A N N A L E S S O C I E T A T I S M A T H E M A T I C A E P O L O N A E S e r i e s I : C O M M E N T A T I O N E S M A T H E M A T I C A E X V I I ( 1 9 7 3 )
R O C Z N I K I P O L S K I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O S é r i a I : P R A C E M A T E M A T Y C Z N E X V I I ( 1 9 7 3 )
M. A. Wo l a n o w s k i (Warszawa)
Since Weierstrass time it is known that integral (1) does not depend on a parametrics representation (x, y) of the curve К if and only if the function / is a positive homogeneous function in respect to x, у and does not depend on t explicitely.
The proof of sufficiency is undoubtful (see e.g. [3], p. 157). The proof of necessity depending on differentiation of (1) in respect to b (it is to find in most handbooks of the calculus of variations, begining at least with [1] till the newest as [2]) is not one that we need. This is because we often consider only classes of curves joining a pair of given points and therefore the differentiation of (1) in respect to Ъ is inadmissible. Other doubts arise from the following observation: if we consider the curves such that their parametric representations are defined in the fixed interval, then the function / may depend on t explicitely. So the necessity of the condition depends on the adopted definition of curve.
In this paper we will introduce a definition in order to have a correct proof also of the necessity of the Weierstrass condition.
I would like to express my thanks to professor K. Tatarkiewicz for drawing my attention to the problem.
2. Suppose that Jc is a natural number and denote by Ek the set of all functions (x,y): T>(x,y) B2 satisfying the following conditions:
(a) D (x,y) is a closed interval of JR1-,
(b) (x, y)e C* and x2(t) + y2(t) > 0 for teT>(x, y).
If Ре В 2, then we denote by E k(P) the set of all (x, y)e Ek such that {x, 2/)(a) = P, where T>(x, y) = <a, b>.
It will be also considered the set Fk of all real functions h of C*-class defined and having positive first derivatives on closed intervals of B 1.
A note on parametrical integrals 1. Let
b
(1)
a
266 M. A. W o l a n o w s k i
We introduce in Ek the equivalence relation “ r-*j ” putting (x, y)
~ (и, v) if and only if there exists heFk such that (x, y) = (u, v)o h(a), h(a) = e, h(b) = d, where <a, b) = D(a?, у), <c , d} = T>{u, v).
The elements of the space E,k[ ~ are called directed curves.
Suppose that f(t, x , y , x , y ) is given real function of 0°-class defined for all real numbers t, x, у, x, у such that x2-\-y2 > 0.
Let us denote by I the functional defined on Ek and having the form :
for each curve K , then f is a positive homogeneous function in respect to x, ÿ and does not depend on t explicitely.
Proof. Suppose that ( x , y ) e E k, m e F k, D( x , y ) = Dm = (r, s>,
t e (Г, S).
In virtue of the change of integral variables theorem it follows from the assumption that
Denoting by m , m . , x , y ?x , y the values of the adequate functions in the point t we obtain
Suppose now that u , v , a , b , c , d , e ( c 2-f-d2> 0 ?e > 0) are the numbers arbitrarily fixed. Take such r, s that ue (r, s) and take such (g , h)e Ek(P) further that ne Fk, Dn = <r, s), n(u) = v, n(n) = e. Putting into (3) (x, y) = (g, h), m = n, t = u we have
Putting e = 1 we conclude that / does not depend explicitely on t.
Then (4) can be written:
ь (2)
a
where (a, b} is the domain of (x , y ).
Th e o r e m I f the functional I takes the same values in the set K n E k(P )
( 3 )
that D (g, h) = <r, s ), <7(u) = a, h{ u) = b, ^(u) = c, h{ u) = d. Suppose
which completes the proof.
Note on parametrical integrals 267
R e f e r e n c e s
[1] A. K n e s e r , Lehrbuch der Variationsrechnung, Braunschweig 1900.
[2] H. R u n d , The H am ilton-Jacobi theory in the calculus o f variations, London, 1966.
[3] K. T a t a r k i e w i c z , BachuneTc w ariacyjny, t. 1, W arszawa 1969.
