Meyer, K. R.

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: 
A Note on Contact Transformations 
by 
K.  R.  Meyer* 
Abstract  
N o t  a l l  contact  transformations a re  of t h e  form p = W (q,P) 
9 
and Q = W (q,P) 
va r i ab le s  any contact  transformation can be wr i t t en  i n  t h i s  form. 
but  t h i s  note  shows t h a t  a f t e r  a l i n e a r  change of P 
0 -  
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* 
This Research was supported by the  NatiQnaL k r o n a u t i c s  and Space 
Administration, Huntsvil le,  under Contract No. NAS 8-11264s. 4 0 .-e 0 M - u / s -  
https://ntrs.nasa.gov/search.jsp?R=19670021961 2020-03-12T11:00:38+00:00Z
. 
It i s  well  known t h a t  a transformation of the  form 
defines  a contact  transformation from the  q,p var iab les  t o  the  
Q,P var iab les  where q,p,Q,P a r e  n-vectors, W i s  a s c a l a r  func- 
t i o n  with continuous second p a r t i a l  der iva t ives  with respect  t o  a l l  
arguments and subscr ipts  denote components of the  vectors .  See f o r  
example [l], and [ 21. It i s  not  always t r u e  t h a t  any contact t r a n s -  
formation can be wr i t ten  i n  the form 1) or  even i n  one of the  other  
th ree  common var ia t ions  of 1). 
[ 3 ]  and the  author recommends [23 as a c a r e f u l  and readable source 
This f a c t  i s  pointed out i n  [2] and 
on contact transformations (see i n  p a r t i c u l a r ,  page 69-70 of [ 2 ] ) .  
This note w i l l  show, however, t h a t  any contact transformation 
be wr i t ten  as a composition of a l i n e a r  orthogonal contact  t r a n s -  
formation and a contact transformation of t h e  form 1). 
say, given any contact transformation one can f i r s t  make a change 
can -
- --  - -  
That i s  t o  - -  -- 
of var iab les  t h a t  i s  l i n e a r ,  orthogonal and preserves Hamiltonian 
form and then wr i te  t h e  transformation i n  t h e  form 1). The above 
i s  t o  be taken as a l o c a l  statement, t h a t  i s ,  the  above statement 
holds only i n  a s u f f i c i e n t l y  small neighborhood of a point .  Also 
we s h a l l  assume t h a t  a l l  functions a r e  s u f f i c i e n t l y  d i f f e r e n t i a b l e  
t h a t  t h e  indicated der ivat ives  a r e  continuous and t h a t  t h e  i m p l i c i t  
funct ion theorem can be applied. The assumption t h a t  a l l  functions 
considered have continuous second p a r t i a l  der iva t ives  with respect  
I .  - 2- 
t o  a l l  arguments w i l l  suf f ice .  
To avoid confusion a contact  transformation w i l l  be taken 
i n  t h e  sense of Whittaker [l], page 293. That i s :  
Definit ion: A transformation 
where q,pJQ,P a r e  n-vectors and * and cp a r e  n-vector valued used 
functions of q and p w i l l  be c a l l e d  a contact  transformation i f  
t h e r e  e x i s t s  a s c a l a r  valued function S(q,p) such t h a t  
n 
n 
c {pidqi + QidPi) 
i= 1 
Observe that 3) i s  of ten wr i t ten  dS = 
and t h a t  t h i s  shor t  notat ion i s  the  cause of some of the confusion 
i n  t h e  l i t e r a t u r e .  The equal i ty  3) s t a t e s  t h a t  S must be con- 
sidered as a function of p and q only. Indeed the whole question 
of when a contact transformation 2) can be wr i t ten  i n  the form 1) 
r e s t s  on t h e  question of when can S be wr i t ten  as  a function of 
qJP. If the  second equation i n  2) can be solved f o r  p i n  terms 
of P and q and t h e  r e s u l t  subs t i tu ted  i n t o  S we would have the  
desired function W.  But when can we solve the  second equation i n  
2) f o r  p i n  terms of q, and P? If t h e  sub-Jacobian d e t (  
i s  non zero then we can solve t h i s  equation, but there  i s  no reason 
t o  suppose t h a t  it i s  nonzero. A t  t h i s  point  a r e s u l t  i n  [ 3 ]  can beused  
8 ' t o  s t ra ighten  things out .  
The formal proof i s  as follows. Let 2) or  3; be a given 
contact  transformation. Without l o s s  of genera l i ty  we can assume 
t h a t  3; takes  t h e  or ig in  i n t o  t h e  or ig in  s ince otherwise we would 
s h i f t  the  or ig in  by a t r a n s l a t i o n .  Let T be the  Jacobian matrix 
of 3; evaluated a t  the  origin,  i . e .  
where A = I ~ ( O , ' O ) l ,  *i 
D = ~ ~ ( 0 , O ) l .  'i 
€3 = ( q o , o ) l ,  +i C ={2(0,0)) and 
Now by a r e s u l t  i n  [ 3 ] ,  page 44 t h e r e  e x i s t s  nonsingular 
contact matrices 0 and R where 0 i s  orthogonal and R i s  
pos i t ive  d e f i n i t e  symmetric such t h a t  T = RO. This r e s u l t  f o r  con- 
t a c t  matrices i s  t h e  analog of t h e  wel l  known r e s u l t  i n  3-dimensions 
t h a t  says t h a t  any matrix of a l i n e a r  transformation i s  the  product 
of a pure r o t a t i o n  (or ro ta t ion  and r e f l e c t i o n )  and a pure d i l a t i o n .  
It should be remarked t h a t  i n  [ 3 ]  as i n  many other  references a con- 
t a c t  matrix i s  c a l l e d  symplectic and i s  sometimes given a d i f f e r e n t  
but equivalent d e f i n i t i o n  (see [ 21). 
Let 6- be the transformation whose representat ion i s  t h e  
matrix 0 .  Define a new transformation b y b =  p o 8 - l  
I 
and so 3 = j 0  e.Observe 
formation 3 i n t o  two operat 
-4- 
t h a t  we 
ons: f 
have "factored" the  trans- 
rs t  apply @ and then 
-1 8 = 3.8 . Another way of looking a t  $ i s  t h a t  we have 
changed coordinates by the  l i n e a r  transformation 0 and now 2G 
has t h e  form $ i n  t h e  new coordinates.  We now want t o  show t h a t  
# can be wr i t ten  i n  the form 1). 
i s  a contact transformation s ince it i s  t h e  compo- 
s i t i o n  of two contact transformations and moreover i t s  Jacobian 
matrix a t  t h e  o r i g i n  i s  T0-l = (R0)O-I = R. Thus i f  I& i s  given 
by Q = a(q ' ,p ' ) ,  P = b(q ' ,p ' )  and R = ( c ,  :l ] where 
A' = (w(O,O)) e t c .  
A' 
i 
J 
aa 
Now R i s  pos i t ive  d e f i n i t e  and symmetric and so  by 
Sylves te r ' s  c r i t e r i o n  [4], page 306, or [ 5 ]  page 94 each p r i n c i p a l  
subdeterminant of R i s  pos i t ive  and so i n  p a r t i c u l a r  
D' = ( T ( ~ , ~ ) ]  i s  nonsingular. i 
a b  
Thus, we can solve the equation P = b(q' ,p' ) f o r  p! 
t o  obtain p' = h(q ' ,P) .  
i s  a contact transformation there  e x i s t s  a Since P 
generating function S' (9' ,pt  ) such t h a t  
. . ,  
I .  
I 
f 
r 
-5 - 
but dW = dS a t  corresponding poin ts  and so  
n 
i= 1 
dW(q',P) = c (pjdqi  + bi(q' ,P')dPi} 
where i n  6) p' = h(q' ,P) , 
i s  nonsingular t h e  d i f f e r e n t i a l s  dqi,  ..., d%,dPl, ..., dPn a r e  
l i n e a r l y  independent and so  we can equate c o e f f i c i e n t s  i n  5 )  and 6) 
t o  obtain 
(7) 
Theref ore  i s  of t h e  form 1). 
Observe t h a t  we can obtain one of the  other  common var ia -  
t i o n s  of 1) when any one of the other  sub-Jacobian matrices i s  non- 
s ingular .  The procedure we have used gives t h a t  A' i s  nonsingular 
so t h i s  gives one var ian t .  By changing var iab les  again with the  
l i n e a r  orthogonal contact matrix 1-1 ' 1 ] then the  Jacobian of 
-B' A' 
t h e  new ,& i s  of the  form c ,  1 and so now the  upper r i g h t  
and lower l e f t  sub-Jacobtan matrices a re  nonsingular and by the  
same procedure you get t h e  other two var ian ts .  
. 
References 
[l] Whittaker, E. T., A Treatise on the Analytic Dynamics of 
Particles and Rigid Bodies, Cambridge University Press, 
4th edition, 1964. 
- -- - 
-- 
[23 Pollard, H., Mathematical Introduction to Celestial Mechanics, 
Prentice Hall, 1966. 
- 
[3] Wintner, A , ,  The Analytic Foundations of Celestial Mechanics, 
Princeton University Press, 1947. 
- -
[4] Gantmacher, F. R., The Theory of Matrices, Vol. 1, Chelsea 
Publishing Company, 1959, p.  306. 
- -
[5] Perlis, Theory of Matrices, Addison-Wesley Publishing Company, - 
1958, P. 94. 


A note on contact transformations

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