cylindrical coordinate motion example of generalized coordinates

(Example)

x = r cos&#981;,   y = rsin &#981;,  z = z,

[              ]
T  = m-  &#729;r2 + r2 &#729;&#981;2 + &#729;z2 .
      2

&#8706;T
--- = m &#729;r,
 &#729;r

T--      &#729;2
&#8706;r =  mr &#981; ,

&#8706;T-      2 &#729;
&#8706;&#981;&#729;=  mr  &#981;,

&#8706;T-
&#8706;z&#729; = m &#729;z.

[       ]
&#948; W  = m   &#168;r &#8722; r &#729;&#981;2 &#948;r = R&#948;r,
 r

(    )
&#948;&#981;W  = m -d  r2&#981;&#729; &#948;&#981; =  &#934;r&#948; &#981;,
         dt

&#948;zW  =  m &#168;z&#948;z = Z &#948;z;

[                ]
    d2r    ( d&#981; )2
m   --2-&#8722; r  ---     = R,
    dt        dt

(      )
m- d-   2d-&#981;
 r dt  r  dt  =  &#934;,

d2z-
m  dt2 = Z.