﻿Interaction 
  of 
  Dynamical 
  Systems. 
  

  

  115 
  

  

  there 
  is 
  no 
  external 
  force 
  tending 
  to 
  alter 
  w, 
  the 
  outcome 
  o£ 
  

   the 
  second 
  group 
  of 
  dynamical 
  equations 
  is 
  that 
  each 
  -ar 
  is 
  

   constant. 
  P 
  then 
  becomes 
  a 
  function 
  of 
  coordinates 
  q, 
  and 
  

   behaves 
  as 
  a 
  potential 
  energy, 
  cf. 
  (7). 
  In 
  (9) 
  the 
  effect 
  of 
  the 
  

   function 
  of 
  interaction 
  is 
  reduced 
  to 
  the 
  last 
  term, 
  viz. 
  : 
  

  

  ■^ 
  /( 
  jCrs 
  O 
  Cms 
  \ 
  

  

  n, 
  s 
  \dq 
  m 
  B?r 
  / 
  

  

  Inf** 
  

  

  (9 
  b) 
  

  

  § 
  5. 
  If 
  the 
  value 
  o£ 
  L 
  is 
  given 
  as 
  a 
  difference 
  of 
  like 
  ex- 
  

   pressions 
  Q( 
  i) 
  — 
  Q'(% 
  ) 
  or 
  P(j?) 
  — 
  P' 
  (w) 
  , 
  we 
  have 
  counterparts 
  

   of 
  the 
  original 
  case. 
  Thus 
  if 
  

  

  Q'(*) 
  

  

  L=Q(« 
  

  

  we 
  form 
  momenta 
  from 
  

  

  E 
  = 
  Q(2)+Q'(X) 
  + 
  I(2,x), 
  

  

  which 
  being 
  homogeneous 
  is 
  a 
  kinetic 
  potential 
  of 
  type 
  (i) 
  

   The 
  equation 
  

  

  3E 
  

  

  (11) 
  

  

  (12) 
  

  

  = 
  30^ 
  3_I_ 
  

   Ws 
  3x. 
  3* 
  3X, 
  

  

  (13) 
  

  

  may 
  be 
  used 
  to 
  eliminate 
  % 
  from 
  (11) 
  and 
  so 
  express 
  L 
  in 
  

   terms 
  of 
  q 
  and 
  w. 
  Here 
  (ZJzyx) 
  in 
  the 
  lemma 
  are 
  replaced 
  

   by 
  (Q', 
  -I, 
  ^ 
  v, 
  ?)> 
  and 
  

  

  BQ'(j.g) 
  ^ 
  BI 
  BQU*0 
  - 
  

   'dir' 
  B^ 
  s 
  

  

  (A) 
  gives 
  

  

  B2r 
  

  

  B^ 
  

  

  X. 
  

  

  ^1 
  

   B0 
  

  

  bq; 
  

  

  B0 
  

  

  (B) 
  gives 
  

   and 
  therefore 
  

  

  aL 
  (j,«r) 
  _ 
  3Q 
  , 
  31 
  _ 
  BE 
  3L(g, 
  OT 
  ) 
  _ 
  3Q 
  , 
  »_y 
  

   32, 
  ~3j, 
  3i, 
  3s/ 
  3^ 
  s 
  ~ 
  3«, 
  s 
  * 
  s 
  

  

  and 
  

  

  BL(?^) 
  _ 
  BQ 
  BQ; 
  BI 
  _ 
  BE 
  

   B<9 
  ~ 
  B# 
  B0 
  B0 
  " 
  30 
  ' 
  

  

  The 
  transformed 
  value 
  of 
  L 
  is 
  therefore 
  a 
  kinetic 
  potential 
  

   of 
  mixed 
  type. 
  The 
  treatment 
  of 
  the 
  case 
  L 
  = 
  P(p) 
  — 
  P'( 
  OT 
  ) 
  

   is 
  similar. 
  I 
  is 
  here 
  subject 
  to 
  the 
  limitation 
  that 
  E 
  must 
  

   be 
  essentially 
  positive. 
  

  

  § 
  6. 
  Some 
  light 
  is 
  thrown 
  on 
  both 
  cases 
  by 
  considering 
  

   the 
  contrary 
  process, 
  the 
  passage 
  from 
  a 
  homogeneous 
  to 
  a 
  

  

  12 
  

  

  Ki±) 
  

  

  