and
$$\frac{d(m \gamma \vec{v})}{dt} = \vec{F}$$

And, I am supposed to make the above equations into polar form.
I think I did it correctly in the solution above. However, if there are any errors with it. Please let me know. Thank you!

 

[PeroK]

Apologies for the confusion. The question, in short, is as follows: "Write down the equations of motion in the system of polar coordinates".

The equations of motion are:
$$\frac{d(m\gamma c)}{dt} = \frac{\vec{v}}{c} \cdot \vec{F}$$
and
$$\frac{d(m \gamma \vec{v})}{dt} = \vec{F}$$

And, I am supposed to make the above equations into polar form.
I think I did it correctly in the solution above. However, if there are any errors with it. Please let me know. Thank you!

Okay. Post #3 doesn't look right at all. Note that the polar unit vectors vary with position, hence with time for your particle's trajectory. The following is invalid:

Part (b)

$$\implies m\gamma \frac{d}{dt} (\dot{r} \hat{e}_r) + m\gamma \frac{d}{dt} (r \dot{\theta} \hat{e}_\theta) = -\frac{GMm}{r^2} \hat{e}_r + 0\cdot \hat{e}_\theta$$
$$\implies \frac{d(m\gamma \dot{r})}{dt} = -\frac{GMm}{r^2}$$
and
$$\frac{d(m\gamma r \dot{\theta})}{dt} = 0$$

You haven't taken into account the time dependence of the polar unit vectors; or the time dependence on $\gamma$. You ought to write $\gamma(t)$ to prevent yourself treating it like a constant.

PS Look up the equations for acceleration in polar coordinates. I'm sure you had them in a previous post, but that might have been a group effort!

 

[Athenian]

Phew, finally got around doing this exercise:

Here's the solution. Please let me know has the issue now been resolved.

$$\frac{d(m\gamma \vec{v})}{dt} = \vec{F}$$

$$m\gamma \begin{pmatrix} \ddot{r} - r \dot{\theta}^2 \\ r \ddot{\theta} + 2 \dot{r} \dot{\theta}\end{pmatrix} + m \dot{\gamma} \begin{pmatrix} \dot{r} \\ r \dot{\theta}\end{pmatrix} = \begin{pmatrix} -\frac{GMm}{r^2} \\ 0\end{pmatrix}$$

Note that this is possible since:
$$\dot{r} = \dot{r} \hat{r} + r \dot{\theta} \hat{\theta}$$
$$\ddot{r} = \ddot{r} \hat{r} + \dot{r} \hat{r} + \dot{r} \dot{\hat{r}} + \dot{r} \dot{\theta} \hat{\theta} + r \ddot{\theta} \hat{\theta} + r \dot{\theta} \dot{\hat{\theta}}$$

Does this solution now work?

 

[PeroK]

I wouldn't have stopped there. It all depends what one means by equations of motion. I would have continued until at least:$$\frac{d}{dt}(\gamma r^2 \dot \theta) = 0$$ $$\frac{d}{dt}(\gamma \dot r) = -\frac{GM}{r^2} + \gamma r \dot \theta^2$$
Those are what I would call equations of motion!

 

[PeroK]

Note that this is possible since:
$$\dot{r} = \dot{r} \hat{r} + r \dot{\theta} \hat{\theta}$$
$$\ddot{r} = \ddot{r} \hat{r} + \dot{r} \hat{r} + \dot{r} \dot{\hat{r}} + \dot{r} \dot{\theta} \hat{\theta} + r \ddot{\theta} \hat{\theta} + r \dot{\theta} \dot{\hat{\theta}}$$

Here's a good reference on polar coordinates.

https://ocw.mit.edu/courses/aeronau...fall-2009/lecture-notes/MIT16_07F09_Lec05.pdf