7/28/2019 B. O. Koopman - Hamiltonian Systems and Transformations in Hilbert Space http://slidepdf.com/reader/full/b-o-koopman-hamiltonian-systems-and-transformations-in-hilbert-space 1/4 MA THEMA TICS: B. 0. KOOPMA N HA MILTONIAN SYSTEMS AND TRANSFORMATIONS IN HILBERT SPACE BY B. 0. KOOPMAN DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY Communicated March 23, 1931 In recent years the theory of Hilbert space and its linear transformations has come into prominence.' It has been recognized to an increasing extent that many of the most important departments of mathematical physics can be subsumed under this theory. In classical physics, for example in those phenomena which are governed by linear conditions- linear differential or integral equations and the like, in those relating to harmonic analysis, and in many phenomena due to the operation of the laws of chance, the essential r6le is played by certain linear transformations in Hilbert space. And the importance of the theory in quantum me- chanics is known to all. It is the object of this note to outline certain investigations of our own in which the domain of this theory has been extended in such a way as to include classical Hamiltonian mechanics, or, more generally, systems defining a steady n-dimensional flow of a fluid of positive density. Consider the dynamical system of n degrees of freedom, the canonical equations of which are formed from the Hamiltonian H(q, p) = H(ql, * a qny ply ... . ps), which we will assume to be single-valued, real, and analytic in a certain 2n-dimensional region R of the real qp-space. Th e solutions, or equations of motion, are qk = fk(q0, p0, t), Pk = gk(q0, po, t), (k = 1, ..., n), these functions being single-valued, real and analytic for all (q°, p°) in R and for t in a real interval containing t = 0 dependent on (q°, p°). It is shown that the transformation St: (q°, po) > (q, p) defined by these equations for suitably restricted t has the formal proper- ties: St1S1, = Si, + ,, So = I. Th e system admits the "integral of energy" H(q, p) = const.; hence, if Ql denote a variety H(q, p) = C of points of R, a path curve of St having one point on Q will remain on Q as long as the curve remains in R. We shall assume that C is such that this is the case for all values of t; this will be the situation, for example, if Q consists of a closed set of interior points of R. It is shown that under these condi- tions fk and gk are analytic for all (q°, p°) on Q and for - co < t < + o, so that SI effectuates a one-parameter group of analytic automorphisms of U. Furthermore, St leaves invariant the value of a certain integral f pdw taken over an arbitrary region of U; here, p is a positive, single- valued, analytic function on Q. This is a consequence of the fact that .dqi.. dqn, dp.. . dp, is an integral invariant of the system. In the special case where there are m further integrals Fj(q, p) = Cj of the system, VOL. 17, 1931 315
Transcript
7/28/2019 B. O. Koopman - Hamiltonian Systems and Transformations in Hilbert Space
DEPARTMENT OF MATHEMATICS, COLUMBIA U N I V E R S I T Y
C o m m u n i c a t e d March 2 3 , 1 9 3 1
I n r e c e n t y e a r s t h e t h e o r y o f H i l b e r t s p a c e a n d i t s l i n e a r t r a n s f o r m a t i o n s
h a s come i n t o p r o m i n e n c e . ' I t h a s b e e n r e c o g n i z e d t o a n i n c r e a s i n g
e x t e n t t h a t m any o f t h e m o s t i m p o r t a n t d e p a r t m e n t s o f m a t h e m a t i c a l
p h y s i c s c a n b e s u b s u m e d u n d e r t h i s t h e o r y . I n c l a s s i c a l p h y s i c s , f o re x a m p l e i n t h o s e phenomena w h i c h a r e g o v e r n e d by l i n e a r c o n d i t i o n s -
l i n e a r d i f f e r e n t i a l o r i n t e g r a l e q u a t i o n s a n d t h e l i k e , i n t h o s e r e l a t i n g t o
h a r m o n i c a n a l y s i s , a n d i n m a n y p h e n o m e n a d u e t o t h e o p e r at io n o f t h e
l a w s o f c h a n c e , t h e e s s e n t i a l r 6 l e i s p l a y e d by c e r t a i n l i n e a r t r a n s f o r m a t i o n s
i n H i l b e r t s p a c e . And t h e i m p o r t a n c e o f t h e t h e o r y i n quantum m e -
c h a n i c s i s known t o a l l . I t i s t h e o b j e c t o f t h i s n o t e t o o u t l i n e c e r t a i ni n v e s t i g a t i o n s o f o u r o w n i n w h i c h t h e d o m a i n o f t h i s t h e o r y h a s b e e n
e x t e n d e d i n s u c h a wa y a s t o i n c l u d e c l a s s i c a l H a m i l t o n i a n m e c h a n i c s ,
o r , m o r e g e n e r a l l y , s y s t e m s d e f i n i n g a s t e a d y n - d i m e n s i o n a l f l o w o f a
f l u i d o f p o s i t i v e d e n s i t y .C o n s i d e r t h e d y n a m i c a l s y s t e m o f n d e g r e e s o f f r e e d o m , t h e c a n o n i c a l
e q u a t i o n s o f w h i c h a r e f o r m e d f r o m t h e H a mi l to ni an H ( q , p ) = H ( q l ,* a q n y p l y . . . . p s ) , w h i c h w e w i l l a s s u m e t o b e s i n g l e - v a l u e d , r e a l , a n d
a n a l y t i c i n a c e r t a i n 2 n - d i m e n s i o n a l r e g i o n R o f t h e r e a l q p - s p a c e . Th e
s o l u t i o n s , o r e q u a t i o n s o f m o t i o n , a r e q k = f k ( q 0 , p 0 , t ) , P k = g k ( q 0 , p o , t ) ,
( k = 1 , . . . , n ) , t h e s e f u n c t i o n s b e i n g s i n g l e - v a l u e d , r e a l a n d a n a l y t i cf o r a l l ( q ° , p ° ) i n R a n d f o r t i n a r e a l i n t e r v a l c o n t a i n i n g t = 0 d e p e n d e n t
o n ( q ° , p ° ) . I t i s s h o w n t h at t h e t r a n s f o r m a t i o n S t : ( q ° , p o ) > ( q , p )
d e f i n e d b y t h e s e e q u a t i o n s f o r s u i t a b l y r e s t r i c t e d t h a s t h e f o r m a l p r o p e r -t i e s : S t 1 S 1 , = S i , + , , S o = I . Th e s y s t e m a d m i t s t h e " i n t e g r a l o f e n e r g y "
H ( q , p ) = c o n s t . ; h e n c e , i f Q l d e n o t e a v a r i e t y H ( q , p ) = C o f p o i n t s o f
R , a p a t h c u r v e o f S t h a v i n g o n e p o i n t o n Q w i l l r e m a i n o n Q a s l o n g a s
t h e c u r v e r e m a i n s i n R . We s h a l l a s s u m e t h a t C i s s u c h t h a t t h i s i s t h e
c a s e f o r a l l v a l u e s o f t ; t h i s w i l l b e t h e s i t u a t i o n , f o r e x a m p l e , i f Q c o n s i s t s
o f a c l o s e d s e t o f i n t e r i o r p o i n t s o f R . I t i s s h o w n t h a t u n d e r t h e s e c o n d i -
t i o n s f k a n d g k a r e a n a l y t i c f o r a l l ( q ° , p ° ) o n Q a n d f o r - c o < t < + o , s o
t h a t S I e f f e c t u a t e s a o n e - p a r a m e t e r g r o u p o f a n a l y t i c a u t o m o r p h i s m s o f
U . F u r t h e r m o r e , S t l e a v e s i n v a r i a n t t h e v a l u e o f a c e r t a i n i n t e g r a lfp d w t a k e n o v e r a n a r b i t r a r y r e g i o n o f U ; h e r e , p i s a p o s i t i v e , s i n g l e -v a l u e d , a n a l y t i c f u n c t i o n on Q . T h i s i s a c o n s e q u e n c e o f t h e f a c t t h a t
. d q i . . d q n , d p . . . d p , i s an i n t e g r a l i n v a r i a n t o f t h e s y s t e m . I n t h e
s p e c i a l c a s e w h e r e t h e r e a r e m f u r t h e r i n t e g r a l s F j ( q , p ) = C j o f t h e s y s t e m ,
V O L . 1 7 , 1 9 3 1 3 1 5
7/28/2019 B. O. Koopman - Hamiltonian Systems and Transformations in Hilbert Space
we t a k e Q a s t h e (2n- m - 1 ) - d i m e n s i o n a l l o c u s i n R o f t h e s e e q u a t i o n s
t o g e t h e r w i t h H ( q , p ) = C ; b u t t h e r e i s n o f u r t h e r c h a n g e t o b e m a d e i nt h e c a s e t r e a t e d a b o v e .
Th e s t a r t i n g p o i n t o f o u r i n v e s t i g a t i o n i s t h e N - d i m e n s i o n a l v a r i e t y
Q a n d t h e g r o u p o f a u t o m o r p h i s m s S t h a v i n g t h e p o s i t i v e i n t e g r a l i n -
v a r i a n t f p d c o , a n d t h e s e a r e c o n s i d e r e d w i t h o u t r e f e r e n c e t o t h e p r o b l e m
w h i c h g a v e t h e m o r i g i n . L e t s o = s o ( A ) b e a c o m p l e x - v a l u e d f u n c t i o n
o f t h e p o i n t A o f Q , r e s t r i c t e d o n l y a s f o l l o w s : ( i ) s p i s s i n g l e - v a l u e d ;( i i ) s o i s m e a s u r a b l e ; ( i i i ) t h e L e b e s g u e i n t e g r a l s , 2 i p | o p d o a n d f 0 p | I s o d w
a r e f i n i t e . Th e t o t a l i t y o f s u c h f u n c t i o n s s o c o n s t i t u t e s t h e a g g r e g a t e o f
p o i n t s o f a H i l b e r t s p a c e S t : t h e m e t r i c o f w h i c h i s d e t e r m i n e d b y t h e" i n n e r p r o d u c t " ( s o , t ) = J a p j I d w .
L e t t h e t r a n s f o r m a t i o n U t o f t h e " p o i n t s " s o o f & b e d e f i n e d b y U t s o ( A ) =
s o ( S t A ) . T h i s t r a n s f o r m a t i o n i s e v i d e n t l y d e f i n e d f o r a l l s o o f a n d i s
c o n t i n u o u s i n s o ; a n d i t i s d e f i n e d a n d c o n t i n u o u s f o r a l l r e a l t . Th e
k i n e m a t i c a l i n t e r p r e t a t i o n o f U t i s i m m e d i a t e l y o b v i o u s : I f t r e p r e s e n t s
t h e t i m e , S t s p e c i f i e s t h e s t e a d y f l o w o f a f l u i d o f d e n s i t y p o c c u p y i n g t h e
s p a c e Q ; a n d i f t h e v a l u e s o f s o ( A ) b e r e g a r d e d a s b e i n g a t t a c h e d t o t h e
r e s p e c t i v e p o i n t s A o f t h e f l u i d w h e n t = 0 , i n t h e c o u r s e o f t h e f l ow t h e s e
v a l u e s w i l l b e c a r r i e d i n t o t h o s e o f t h e f u n c t i o n U _ t s o ( A ) . T h u s , U s ph a s a t A t h e v a l u e w h i c h s o h a s a t t h e p o i n t S t A i n t o w h i c h A f l o w s a f t e r
t h e l a p s e o f t h e t i m e t .
Th e t r a n s f o r m a t i o n U 1 i s u n i t a r y , t h a t i s , i t i s a o n e t o o n e t r a n s f o r m a -
t i o n o f t i n t o i t s e l f w h i c h i s l i n e a r :
U 1 [ a p ( A ) + b t ( A ) I = a s o ( S A ) + b p ( S I A )= a U 1 s p ( A ) + b U 1 p ( A )
a n d s u c h t h a t
( U s o ,U 1 ' / ' ) = ( s o ,
4 ) .T h i s l a s t i s an e a s i l y e s t a b l i s h e d
c o n s e q u e n c e o f t h e i n v a r i an c e o f f p d w u n d e r S t . An d s i n c e 2 U t i s a
o n e - p a r a m e t e r g r o u p o f u n i t a r y t r a n s f o r m a t i o n s i n H i l b e r t s p a c e , i t w i l l
h a v e o n i n f i n i t e s i m a l g e n e r a t o r P :
_ 6 _
[ a U t ( s ( A ) ] i P p ( A ) ,
i = + / - 1 , a n d P i s a s e l f - a d j o i n t l i n e a r o p e r a t o r d e f i n e d i n a s e t o f
p o i n t s e v e r y w h e r e d e n s e i n t : ( P s o , V / )=
( s o , P t ) .I n a s u f f i c i e n t l y s m a l l r e g i o n o f Q , P m a y b e g i v e n a s i m p l e i n t e r p r e t a -t i o n . L e t t 1 , . . . , N b e a s e t o f G a u s s i a n c o 6 r d i n a t e s o f t h e r e g i o n , t h e
d i f f e r e n t i a l e q u a t i o n s o f t h e f l o w w i l l b e d i k / d t = = 3 k ( 1 , . . . , N ) ( k = 1 ,
. . , N ) ; t h e n f o r a n y d i f f e r e n t i a b l e s p ( A ) = s o ( t 1 , . . . , { N )
3 1 6 P R O C . N . A . S .
7/28/2019 B. O. Koopman - Hamiltonian Systems and Transformations in Hilbert Space
We r e c o g n i z e t h e L i e o p e r a t o r . T h e p r o p e r t y o f p = p ( t l , . . * , N ) i s
e x p r e s s e d b y t h e " e q u a t i o n o f c o n t i n u i t y "
b ( P V k )E = O ,k =1 C ( k
w h i c h m i g h t h a v e b e e n u s e d t o p r o v e f or ma l l y t h at P i s s e l f - a d j o i n t .Th e r e c e n t r e s u l t s i n t h e s p e c t r a l t h e o r y o f l i n e a r o p e r a t o r s e s t a b l i s h
t h e e x i s t e n c e o f a " c a n o n i c a l r e s o l u t i o n o f t h e i d e n t i t y " E l c o r r e s p o n d i n g
w i t h U t , w h i c h c o n s i s t s o f a f a m i l y o f s e l f - a d j o i n t o p er a to r s d ef i ne d t h r o u g h -
o u t ! , w i t h t h e p r o p e r t i e s t h a t
E _ O D 0 , E 0 , = I , EEh=EEl = E B , when h < 1 ( 1 ) .
T I h i s E l e f f e c t u a t e s t h e s p e c t r a l r e s o l u t i o n o f U 1 a n d P :
U tp
= J X e " ' d E p p ( 2 ) , Pp = l d E g 4 p .
T h e s e a r e L e b e s g u e - S t i e l t j e s i n t e g r a l s a n d t h e y e x i s t i n t h e s e n s e o f c o n -
v e r g e n c e i n t h e m e a n . T h u s , e . g . ,
J
f p I E e ' i ( E j 3 p - E l _ , p ) - U 1 t o I 2 d w > 0j = l
a s t h e n e t d e t e r m i n e d b y t h e p o i n t s ( l o , 1 1 , . . . , l I J ) b e c o m e s " i n d e f i n i t e l yf i n e " i n a n a r b i t r a r y i n t e r v a l o f t h e i - a x i s . I t f o l l o w s f r o m a c o r o l l a r y
o f t h e R i e s z - F i s c h e r T h e o r e m t h a t a s e t Q C Q e x i s t s h a v i n g f 2 , p d w =J f Q p d w , a t e v e r y p o i n t A o f w h i c h
J
U 1 ( A ) = l i m e i " ' i [ E l g s o ( A ) - s o ( A ) ]j=1
a s t h e s e t o f v a l u e s ( l o , 1 1 , . . . , j ) i s r e p l a c e d s u c c e s s i v e l y b y t h e m e m b e r s
o f a s e q u e n c e o f s e t s , w h i c h s e q u e n c e f o r m s a p r o p e r l y c h o s e n s u b - s e q u e n c e
o f t h e t o t a l i t y o f s e t s c o n s i d e r e d i n t h e p r e v i o u s f o r m u l a . I n p a r t i c u l a r ,s i n c e t h e v a l u e s on Q o f t h e d y n a m i c a l v a r i a b l e s
q ka n d P k a r e among t h e
p o i n t s o f S & , i t f o l l o w s t h a t t h r o u g h o u t s u c h a . s e t Q 1 t h e s e v a r i a b l e s m a y
b e r e p r e s e n t e d a s l i m i t s o f t r i g o n o m e t r i c a l sums o f t h e a b o v e t y p e .
An e v i d e n t p r o p e r t y o f U t i s t h a t , f o r an a r b i t r a r y s i n g l e - v a l u e d f u n c t i o n
o f s e v e r a l v a r i a b l e s , F ,
V O L . 1 7 , 1 9 3 1 3 1 7
7/28/2019 B. O. Koopman - Hamiltonian Systems and Transformations in Hilbert Space
a n e q u a t i o n w h i c h p l a y s a n i m p o r t a n t p a r t i n t h e d e v e l o p m e n t s o f o u r
t h e o r y .I f s o i s a c h a r a c t e r i s t i c f u n c t i o n o f U t : U 1 ' p = e i X t p , i t w i l l f o l l o w f r o m
t h e p r e ce di ng e q u a ti on w i t h F ( p ) = I p I t h a t U I p I = U ( = I p | ;a n d h e n c e , w h e n P I ' P i s d e f i n e d , P I p = 0 ; t h u s j o | , w h i c h r e m a i n s
c o n s t a n t a l o n g t h e p a t h c u r v e s o f S i , h a s t h e p r o p e r t y o f a s i n g l e - v a l u e d
i n t e g r a l o f t h e e q u a t i o n s o f m o t i o n . I n t h e i m p o r t a n t c a s e w h e r e n o
s u c h i n t e g r a l e x i s t s w e m a y t a k e s p - e ' . T h e n 0 = 0 ( A ) i s a r e a l f u n c t i o n
o n Q , i n g e n e r a l i n f i n i t e l y m u l t i p l y - v a l u e d , i t s b r a n c h e s d i f f e r i n g b ym u l t i p l e s o f 2 7 r , e a c h b r a n c h b e i n g m e a s u r a b l e ; a n d s i n c e U t 1 p = e i U t O =
e ' x t + i @ , w e s h a l l h a v e U t 0 = 0 + X t a n d w h e n P O i s d e f i n e d , P O = X . T h u s ,
( p a n d 0 a r e a n a l y t i c a l o n g e a c h g i v e n p a t h c u r v e . I f t h e y a r e c o n t i n u o u s
t h r o u g h o u t Q , 0 = c o n s t . w o u l d r e p r e s e n t a " s u r f a c e o f s e c t i o n " o f t h e
d y n a m i c a l s y s t e m .
I f ( P , . . . , P k a r e c h a r a c t e r i s t i c f u n c t i o n s c o r r e s p o n d i n g t o t h e c h a r -a c t e r i s t i c n u m b e r s X l , . . . , X k , w e s h a l l h a v e f o r a n y s e t o f i n t e g e r s m i ,
U g ( Pm l
i k " ) = ( P U 1 O ) U t .P U t ' k )= e i ( m l x l + + M k ' 1 k ) Ml Mk
s o t h a t s p = m l . . . ° k m ' ( i f i t i s i n S , a s w i l l b e t r u e i f , a s a b o v e , y = e i 0 i ) i s
a c h a r a c t e r i s t i c f u n c t i o n o f c h a r a c t e r i s t i c n u m b e r X = m b , X 1 +* +Mk. k
I f i n p a r t i c u l a r t h e r e i s t h e D i o p h a n t i n e r e l a t i o n m 1 X l + . . . + m k X k = 0 ,s o w i l l b e a n i n t e g r a l i n t h e a b o v e s e n s e o f t h e e q u a t i o n s o f m o t i o n . H e n c e ,
i n t h e n o n - i n t e g r a b l e c a s e , s p o " . . 'k = c o n s t a n t ; h e n c e , i f t h e X ' s h a v e
t h e " b a s i s " X 1 , . . . , X i , : X i = m l i x + . . . + m j X h , w e h a v e ' p j = 1 s o j ( P h
I n t h e c a s e w h e r e U 1 h a s a p u r e p o i n t s p e c t r u m , t h e c h a r a c t e r i s t i c
f u n c t i o n s f o r m a c o m p l e t e s y s t e m , a n d t h e f o r m u l a U t 1 p = f i ' , . e " ' 1 d E 1 e ca b o v e p r o v i d e s u s w i t h a t r i g o n o m e t r i c a l s e r i e s c o n v e r g e n t i n t h e m e a n
o n Q b y means o f w h i c h t o r e p r e s e n t t h e c o o r d i n a t e s a n d m o m e n t a ( q , p ) .A s t u d y i s made o f t h e s t r u c t u r e o f E l i n t h e g e n e r a l c a s e , w i t h a v i e w
t o r e l a t i n g t h e d y n a m i c a l p r o p e r t i e s w i t h t h e s t r u c t u r e o f t h e s p e c t r u m .
I n p a r t i c u l a r , t h e g r o u p o f u n i t a r y t r a n s f o r m a t i o n s V o f & i s c o n s i d e r e d
w h i c h t r a n s f o r m E l a s f o l l o w s : V F 1 V - I = E l + V a r i o u s g e n e r a l i z a t i o n so f t h e a b o v e r e s u l t s a r e m a d e .
1 C f . J . v . N e u m a n n , M a t h . A n n a l e n , 1 0 2 , 4 9 - 1 3 1 ( 1 9 2 9 ) .
2 C f . M . H . S t o n e , t h e s e P R O C E E D I N G S , 1 6 , 1 7 3 - 1 7 4 ( 1 9 3 0 ) .
