Jump to ContentJump to Main Navigation
Isochronous Systems$

Francesco Calogero

Print publication date: 2008

Print ISBN-13: 9780199535286

Published to Oxford Scholarship Online: May 2008

DOI: 10.1093/acprof:oso/9780199535286.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 26 February 2017

(p.213) APPENDIX A SOME USEFUL IDENTITIES

(p.213) APPENDIX A SOME USEFUL IDENTITIES

Source:
Isochronous Systems
Author(s):

Francesco Calogero

Publisher:
Oxford University Press

In this Appendix—which is lifted essentially verbatim from [21]—we list certain useful relations; we consider their derivation sufficiently straightforward not to require any proof. Not all the formulas reported here are used in this monograph, but we thought it useful to report a rather extensive compilation as a convenient tool for future utilizations.

Let ψ (z, t) be a monic polynomial of degree N in z, and let us indicate as z nz n (t) its N zeros and as c mc m (t) its N coefficients:

(A.1)
ψ ( z , t ) = n = 1 N [ z z n ( t ) ] = z N + m = 1 N c m ( t ) z N m = m = 0 N c m ( t ) z N m , c 0 = 1.
Then clearly
(A.2)
ψ z ( z , t ) = ψ ( z , t ) t n = 1 N [ z z n ( t ) ] 1 ,
(A.3)
ψ t ( z , t ) = ψ ( z , t ) n = 1 N [ z z n ( t ) ] 1 [ z ˙ ( t ) ] .
Here (and throughout) subscripted variables denote partial differentiations, ψ z ψ / z , ψ t ψ / t , and so on.

To conveniently streamline the look of these two, and of all the following, formulas, we rewrite these two relations via the self-evident notation

(A.4)
ψ z 1 ,
(A.5)
ψ t z ˙ n ,
and we write accordingly the following relations:
(A.6a)
z ψ z N ψ z n ,
(A.6b)
z 2 ψ z + ( c 1 N z ) ψ z n 2 ,
(A.6c)
z 3 ψ z ( N z 2 c 1 z + c 1 2 2 c 2 ) ψ z n 3
(A.6d)
1 z [ ψ z c N 1 c N ψ ] 1 z n ,
(p.214)
(A.7a)
z ψ t c ˙ 1 ψ z ˙ n z n ,
(A.7b)
1 z [ ψ t c ˙ N c N ψ ] z ˙ n z n ,
(A.8a)
ψ z z m = 1 , m n N 2 z n z m ,
(A.8b)
z ψ z z 2 ( N 1 ) ψ z m = 1 , m n N 2 z m z n z m ,
(A.8c)
z ψ z z m = 1 , m n N 2 z n z n z m ,
(A.8d)
z ψ z z ( N 1 ) ψ z m = 1 , m n N z n + z m z n z m ,
(A.8e)
z 2 ψ z z N ( N 1 ) ψ m = 1 , m n N 2 z n 2 z n z m ,
(A.8f)
z 2 ψ z z 2 [ ( N 2 ) z c 1 ] ψ z + N ( N 3 ) ψ m = 1 , m n N 2 z m 2 z n z m ,
(A.8g)
z 2 ψ z z [ ( N 2 ) z c 1 ] ψ z + N ψ m = 1 , m n N z n 2 + z m 2 z n z m ,
(A.8h)
z 2 ψ z z 2 ( N 1 ) z ψ z + N ( N 1 ) ψ m = 1 , m n N 2 z n z m z n z m ,
(A.8i)
z 3 ψ z z N ( N 1 ) z ψ + 2 ( N 1 ) c 1 ψ m = 1 , m n N 2 z n 3 z n z m ,
(A.8j)
z [ z 2 ψ z z 2 ( N 1 ) z ψ z + N ( N 1 ) ψ ] m = 1 , m n N 2 z n 2 z m z n z m ,
(A.8k)
z 3 ψ z z 2 ( N 2 ) z 2 ψ z + 2 c 1 z ψ z + [ N ( N + 1 ) z 2 ( N 1 ) c 1 ] ψ m = 1 , m n N 2 z n z m 2 z n z m ,
(p.215)
(A.8l)
z 3 ψ z z ( 2 N 3 ) z 2 ψ z + c 1 z ψ z + [ N 2 z ( N 1 ) c 1 ] ψ m = 1 , m n N z n z m 2 + z n 2 z m z n z m ,
(A.8m)
z 4 ψ z z [ N ( N 1 ) z 2 2 ( N 1 ) c 1 z + 2 ( N 1 ) c 1 2 2 ( 2 N 3 ) c 2 ] ψ m = 1 , m n N 2 z n 4 z n z m ,
(A.8n)
z 4 ψ z z 2 z 2 [ ( N 2 ) z c 1 ] ψ z + [ N ( N 3 ) z 2 2 ( N 1 ) c 1 z + 2 c 2 ] ψ m = 1 , m n N 2 z n 2 z m 2 z n z m ,
(A.9a)
ψ z t m = 1 , m n N z ˙ n + z ˙ m z n z m ,
(A.9b)
z ψ z t m = 1 , m n N ( z ˙ n + z ˙ m ) z n z n z m ,
(A.9c)
z ψ z t ( N 1 ) ψ t m = 1 , m n N z ˙ n z m + z ˙ m z n z n z m ,
(A.9d)
z [ z ψ z t ( N 1 ) ψ t ] m = 1 , m n N ( z ˙ n z m + z ˙ m z n ) z n z n z m ,
(A.9e)
z 2 ψ z t + [ c 1 ( N 2 ) z ] ψ t c ˙ 1 ψ m = 1 , m n N z ˙ n z m 2 + z ˙ m z n 2 z n z m ,
(A.10)
ψ t t z ¨ n ( t ) + m = 1 , m n N 2 z ˙ n z ˙ m z n z m .
To obtain some of the formulas written above we used the relations
(A.11)
c N 1 c N = n = 1 N 1 z n , c ˙ N c N = n = 1 N z ˙ n z n ,
(p.216) which are obvious consequences of the formulas
(A.12)
c N ( t ) = n = 1 N [ z n ( t ) ] = ψ ( 0 , t ) , c N 1 ( t ) = n = 1 N m = 1 , m n N [ z m ( t ) ] = ψ z ( 0 , t )
themselves direct consequences of (A.1).

Likewise, we introduce the following notation whereby in the formulas written below (which are also straightforward consequences of (A.1)) the expression in the right-hand side identifies the coefficient of z N −m in the polynomial (of degree N or less) appearing in the left-hand side:

(A.13a)
ψ c m ,
(A.13b)
ψ c N z c m 1 ,
(A.13c)
ψ z ( N m + 1 ) c m 1 ,
(A.13d)
z ψ z ( N m ) c m ,
(A.13e)
ψ z c N 1 z ( N m + 2 ) c m 2 ,
(A.13f)
ψ z z ( N m + 2 ) ( N m + 1 ) c m 2 ,
(A.13g)
z ψ z z ( N m + 1 ) ( N m ) c m 1 ,
(A.13h)
z 2 ψ z z ( N m ) ( N m 1 ) c m ,
(A.13i)
ψ t c ˙ m ,
(A.13j)
z ψ t c ˙ m + 1 ,
(A.13k)
ψ t c ˙ N z c ˙ m 1 ,
(A.13l)
ψ z t ( N m + 1 ) c ˙ m 1 ,
(A.13m)
z ψ z t ( N m ) c ˙ m ,
(A.13n)
ψ t t c ¨ m ;
(A.13o)
N ψ z ψ z m c m ,
(A.13p)
N 2 ψ ( 2 N 1 ) z ψ z + z 2 ψ z z m 2 c m ,
(A.13q)
( N + 1 ) ψ c N z ψ z m c m 1 ,
(A.13r)
( N + 1 ) 2 ψ c N z ( 2 N + 1 ) ψ z + z ψ z z m 2 c m 1 ,
(A.13s)
ψ c N c N 1 z z 2 c m 2 ,
(p.217)
(A.13t)
( N + 2 ) ψ z ψ z ( N + 2 ) c N ( N + 1 ) z c N 1 z 2 m c m 2 ,
(A.13u)
( N + 2 ) 2 ( ψ c N ) z [ ( 2 N + 3 ) ψ z + ( N + 1 ) 2 c N 1 ] + z 2 ψ z z z 2 m 2 c m 2 ,
(A.13v)
( N + 1 ) ψ t c ˙ N z ψ z t m c ˙ m 1 .