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Tuesday, 25 July 2023

Back to de Broglie, II: Polar twist

Recall r2=x2+y2,

and θ=tan1yx.

If

ω=dθdt,

we have

ω=(dxdt,dydt)(yx2+y2,xx2+y2)=vk,

where k:=(yx2+y2,xx2+y2)

so

ω=vkcosϕ,

which gives 

v=ωk,

for ϕ=2nπ.


As k=1x2+y2=1r,

this means to each trajectory  x(t)=(x(t),y(t)), we can associate a harmonic wave with frequency ω=v(t)x(t).

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