Schulz M. Control Theory in Physics and Other Fields of Science: Concepts, Tools, and Applications
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4.2 Control by External Sources 115
This formal solution may be interpreted as an alternative version of the field
equations. We remark that the homogeneous solution is mainly determined
by the initial conditions. We use again the quadratic performance function
(4.66) in order to construct the generalized action
S =
T
0
dt
G
d
d
rφ (r,t,Ψ,∇Ψ, u)
+
T
0
dt
G
d
d
rΠ (r,t)[Ψ (r,t) − Ψ
hom
(r,t)
−
t
0
dτ
G
d
d
xΓ (r,t| x,τ)[Au(r,t)+BJ(r,t)]] , (4.72)
where we have considered a final control horizon T . The variation of the
generalized action with respect to the field momentum Π again leads to the
dynamic equation, while the variation with respect to the field leads to
Π (r,t)=∇(Ω (r,t) ∇Ψ (r,t)) − Q (r,t) Ψ (r,t) (4.73)
and the variation with respect to the control field gives
u (r,t)=R
−1
(r,t)
T
t
dτ
G
d
d
xA
T
(r,t) Γ
T
(x,τ | r,t) Π (x,τ) . (4.74)
The substitution of (4.73)in(4.74) eliminates the momentum field from the
control equations and we obtain
u (r,t)=R
−1
(r,t)
T
t
dτ
G
d
d
xA
T
(r,t) Γ
T
(x,τ | r,t)
× [∇(Ω (x,τ) ∇−Q (x,τ)] Ψ (x,τ) . (4.75)
This integral relation and the formal solution (4.70) are a complete set of
integral equations determining the linear field control problem with passive
boundaries. The procedure is applicable to all field equations if the Green
function is available. But this is, by no means, a trivial problem, and it requires
often a large amount of numerical and analytical tools [82, 83, 84].
A practical problem with passive boundary conditions is the optimal con-
trol of soil venting [85] used for remediation of soil contaminations with volatile
organic components, such as chlorinated organic hydrocarbon. Soil venting ba-
sically means that the soil air is removed from the contaminated soil region
by pumping while the volatile contaminants in the extracted air are collected
and annihilated [86, 87]. In principle, the corresponding physical model is a
diffusion-reaction process (Fig 4.4). It considers (i) the diffusion of soil air
from several boreholes filled with fresh air under pressure (sources) to other
116 4 Control of Fields
APAPA
Fig. 4.4. The principle of soil venting: fresh air is pumped into the active boreholes
(A) and moves by diffusion processes to the passive boreholes (P). Volatile organic
materials follow the diffusion flow to the passive holes. The contaminated air is
finally filtered and the organic components are collected in special tanks
boreholes sucking off the contaminated air (sinks) (ii) the transport of organic
components following the diffusion gradient of the air, and (iii) a spontaneous
reduction of the organic components by the presence of microorganisms. Ad-
sorbing boundary conditions are assumed for the interface between the af-
fected region and the atmosphere as well as the ground water or surface water
while reflecting boundary conditions are taken into account for the interface
between the soil and rocky grounds.
The optimal field control problem consists in an optimal choice of the dis-
tribution of sources and sinks in order to clean the soil in a minimal time with
as small as possible costs. The practical solution requires various numerical
methods, for example, genetic simulation methods [88] and modified versions
of the Powell algorithm [89].
4.3 Control via Boundary Conditions
A very important method controlling the dynamics of fields is the so-called
boundary control. This concept is applied in practice very often. For example,
the complicated aerodynamic and chemical evolution of a mixture of air and
natural gas in a combustion chamber of power station is controlled via the
boundaries, e.g., by the change of the current injection rates of the air or the
amount of outgoing waste air.
Unfortunately, the control of nonlinear field equations via boundaries is
more or less an empirical and essential field for a broad experimental research.
Another situation occurs for linear field equations. Here, boundary control can
be mapped onto the source control. To this aim, let us assume that the field
equation is given by the homogeneous version of (4.64)
∂Ψ(r,t)
∂t
=
'
FΨ(r,t) (4.76)
4.3 Control via Boundary Conditions 117
G
(t)
G
(t,u(t))
active boundaries
=const.
G
passive boundaries
G
Fig. 4.5. Passive boundaries are defined by time-independent or uncontrolled time-
dependent functions, active boundary conditions depend on free changeable control
functions u(t)
defined over the region G of the space R
d
while the boundary conditions
are given by some constraints at the border ∂G, see Fig. 4.5. Furthermore,
the boundary conditions should be changeable by several external control
fields w(x,t) which are declared only at the border, x ∈ ∂G. Therefore, we
assume that at the boundaries the values of the field Ψ(x,t) and possibly
its derivatives ∇Ψ (x,t), ... are well defined functions of the control fields,
Ψ(x,t)=Ψ (x,t,w(x,t)), ∇Ψ (x,t)=∇Ψ(x,t,w(x,t)),... . We split the field
Ψ(r,t) in two parts,
Ψ(r,t)=Ψ
0
(r,t)+
Ψ(r,t) (4.77)
with passive boundary conditions for Ψ
0
(r,t), Ψ
0
(x,t)=0forx ∈ ∂G.The
field
Ψ(r,t) satisfies the equation
'
F
Ψ(r,t) = 0 (4.78)
and have the same boundary conditions as the original field Ψ(r,t), i.e.,
Ψ(x,t)=Ψ (x,t), ∇
Ψ(x,t)=∇Ψ (x,t), ... . Obviously, (4.78) is no evolution
equation with respect to time. That means the time t is here only a simple
parameter. Now we assume that we are able to solve these equations for ar-
bitrary boundary conditions. That is not a trivial task. The structure of the
solution is mainly defined by the structure of the operator
'
F and the bound-
ary conditions. For example, the one-component equation ∆ ϕ =0requires
for a given region G of R
3
the solution
ϕ(r,t)=
∂G
dA
4π
1
|x − r|
∇ϕ(x,t,w(x,t))+
x − r
|x − r|
3
ϕ(x,t,w(x,t))
, (4.79)
118 4 Control of Fields
where dA is the oriented infinitesimal surface element of the border ∂G.
However, the explicit solution of (4.78) considering the boundary conditions
correctly is a functional of the boundary conditions and therefore of the con-
trol functions,
Ψ(r,t)=
Ψ[r,t,w]. We substitute this solution and (4.77)in
(4.76) and obtain because of (4.78)
∂Ψ
0
(r,t)
∂t
=
'
FΨ
0
(r,t) −
˙
Ψ[r,t,w] . (4.80)
This is an inhomogeneous field equation for the function Ψ
0
(r,t) which satisfies
passive boundary conditions and which is controlled by an apparently external
source
˙
Ψ[r,t,w]. Thus, the above-discussed formalism can now be used for the
control problem.
The most currently intensive discussed field control problems with active
boundary conditions are related to the scalar wave equation [5, 90], ¨ϕ−∇
2
ϕ =
0, generalized Klein–Gordon equations [91], ¨ϕ −∇a(r)∇ϕ + m(r)ϕ =0,or
vibrating plates [91, 92]¨ϕ + ∇
4
ϕ = 0. It is surprising that these apparently
simple problems also offer a lot of mathematical problems related to the con-
trollability and stability of the control.
References
1. W. Thirring: A Course in Mathematical Physics II. Classical Field Theory
(Springer, Berlin Heidelberg New York, 1998) 93
2. E. Schmutzer: Grundprinzipien der klassischen Mechanik und der klassischen
Feldtheorie (Verlag der Wissenschaften, Berlin, 1973) 93
3. M. Kaku: Quantum Field Theory (Oxford University Press, Oxford, 1993) 93, 95
4. N. Sanchez: Non-Linear Equations in Classical and Quantum Field Theory
(Springer, Berlin Heidelberg New York, 1985) 93
5. W. Krabs: Math. Meth. Appl. Sci. 1, 322 (1979) 97, 118
6. R. Triggiani: Appl. Math. Optim. 18, 241 (1988) 97
7. W. Krabs: On time-optimal boundary control of vibrating beams. In: Control
Theory for Distributed Parameter Systems and Applications, Lecuture Notes in
control and Information Sciences, vol. 54, ed by F. Kappel, K. Kunisch, W.
Schappacher (Springer, Berlin Heidelberg New York, 1983), p. 127 97
8. V. Komornik: Exact Controllability and Stabilization: The Multiplier Method
(Wiley, Chichester, 1994) 97
9. N.J. Lynch-Aird: IEEE J. Select. Areas Commun. 9, 830 (1991) 99
10. A. Asamitsu, Y. Moritomo, Y. Tomioka, T. Arima, Y. Tokura: Nature 373,407
(1995) 99
11. Y. Tomioka, A. Asamitsu, Y. Moritomo, H. Kuwahara, Y. Tokura: Phys. Rev.
Lett. 74, 5108 (1995) 99
12. H. Risken: Z. Phys. 186, 85 (1965) 99
13. H. Haken: Laser Theory (Springer, Berlin Heidelberg New York, 1985) 99
14. W. Thirring: Classical Dynamical Systems and Classical Field Theory (Springer,
Berlin Heidelberg New York, 1992) 99
15. F.E. Low: Classical Field Theory: Electromagnetism and Gravitation (Wiley-
Interscience, New York, 1997) 99
References 119
16. A.O. Barut: Electrodynamics and Classical Theory of Fields and Particles
(Dover Publications, New York, 1981) 99
17. E. Binz, J. Sniatycki, H. Fischer: Geometry of Classical Fields (Elsevier, North-
Holland, 1988) 99
18. D.E. Soper: Classical Field Theoryn (Wiley, Chichester, 1976) 99
19. E.F. Kaasschieter, A.J.M. Huijben: Report PN 90-92-A, TU Delft (1990) 100
20. P.G. Ciarlet, J.-L. Lions: Finite Element Methods, Vol. II of Handbook of Nu-
merical Analysis (North-Holland, Amsterdam, 1991) 100
21. F. Brezzi, M. Fortin: Mixed and Hybrid Finite Element Methods (Springer,
Berlin Heidelberg New York, 1991) 100
22. H.-C. Huang: Finite Element Analysis of Non-Newtonian Flow (Springer, Berlin
Heidelberg New York, 1999) 100
23. G. Dhondt: The Finite Element Method for Three-Dimensional Thermomechan-
ical Applications (Wiley, Chichester, 2004) 100
24. A. Munjiza: The Combined Finite-Discrete Element Method (Wiley, Chichester,
2004) 100
25. T.A. Nooren, H.A. Wouters, T.W.J. Peeters, D. Roekarts, U. Maas, D. Schmidt:
Combust. Theory Model. 1, 79 (1997) 100
26. J.O. Hirschfelder, C.F. Curtiss: Theory of Propagation of Flames, Part I: Gen-
eral Equations (Williams and Wilkins, Baltimore, 1949) 100
27. N. Peters, K. Seshadri: Combust. Flame 187, 197 (1988) 100
28. T. Schmidt, T. Blasenbrey, U. Maas: Combust. Theory Model. 2, 135 (1998) 100
29. B. Ruf, O. Deutschmann, F. Behrendt, J. Warnatz: J. Appl. Phys. 79,7256
(1996) 101
30. D.G. Zeitoun, G.F. Pinder: Water Resour. Res. 29, 217 (1993) 101
31. O.M. Aamo: Flow Control by Feedback (Springer, Berlin Heidelberg New York,
2003) 101
32. R.M. Christensen: Theory of Viscoelasticity (Dover Publications, New York,
1982) 101
33. J.D. Ferry: Viscoelastic Properties of Polymers, 3rd edn (Wiley, Chichester,
1980) 101
34. K.L. Dorrington: Symp. Soc. Exp. Biol. 34, 289 (1980) 101
35. K. Kawashima, S. Unjoh: J. Struct. Eng. 120, 2583–2600 (1994) 101
36. R.M. Mutobe, T.R. Cooper: Comput. Struct. 72, 279 (1999) 101
37. L.L. Chung, L.Y. Wu, T.G. Jin: Eng. Struct. 20, 62 (1998) 101
38. I.I. Blekhman: Vibrational Mechanics (World Scientific Publishing, Singapore,
1999) 101
39. K.J. Laidler: Chemical Kinetics (McGraw-Hill, New York, 1965) 101
40. S.W. Benson: The Foundations of Chemical Kinetics (McGraw-Hill, New York,
1960) 101
41. N.G. van Kampen: Stochastic Processes in Physics and Chemistry (North-
Holland, Amsterdam, 1981) 102
42. H. Haken: Synergetics (Springer, Berlin Heidelberg New York, 1978) 102
43. G. Nicolis, I. Prigonine: Self-Organization in Non-Equilibrium Systems (Wiley,
New York 1980) 102
44. T.M. Liggett: Interacting Particle Systems (Springer, Berlin Heidelberg New
York, 1985) 102
45. V. Kuzovkov, E. Kotomin: Rep. Prog. Phys. 51, 1479 (1988) 102
46. K. Kang, S. Redner: Phys. Rev. A 32, 435 (1985) 102
120 4 Control of Fields
47. V.I. Yudson, M. Schulz, S. Stepanow: Phys. Rev. E 57, 5063 (1998) 102
48. B. Bock, H. H¨otzel, M. Nahold: Untergrundsanierung mittels Bodenluftab-
saugung und In-Situ-Strippen, vol. 9 (Schriftenreihe Angewandte Geologie, Karl-
sruhe, 1990) 102
49. U. Fischer: Experimental and numerical investigation of soil vapor extraction.
Thesis no. 11277, ETH Z¨urich (1995) 102
50. J.S. Gierke, N.J. Hutzler, D.B. McKenzie: Water Resour. Res. 28, 323 (1992) 102
51. V. Privman: Nonequilibrium Statistical Mechanics in One Dimension (Cam-
bridge University Press, Cambridge, 1997) 102
52. Y. Smagina, O. Nekhamkina, M. Sheintuch: Ind. Eng. Chem. Res. 41, 2023
(2002) 102
53. A.A. Alonso, B.E. Ydstie: Automatica 37, 1739 (2001) 102
54. H. Motz, H. Wise: J. Chem. Phys. 32, 1893 (1960) 102
55. B.M. Schulz, S. Trimper, M. Schulz: Eur. Phys. J. B 15, 499 (2000) 103
56. B.M. Schulz, M. Schulz, S. Trimper: Phys. Lett. A 291, 87 (2001) 103
57. B.M. Schulz, P.Reineker, M.Schulz: Phys. Lett. A 299, 337 (2002) 103
58. M. Schulz, P. Reineker, B.M. Schulz, S. Trimper: Phys. Chem. 282, 379 (2002) 103
59. B.M. Schulz, S. Trimper, M. Schulz: Phys. Rev. E 66, 031106 (2002)
60. M. Schulz, S. Stepanow: Phys. Rev. B 59, 13528 (1999) 103
61. M. Schulz, K. Zabrocki, S. Trimper: Phys. Rev. E 65, 056106 (2002) 103
62. S. Trimper, K. Zabrocki, M. Schulz: Phys. Rev. E 70, 056133 (2004) 103
63. M. Schulz, S. Trimper: Phys. Rev. B 64, 233101 (2001) 103
64. J.W. Nunziato: Quart. Appl. Math. 29, 187 (1971) 103
65. W. Jung:
¨
Uber ein pseudoparabolisches Rand-Kontroll-Problem aus der
W¨armeleitung. Thesis, Universit¨at Frankfurt/M. (1982) 103
66. W. Scondo: Randsteuerung einer evolutionsgleichung mit Ged¨achtnis. Thesis,
Universit¨at Frankfurt/M. (1982) 103
67. M. Schulz: Statistical Physics and Economics (Springer, Berlin Heidelberg New
York, 2003) 107
68. S. Angel, G.M. Hyman: A Geometry of Movement for Regional Science (Pion,
London, 1976) 107
69. T. Puu: Reg. Sci. Urban Econ. 11, 317 (1981) 107
70. M.J. Beckmann: Econometrica 20, 643 (1952) 107, 108
71. M.J. Beckmann, T. Puu: Spatial Structures (Springer, Berlin Heidelberg New
York, 1990) 107, 108
72. T. Puu: Reg. Sci. Urban Econ. 8, 225 (1978) 107
73. T. Puu: Chaos, Solitons and Fractals 5, 35 (1995)
74. J.H.V. Th¨unen: Der isolierte Staat in Beziehung auf Nationaleinkommen und
Landwirtschaft, reprint of the 1826 edition (Gustav Fischer, Stuttgart, 1966) 107
75. W. Launhardt: Mathematische Begr¨undung der Volkswirtschaftslehre (B.G.
Teubner, Leipzig, 1885) 107
76. A. Weber:
¨
Uber der Standort der Industrien (T¨ubingen, 1909) 107
77. T. Puu: The Allocation of Road Capital in Two-Dimensional Space: A Contin-
uous Approach (North-Holland, Amsterdam, 1979) 107
78. T. Puu: Chaos, Solitons and Fractals 3, 99 (1993) 107, 108
79. M.J. Beckmann, T. Puu: Spatial Economics: Potential, Density and Flow
(North-Holland, Amsterdam, 1979) 108
80. T.F. Palander: Beitr¨age zur Staatstheorie (Almquist and Wiksells, Uppsala,
1935) 111
References 121
81. H.V. Stackelberg: Jahrb. National¨okon. Stat. 148, 680 (1938) 111
82. E. Zauderer: Partial Differential Equations of Applied Mathematics (Wiley-
Interscience, Chichester, 1998) 115
83. Y. Pinchover, J. Rubinstein: An Introduction to Partial Differential Equations
(Cambridge University Press, Cambridge, 2005) 115
84. A. Broman: Introduction to Partial Differential Equations: From Fourier Series
to Boundary-Value Problems (Dover Publications, New York, 1990) 115
85. H.H. Gerke, U. Hornung, Y. Kelanemer, M. Slodiˆcka, S. Schumacher: Optimal
Control of Soil Venting: Mathematical Modeling and Applications (Birkh¨auser,
Basel, 1999) 115
86. F. Schwille: Besond. Mitt. Deutsch. Gew¨asserkundl. Jahrb. 46, 72+XII (1984) 115
87. F. Schwille: Dense Chlorinated Solvents in Porous and Fractured Media (Lewis
Publishers, Chelson, 1988) 115
88. D.E. Goldberg: Genetic Algorithms in Search, Optimizationand Machine Learn-
ing (Addison-Wesley Publishing Co., New York, 1989) 116
89. R.P. Brent: Algorithms for Minimization without Derivatives (Prentice-Hall,
New York, 1973) 116
90. W. Krabs: Z. Operations Res. 26, 63 (1982) 118
91. W. Krabs: Optimal Control of Undamped Linear Vibrations (Heldermann-
Verlag, Lemgo, 1995) 118
92. M. Niezg˝odka: Stability of a class of nonlinear evolution free boundary problems
with respect to domain variations. In: Optimal Control of Partial Differential
Equations, ed by K.-H. Hoffmann, W. Krabs (Birkh¨auser, Basel, 1984), p. 173 118
5
Chaos Control
5.1 Characterization of Trajectories in the Phase Space
5.1.1 General Problems
In the previous chapters we have discussed the control theory from a me-
chanical and field theoretical point of view. This control theory is, of course,
applicable to all deterministic evolution equations and leads, under correct
formulated initial and boundary conditions and under a reasonable coupling
between the control functions and the system, to an optimal control. The scope
of these control methods is very large; however these methods require three
important considerations: (i) the complete knowledge about the dynamics of
the system (ii) accurately determined initial conditions, and (iii) a permanent
and nondelayed control by a control function u(t) continuously varying over
the whole control period.
In the following chapters we will give up successively these conditions.
This necessitates a certain adjustment of the control concepts, since all three
points are related to a more or less strong lack of information. In this chapter
we focus on a discontinuously and delayed control of unstable system states.
Simultaneously we take into account a complete knowledge about the char-
acteristic properties of the system dynamics. But we will consider a weak
inaccuracy of the initial conditions and several system parameters so that the
uncontrolled system becomes more or less unpredictable at least for large time
scales.
A sensitive analysis of the nature of this type of control requires a more
detailed knowledge about possible general scenarios concerning the motion of
system with a nontrivial degree of complexity. Especially, it is necessary to
know some general aspects about the structure of a trajectory through the
phase space of a system under a constant or zero control.
M. Schulz: Control Theory in Physics and other Fields of Science
STMP 215, 123–148 (2006)
c
Springer-Verlag Berlin Heidelberg 2006
124 5 Chaos Control
5.1.2 Conservative Hamiltonian Systems
To this aim we assume that the motion of the mechanical system may
be confined to a finite volume of the phase space. That is, for example, the
case if a system of finite energy is trapped in a sufficiently deep potential or
if we observe a system from its center of mass while all degrees of freedom
of the system are coupled by such interaction terms that no component can
escape the system. In principle, such bounded systems are the standard situa-
tion observed in the majority of physical problems. Furthermore, we take into
account that the uncontrolled Hamiltonian is explicitly time-independent.
We know from classical mechanics that the definition of spatial coordinates
q
i
(i =1,...,N) and momenta p
i
and (i =1,...,N) are not fixed
1
.Itis
always possible to find another set of degrees of freedom
Q
i
= Q
i
(q
1
,...,q
N
,p
1
,...,p
N
)andP
i
= P
i
(q
1
,...,q
N
,p
1
,...,p
N
) (5.1)
by a canonical transformation
p
i
=
∂F (q
1
,...,q
N
,P
1
,...,P
N
)
∂q
i
Q
i
=
∂F (q
1
,...,q
N
,P
1
,...,P
N
)
∂P
i
(5.2)
with an arbitrary generating function F which allows the unique representa-
tion of the new coordinates Q and momenta P as functions of the old degrees
of freedom (q,p)via(5.1) and vice versa. Each transformation of this type
conserves the structure of the canonical equations of motion (1.1) for the new
Hamiltonian
2
H
(Q, P )=H(q(Q, P ),p(Q, P )) . (5.3)
One special choice of the generating function is the identification of F with
the explicitly time-independent part of a complete integral I of the mechanical
Hamilton–Jacobi equation
∂I
∂t
+ H
q,
∂I
∂q
=0, (5.4)
where the momenta of the original Hamiltonian H = H(q, p) are replaced by
∂I/∂q. A complete integral is a special solution of the Hamilton–Jacobi equa-
tion consisting of the N coordinates q and N +1 independent constants which
are denoted as C
0
and J
1
,...,J
N
; see also Sect. 2.5. The structure of (5.4)
1
In this chapter we follow the traditional mechanical notation and consider a 2N-
dimensional phase space.
2
If the generating function is explicitly time dependent, the Hamiltonian must be
transformed via
H(q, p) → H
(Q, P, t)=H(q(Q, P, t),p(Q, P, t),t)
+
∂F(q, P, t)
∂t
q=q(Q,P,t)
.
This case is not considered here.
5.1 Characterization of Trajectories in the Phase Space 125
suggests the general form I = C
0
+ F (q, J) −E(J)t. Identifying the invariant
quantities of the motion, J, with the new momenta P and F (q, J) with the
generating function F (q, P) of the canonical transformation, we obtain the
new Hamiltonian from (5.4) and (5.3)
H
(P, Q)=H (q, p)=E(J)=E(P ) (5.5)
i.e., the transformed Hamiltonian H
depends only on the new momenta P
but not on the coordinates. Thus, we obtain the new equations of motion
˙
P
i
=
∂H
∂Q
i
=0 and
˙
Q
i
=
∂H
∂P
i
= Ω
i
(P ) (5.6)
with the frequencies Ω
i
depending only on the new momenta P . Consequently,
the trajectories are given by
P
i
=const. and Q
i
= Ω
i
(P ) t + Q
(0)
i
(5.7)
in the new coordinate system spanned by the new coordinates Q and the new
momenta P . Since, on the one hand, the motion of the system was assumed to
be constrained on a finite domain of the phase space, but, on the other hand,
Q
i
increases monotonously with increasing time, the new coordinates must
be interpreted as angles. That means the phase space is structured in a set
of imbedded N -dimensional tori. Any possible trajectory lies on one of these
tori. Obviously, all tori must be covered by closed trajectories or by dense
ergodic curves. The decision wether a trajectory forms closed loops or not is
determined by the frequencies Ω
i
(P ) and therefore by the momenta P .Ifthe
equation
N
i=1
Ω
i
(P ) m
i
= 0 (5.8)
can be satisfied by nonzero integers m
i
, the trajectories form closed loops,
and the motion of the system has a periodic character. Such a solution is an
exception which requires that all frequency ratios are rational numbers. In
other words, a closed trajectory may be expected only for special momenta
P and therefore only for special initial conditions of the system. All other
initial conditions should lead to conditionally periodic motion on trajectories
covering the corresponding torus completely.
However, this nice geometrical picture is too simple for a general discus-
sion of the motion of a system through the phase space. The Hamilton–Jacobi
equation is, in general, an extremely complicated mathematical object. Thus,
the actual structure of the complete integrals and therefore of the generat-
ing function is open. In particular, it is not clear if the conservation laws
P
i
=const. leads to isolated or nonisolated tori. If they are not isolated, the
trajectories are not necessary confined to one torus. Thus, we would expect
that the system moves through a domain of higher dimension than that of
the simple torus by successive “jumps” at certain contact points between tori
which belong together.