 #1
 103
 17
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 ##xy## 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...insidesunsgravitationalfield.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!
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!