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## Homework Statement

Two-dimensional SHM: A particle undergoes simple harmonic motion in both the x and y directions

simultaneously. Its x and y coordinates are given by

x = asin(ωt)

y = bcos(ωt)

Show that the quantity x[itex]\dot{y}[/itex]-y[itex]\dot{x}[/itex] is also constant along the ellipse, where here the dot means the derivative with respect to time. Show that this quantity is related to the angular momentum of the system.

## Homework Equations

L = mv x r

## The Attempt at a Solution

Hi, so for the first part it is pretty simple and my answer is -abω, unless i made a dumb mistake which I don't think I did.

It's the second part that is giving me issues. How do I show that it is related to angular momentum? I tried doing this

L = [itex]\sqrt{L^{2}_{x}+L^{2}_{y}}[/itex]

then L[itex]_{x}[/itex] = m[itex]\frac{∂x}{∂t}[/itex] x r

where r = [itex]\sqrt{x^{2}+ y^{2}}[/itex]

and then plugging everything in. I was hoping all the cos and sin were going to cancel out but it got really huge and messy. I didn't think it was supposed to be that hard so can anyone tell me if I am going in the right direction or if there is something I am missing?

Thanks