SOLUTION OF AN INFINITE NUMBER OF INEQUATIONS DEPENDING ON A CONTINUOUS
PARAMETER AND APPLICATION TO THE SOLUTION OF EQUATIONS IN DYNAMICAL
ASTRONOMY
J.-P. J. LAFON
Observatoire de Paris-Meudon
DASGAL - URA CNRS 663
92195 Meudon Principal Cedex
France
ABSTRACT. The investigation of a large class of problems in physics
and astrophysics requires the dtermination of the ranges of some parameters z, E, ... for which inequations of the form F(r;z,E,...) ^? 0
are satisfied for all r in some interval )a,b(. The solution of this
problem is given under the form of three general theorems and, resulting
from them, a very simple numerical procedure. This can also be used to
solve equations of the form df/dt = 0 where f is some function of variables and derivatives of these variables (functions of t) with respect
to t, for instance Vlasov-type equations in dynamics of flat stellar
disks.
1. THE PROBLEM
Consider a set of inequations of the form
i
r e )p,R(
(l)
F(rAx,y,...)^r 0
with (x,y,...)(= R
Tiiere are an infinite number of inequalities indexed by a continuous parameter r and the (x,y,.. .) belong to continuous sets (the
variables x, y, ... belong to intervals of real numbers).
The solution lies in two theorems (Lafon , 1977) with which we
can reduce the problem to a similar but much simpler problem i.e. with
a set of countable values of r and, in most cases, with a finite number
of values of r.
The resulting method of solution has many advantages over the usual
empirical methods because it is universal and systematic, well fitted
to numerical analysis and rigorous :
Such inequalities are usually solved with empirical methods depending on the particular form of the function, one for each case. The
method proposed is independent of F for a large class of functions satisfying a few simple conditions (continuity and derivability) chosen
so that they are satisfied in most cases. The method can easily be generalized even when some of these conditions are no longer satisfied.
The solutions are found progressively i.e; successively for all
the intervals )p,a( or )a,R( with p^ca,<c R. Thus, the number of nume-
Celestial Mechanics 45: 105-110, 1989.
© 1989 Kluwer Academic Publishers. Printed in the Netherlands.
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106
J.-P. J. LAFON
rical tests is greatly reduced since the interval )p,R( is traversed
only once whereas other methods require many traverses. As a consequence, the numerical determination of the solutions is also more easy
and rigorous.
For instance, inequations like (1) have to be solved when building models of finite, flat, a>.isymmetric, self-gravitating disks of
stars (Lafon, 1976). The star motion is governed by Vlasov equations;
in other words, the stars move in the gravitational potential that
they generate collectively, along non collisional orbits characterized
by two constant parameters, the angular momentum z and the energy E.
A star belonging to the system travels along its trajectory between
two radial distances lower than the radius of the system. A non collisional stellar orbit in the self gravitation field of the system is
populated only if it does not pass beyond the external limit of the
system; otherwise the 'potential' orbit is unpopulated.
Thus, the distribution functions of the stars which satisfy
the Vlasov equations are functions f(r;z,E) equal to zero within the
ranges of z and E for which inequalities like (1) are satisfied : the
functions F(r;z,E) depend
on the potential profile; each condition
Fa» 0 expresses that a f.tar moving along the orbit characterized by z
and E could overcome the local potential barrier; at least R is greater than the radius of the system.
Details are given elsewhere (Lafon, 1976, 1986)
2: DEFINITIONS AND HYPOTHESES
For simplicity and to illustrate the theorems easily with figures, the
problem will be concerned hereafter with two variables only x and y.
However the method does not have to be always restricted to two variables (Lafon, 1977).
We shall assume that :
- F(r;x,y) is continuously derivable with respect to r,x,y
- F' = J F / ^ X and F' = >)F/3y are never simultaneously zero
x
y
- )/ rg= )p,R( the sets of points x,y such that F ^? 0 and
F <:0 are connected.
- the solutions, if any, of the system F = 0 and F 1 = 0 are
bounded and F' = 0 )/ x and y such that F = 0 for at most a finite
number of r ^ )p>R(
These conditions are fairly weak; the generalisation to the case
where they are not satisfied for countable sets of r,x,y is straightforward. The third assumption is used only to illustrate more easily
the solutions by figures.
In terms of geometrical representation in the x,y plane these a s sumptions mean that, for each r, F = 0 is the equation of a continuous,
differentiable, non singular curve P .
The following notations will be used : for F(s;X,y)
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SOLUTION OF AN INFINITE NUMBER OF INEQUATIONS
w+
ss
ss
set of the points x,y such that F > 0
set of the points x,y such that F < 0
s
pr
set of the points x,y such that F = 0
set of the points x,y such that F > 0
p
y
s £ )p,r(
W
set of the points *,y such that F < 0
y s ^ )p,r(
set of the points x,y such that F = 0 for at least
o
interior of N
pr
N
pr
N
107
in )p,r(
pr
pr
pr
0 W
boundary of W +
pr
J
pr'' pr
such that F^>0
set of the points x,y
set of the points x,y such that F<_0
W
rR
rR
y
y
s<=. )r,R(
s ^ )r,R(
In terms of geometrical representation in the x,y plane the
curve P divides the plane into two connected regions W
and W
S
r
+
°
J.
GC
SS
The regions W
,W
, W _ and W _ are the intersections of
infinite sets of such regions.
Interms of 'shadow' ( F ? > 0 ) and 'light' (F-"=£0), W + is the zone
of full shadow, N is the zone of penumbra and W is ?ne zone of
pr
pr
full light.
NOTE
The graphical representations are useful to visualize the results, in which there are only a few possible cases; this should not
suggest that so it is with the demonstrations, for which many situations have to be envisaged before they can be reduced to these cases.
Thus, we shall not go into details here : the demonstrations are
detailed elsewhere (Lafon, 1977)- We only display the results and the
way to use them.
3. THEOREM 1
Lafon (1977) proved that,
are satisfied :
N
(
pr
when the conditions stated in section 2
„
u
+
U w+ \ n
\p^ r ^ R
w
rr / ! II PR
o_
pR
Npr = (
U <.R„ W"rr )| []
\ p ^.r
I «
W
T>
I)
N
pr
£r
0
4. THEOREM 2
Lafon (1977) also proved that, when the conditions stated in section 2
are satisfied, two exclusive cases are possible :
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108
J.-P. J. LAFON
l / There i s some Lg= )p,R) and 1 £= (p>R( such t h a t
Wpr = 4>
y r 52 L
W+ R = <t>
Vr ^ 1
2/ The interval )p,R( can be divided into a countable set of
adjacent subintervals )a.,a. 1 ( such that a.. = p , a. . ^. R and such
that one of the following four cases describes what occurs for
a. <c.r -<a. ^
1
1+1
-!HBB:- a/ F = 0 and F' ^ 0 except for a finite number of values of
r say r-, r„, ...r. for which F' = 0 ]/ x and y such that F = 0
(extremal curves P ). Then,
+
W
pr
= w+ n w+ ( . , n . w+ \
pa ±
rr
k=l,2,...j
r ^ '
Fi
!
**-"-"- b/ F = 0 and F' = 0 y r for a finite number of fixed points
x,y ('envelope points') . Then,
W+
= W+
0
W+
Fig. 2
pr
pa.
rr
-x-x-x-* c/ F(s;X,y) = 0 and F'(s5X,y) = 0 define some bounded continuous functions x(s) and y(s) (corresponding to arcs of envelopes of
the curves P ). Then,
s
Fi s
W + = W„ r— W +
0 w+
§ 3, 4
v
pa. "
rr
°
'
pr
E -—
where W„ denotes that part of the region W
I] W
bounded by the
set of the points x(s),y(s) for all s
i
between a. and
r; in terms of geometry, W
is that part of the region
W
H W
reduced by the^ arcs of the envelope corresponding to s
i
between a. and r.
1
-;BBB:- d/ Cases b and c appear simultaneously within the same interval a.,a. . Then W
is the intersection of the solutions obtained assuming pure b and c cases separately.
NOTES Other theorems :
There is also a countable set of adjacent subintervals h, .,,h,
such that h 1 = R and h, S: p with similar properties for W
Similar theorems can also be proven for W
and W
In most of the practical cases, the countabxe sets of a. and
h, are sets with a finite number of values.
k
CONCLUSION
0000000000
The determination of W _ and W _ and, consequently, of N _ is finally
reduced to intersections of countable and usually finite sets of regions in the x,y plane.
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SOLUTION OF AN INFINITE NUMBER OF INEQUATIONS
Fig 1 Interval of type a
Fig 2
Interval of type b
Here P is an
a
extremal curve
Fig 3 Interval of type c
Fig 4
Interval of tvne c
WpR
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J.-P. J. LAFON
110
5. NUMERICAL TESTS FOR DETECTING THE LIMITS OF THE SUBINTERVALS
It may seem difficult to determine the a. or h, and so one could
think that , in spite of its generality, the method that can be
deduced from the above mentioned theorems requires complicated analyses for numerical computations. This is far from beeing true.
Indeed, in a numerical problem, it is never necessary to determine a. or h, with an accuracy better than the step characterizing
the finite differences(Jr. Moreover, it is very easy to detect when
one steps into a new subinterval.
In practice, it is only important to detect :
1/ cases of type a; one steps into such an interval when
Q f\ P
r
=$
2/ other cases when some arcs of envelopes bound W^ and effectively1 reduce W
(1 1
. This occurs as long
as () fl P c " <
e
b
pa.I ' rr
r
rt= r+or,r+ar
6. REFERENCES
LAFON J.-P. J., Astron. Astrophys., 'On self gravitating stellar
disks I A new model', 1976, ^ 6 , 461
LAFON J.-P. J., J. MathT Phys. , ' On a topological problem
arising in physics', 1977, 18," 1178. '
LAFON J.-P. J., Hevetica Physica Acta,'Missing mass and stellar
dynamics in flat galaxies', 59, 1142
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