Page 18 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 2: Complex Functions

Thus, Notes

d (re ) = r d i d e i
i
dt dt (e )  dt


= r i d e i    dr e i

 dt  dt
=   dr  ir d   e .
i
 dt dt 
Now,

d 2 i  d r dr d d   i  dr d  d i
2
2
2 
dt 2 (re ) =   dt 2  i dt dt  ir dt  e    dt  ir dt   i dt e
2
 d r  d  2   d  dr d  
2

=   2  r      i r 2  2 e   i

  dt  dt    dt dt dt  
d 2 k
Now, the equation (re )   e becomes
i
i
dt 2 r 2
2
2
 d r  d  2   d  dr d  k

  2  r      i r  2  2    2 .
 dt  dt    dt dt dt  r
This gives us the two equations
d r r  d  2 k ,
2
dt 2    dt     r 2
and,
2
r d   2 dr d  0.
dt 2 dt dt
Multiply by r and this second equation becomes

d  r 2 d   0
dt   dt  
This tells us that

  r 2 d
dt
is a constant. (This constant  is called the angular momentum.) This result allows us to get rid
d
of in the first of the two differential equations above:
dt
d r  r    2   k
2
dt 2   r   r 2
or

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