﻿Interaction 
  of 
  Dynamical 
  Systems. 
  Ill 
  

  

  expressed 
  in 
  terms 
  of 
  velocities 
  only 
  is 
  denoted 
  by 
  Q('j) 
  ; 
  

   from 
  it 
  are 
  derived 
  

  

  momentum 
  p 
  r 
  =^- 
  ? 
  and 
  force 
  =p 
  r 
  —^~- 
  . 
  . 
  (i) 
  

  

  A 
  kinetic 
  energy 
  expressed 
  in 
  terms 
  of 
  momenta 
  only 
  is 
  

   denoted 
  by 
  P(^>) 
  ; 
  from 
  it 
  are 
  derived 
  

  

  velocity 
  q 
  r 
  = 
  ^— 
  , 
  and 
  force 
  =iv+ 
  ^— 
  . 
  . 
  . 
  (ii) 
  

  

  Opr 
  OPr 
  

  

  It 
  will 
  then 
  be 
  shown 
  that 
  a 
  composite 
  system 
  for 
  which 
  the 
  

   kinetic 
  energy 
  is 
  given 
  in 
  the 
  form 
  

  

  E 
  = 
  Q(£)+P(*0, 
  ( 
  iiia 
  ) 
  

  

  may 
  be 
  treated 
  with 
  the 
  aid 
  of 
  a 
  kinetic 
  potential 
  of 
  mixed 
  

   type, 
  viz. 
  : 
  — 
  

  

  L 
  = 
  Q($)-P(*r) 
  + 
  I(}, 
  W 
  ) 
  .... 
  (iii 
  6) 
  

  

  where 
  I 
  is 
  a 
  bilinear 
  function 
  of 
  q 
  and 
  ot 
  with 
  coefficients 
  

   dependent, 
  like 
  those 
  of 
  P 
  and 
  Q, 
  on 
  the 
  coordinates 
  q 
  and 
  

   %. 
  In 
  this 
  mixed 
  form 
  L 
  is 
  to 
  he 
  taken 
  as 
  a 
  function 
  of 
  type 
  

   (i) 
  for 
  the 
  coordinates 
  q, 
  and 
  — 
  L 
  as 
  a 
  function 
  of 
  type 
  (ii) 
  

   for 
  the 
  variables 
  %. 
  From 
  (iii 
  b) 
  are 
  derived 
  in 
  this 
  way 
  

  

  momentum 
  ^ 
  =M;= 
  :_ 
  +_ 
  , 
  . 
  . 
  . 
  ( 
  m 
  c) 
  

  

  and 
  forces 
  p 
  r 
  - 
  |£, 
  OTs+ 
  ^L 
  (_ 
  L 
  ) 
  

   or 
  <i 
  dL 
  ^L 
  ■ 
  BL 
  

  

  (iii 
  <?) 
  

  

  dfdir 
  ~dq 
  r 
  ' 
  "dX 
  8 
  ' 
  ' 
  ' 
  ' 
  

  

  Since 
  I 
  is 
  a 
  bilinear 
  function 
  of 
  ( 
  q, 
  w), 
  we 
  have 
  at 
  once 
  

   tp£r=2Q 
  + 
  I, 
  and 
  S^ 
  S 
  % 
  S 
  = 
  2Q-I 
  ; 
  so 
  that 
  

  

  ZipAr 
  + 
  2i^ 
  s 
  % 
  s 
  = 
  Q 
  + 
  P 
  = 
  E, 
  

  

  (1) 
  

   and 
  2iMr— 
  Sizars 
  = 
  Q-P 
  + 
  I 
  = 
  L. 
  

  

  }• 
  

  

  The 
  latter 
  equation 
  gives 
  the 
  reason 
  for 
  the 
  use 
  of 
  — 
  L 
  in 
  

   the 
  case 
  of 
  the 
  variables 
  %. 
  

  

  Now 
  E 
  can 
  be 
  made 
  homogeneous 
  in 
  two 
  ways, 
  (a) 
  as 
  a 
  

   function 
  of 
  velocities 
  by 
  eliminating 
  «r 
  through 
  the 
  linear 
  

  

  