Equation of motion in polar coordinates for charged particle

In summary, the conversation discusses the challenge of solving equations of motion for a charged particle in a uniform magnetic field using only mathematical background. The equations of motion are well known in polar coordinates, but they become entangled and difficult to solve without physical reasoning. The conversation suggests various approaches, including finding a solution with fixed r, using the general equation of a circle in polar coordinates, and trying a change of coordinates. Ultimately, it is agreed that Cartesian coordinates may be the best choice for solving this problem.
  • #1
sergiokapone
302
17
A solution of equations of motion for charged particle in a uniform magnetic field are well known (##r = const##, ## \dot{\phi} = const##). But if I tring to solve this equation using only mathematical background (without physical reasoning) I can't do this due to entaglements of variables. What trick should I know?

\begin{align}
\ddot{r} - r\dot{\phi}^2 &= \frac{q}{m} r\dot{\phi}B\\
r \ddot{\phi} + 2\dot{r}\dot{\phi} &= -\frac{q}{m}\dot{r}B
\end{align}
 
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  • #2
sergiokapone said:
Summary: How to solve equations of motion for charged particle in a uniform magnetic field in a polar coordinates?

A solution of equations of motion for charged particle in a uniform magnetic field are well known (##r = const##, ## \dot{\phi} = const##). But if I tring to solve this equation using only mathematical background (without physical reasoning) I can't do this due to entaglements of variables. What trick should I know?

\begin{align}
\ddot{r} - r\dot{\phi}^2 &= \frac{q}{m} r\dot{\phi}B\\
r \ddot{\phi} + 2\dot{r}\dot{\phi} &= -\frac{q}{m}\dot{r}B
\end{align}

You could look for a solution with fixed ##r##. That might be a quick way.

Alternatively, you could find the general equation of a circle in polar coordinates and see what that looks like.
 
  • #3
PeroK said:
You could look for a solution with fixed rrr. That might be a quick way
As I mentioned I know the silution. I want to solve equations and get the constant r.
 
  • #4
sergiokapone said:
As I mentioned I know the silution. I want to solve equations and get the constant r.

Why do you think those equations yield a constant ##r##? Most circles in polar coordinates do not have constant ##r##.
 
  • #5
PeroK said:
Why do you think those equations yield a constant ##r##? Most circles in polar coordinates do not have constant ##r##.

Ah, ok, this depend on initial condition. I can always choose so.
 
  • #6
But what if I do not know is a circle? I need to solve equations in a right way, without any hypothesis.
 
  • #7
sergiokapone said:
Ah, ok, this depend on initial condition. I can always choose so.

There is nothing in your equations so far that implies that the origin must be at the centre of your circle. What you get from those equations must be the general equation of uniform circular motion.

It might be interesting to calculate this equation and see how close it is to what you have already.
 
  • #8
sergiokapone said:
But what if I do not know is a circle? I need to solve equations in a right way, without any hypothesis.

A "guess" is different from a "hypothesis". If you have an equation such as:

##x^3 - 3x^2 - 2x + 4 = 0##

Then, you might try ##x = 1## and find it is a solution. That's very different from assuming ##x = 1## is a solution. Guessing a solution is perfectly legitimate, even in pure mathematics.
 
  • #9
PeroK said:
Guessing a solution is perfectly legitimate, even in pure mathematics.
Yes. But it doesn't have much values, I think. For example I know how to get solution for the problem in Cartesian coordianates, using method of complex velocity (known from Landau and Lifshitz, V2).
 
  • #10
sergiokapone said:
Yes. But it doesn't have much values, I think.

Don't underestimate the value of educated guesswork. Especially in physics. Didn't Dirac just guess the form of his famous equation?

In general, trajectories in polar coordinates are difficult. Even the equation of a straight line is complicated. Another good approach to this problem would be a change of coordinates:

##x = r \cos\phi, \ \ y = r\sin \phi##

Don't underestimate the value of a well-chosen coordinate transformation. Especially in physics.
 
  • #11
PeroK said:
Don't underestimate the value of a well-chosen coordinate transformation. Especially in physics.

Perhaps, I agree with you.
 
  • #12
Well, cylinder coordinates are simply not the best choice of generalized coordinates for this problem. Try again with Cartesian coordinates. If you want to use the Hamilton principle also the choice of gauge can help a lot. E.g., you can choose a gauge, where ##\vec{A}## depends only on one coordinate so that two become cyclic!
 
  • #13
I had already solve similar, but slightly harder problem in Cartesian coordinates. The charged particle in uniform magnetic field with friction.

Here my solution, but the text in Ukrainian, unfortunately

 
  • #14
Looks good, though I can only read the formulas. You have everything solved since this problem is just a special case of your more general problem. Just set ##k=0##.
 
  • #15
Yes, I checked ##k=0## case. My post originated from this problem, I thought I can obtain the solution in polar coordinates. But when I was simplified problem (removed the friction) I realized I can't solve even this simple case. Yes, I agree with you, the Cartesian coordinates is a best choice for this problem.
 
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Related to Equation of motion in polar coordinates for charged particle

1. What is the equation of motion in polar coordinates for a charged particle?

The equation of motion in polar coordinates for a charged particle is given by m(r̈ - rθ̇²) = q(E + rθ̇B), where m is the mass of the particle, r is the distance from the origin, θ is the angle from the x-axis, q is the charge of the particle, E is the electric field, and B is the magnetic field.

2. How is the equation of motion in polar coordinates derived?

The equation of motion in polar coordinates is derived by applying Newton's second law of motion to a charged particle moving in a polar coordinate system. This involves converting the Cartesian coordinates (x, y) to polar coordinates (r, θ) and considering the forces acting on the particle in these coordinates.

3. What is the significance of the terms in the equation of motion in polar coordinates?

The term m(r̈ - rθ̇²) represents the acceleration of the particle in the radial direction, while the term rθ̇B represents the force due to the magnetic field acting on the particle. The term qE represents the force due to the electric field acting on the particle. Together, these terms describe the motion of a charged particle in a polar coordinate system.

4. Can the equation of motion in polar coordinates be used for any type of motion?

The equation of motion in polar coordinates can be used for any type of motion as long as the motion can be described in terms of polar coordinates. This includes circular motion, elliptical motion, and any other type of motion that can be described using a polar coordinate system.

5. How is the equation of motion in polar coordinates different from the equation of motion in Cartesian coordinates?

The equation of motion in polar coordinates is different from the equation of motion in Cartesian coordinates because it takes into account the changing direction of the particle's velocity. In Cartesian coordinates, the acceleration is only in the x and y directions, while in polar coordinates, the acceleration can have both radial and tangential components due to the changing angle θ.

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