Elliptical Orbits and Angular Momentum

[WWCY]

Homework Statement

Why is the magnitude of Angular Momentum for an elliptical orbit as such?
$$l = mr^2\dot{\phi}$$
where ##\dot{\phi}## represents angular momentum.

I have always assumed that angular momentum was $$l = r \times P = mr \times V = mrVsin(\theta) = mr^2\dot{\phi} sin(\theta)$$
And since the angle (taken to be ##\theta##) between the ##r## and ##V## vectors isn't always a right angle, shouldn't angular momentum for elliptical orbits be defined with the sine term?

Assistance is greatly appreciated!

Homework Equations

The Attempt at a Solution

 

[Ray Vickson]

Homework Statement

Why is the magnitude of Angular Momentum for an elliptical orbit as such?
$$l = mr^2\dot{\phi}$$
where ##\dot{\phi}## represents angular momentum.

I have always assumed that angular momentum was $$l = r \times P = mr \times V = mrVsin(\theta) = mr^2\dot{\phi} sin(\theta)$$
And since the angle (taken to be ##\theta##) between the ##r## and ##V## vectors isn't always a right angle, shouldn't angular momentum for elliptical orbits be defined with the sine term?

Assistance is greatly appreciated!

Homework Equations

The Attempt at a Solution

The correct expression is
$${\mathbf V} = \dot{r}\, {\mathbf e}_r + r \dot{\phi} \, {\mathbf e}_{\phi}, $$
so
$$ {\mathbf r \times V} = r \dot{\phi}\, {\mathbf e}_r \times {\mathbf e}_{\phi}$$

Here, ##r = |{\mathbf r}|,## and ##{\mathbf e}_r##, ##{\mathbf e}_{\phi}## are the unit vectors in the ##r## and ##\phi## directions (so ##{\mathbf e}_r \times {\mathbf e}_{\phi} = {\mathbf e}_z##).

Be very careful to write vectors when you should have vectors, and scalars when you should have scalars.

 

[Chandra Prayaga]

WWCY, your expression is correct but not very useful. Depending on the path of the particle, θ could be a very complicated function of r and ∅. Ray Vickson's expression is always correct.

 

[WWCY]

The correct expression is
$${\mathbf V} = \dot{r}\, {\mathbf e}_r + r \dot{\phi} \, {\mathbf e}_{\phi}, $$

Could you elaborate on the meaning of this expression? I don't think I've seen something similar before.

$$ {\mathbf r \times V} = r \dot{\phi}\, {\mathbf e}_r \times {\mathbf e}_{\phi}$$

How do we arrive at this expression? Also, is there not a ##r^2## in the final term for magnitude?

Thanks both for the replies. Apologies if this is an elementary question.

 

[Chandra Prayaga]

l=mr²˙ϕ

There is (should be) an r² in Ray Vickson's expression. The theory involves the use of polar coordinates and corresponding unit vectors (radial and transverse coordinates). If you are not familiar with that, then how did you arrive at your original expression for the angular momentum, which I quoted above?

 

[Ray Vickson]

There is (should be) an r² in Ray Vickson's expression. The theory involves the use of polar coordinates and corresponding unit vectors (radial and transverse coordinates). If you are not familiar with that, then how did you arrive at your original expression for the angular momentum, which I quoted above?

Yes, of course ##{\mathbf l} = r^2 \dot{\phi} \, {\mathbf e}_z##.

 

[WWCY]

There is (should be) an r² in Ray Vickson's expression. The theory involves the use of polar coordinates and corresponding unit vectors (radial and transverse coordinates). If you are not familiar with that, then how did you arrive at your original expression for the angular momentum, which I quoted above?

My first exposure to angular momentum came from introductory texts that did not use such a theory.

Do you mind guiding me through the steps?

Thank you!

 

[Chandra Prayaga]

That treatment is available in any standard textbook on Classical Mechanics, such as the one by Taylor. It takes some time to work through it.