Variational Integration of Hamiltonian Systems by Discontinuous Approximations

135 Pages·2016·2.92 MB·English

ETH Library
Variational Integration of
Hamiltonian Systems by
Discontinuous Approximations
Doctoral Thesis
Author(s):
Heimsch, Thomas Franz
Publication date:
2015
Permanent link:
https://doi.org/10.3929/ethz-a-010666504
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In Copyright - Non-Commercial Use Permitted
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Diss. ETH NO. 22952
Variational Integration of Hamiltonian
Systems by Discontinuous Approximations
A thesis submitted to attain the degree of
DOCTOR OF SCIENCES of ETH ZURICH
(Dr. sc. ETH Zurich)
presented by
Thomas Franz Heimsch
MSc ETH Masch.-Ing., ETH Zurich
born September 8, 1981
citizen of
St.Gallen (Switzerland)
accepted on the recommendation of
Prof. Dr.-Ing. Dr.-Ing. habil. Ch. Glocker, examiner
Prof. Dr. J. Dual, co-examiner
2015

Acknowledgements
The work presented in this thesis has been carried out at the Center of Mechanics in the
Department of Mechanical Engineering at the ETH Zurich.
First of all, I want to express my gratitude to my supervisor Prof. Dr.-Ing. Dr.-Ing habil.
Christoph Glocker for enabling this thesis and his unconditional support with amazing
patience. As a student of Mechanical Engineering, I am still impressed about how I
experienced the first encounter with him in the lecture Mechanics III at the beginning
of the third semester of my studies. The clear explanations and the neat lecture notes
motivated me to dive deeper into the subject, looking for further courses under his wing
to attend. As a passionate tutor myself, I have always been interested in teaching and
Professor Glocker has definitely set the standards. I always enjoyed our discussions on
the rooftop on various topics, thank you for being a passionate, patient and benevolent
supervisor.
Next, I want to thank Prof. Dr.-Ing. Ju¨rg Dual for reviewing my thesis. I remember me
and my twin brother summing up your lecture in the first two semesters in Mechanics in
the evenings, appreciating it for being self contained and interesting. Your lecture was a
big motivation for us at the beginning of our studies in Mechanical Engineering, thank
you very much also for that.
I express Georg, my office mate from Appenzell, full gratitude for the time spent to-
gether. I really enjoyed our talks and to share the office with you. You are a great help
and mental backup for me in so many aspects. There are also Adrian, Christian, Gabriel,
Marc, Michael, Remco, Rolf, Simon and Ueli who have always been there for a word and
more. Adrian, thanks for reviewing parts of my thesis.
Patrick, thanks for cycling various Alpine passes together, this was the most important
medicine I needed to stay on track and to release tension. This and also our common
passion for hiking and cooking has helped me through such a lot.
Another important person in touch with this thesis is Michael Mu¨hlebach who I su-
pervised during his Master’s thesis. Michael, thank you for your effort, your ideas, your
enthusiasm and your ingenuity. Your motivation has helped me back on my path to finish
this thesis. You, Professor Glocker and Simon were the reason to engage myself with
variational numerics.
My parents, my twin brother Fabian, and Prisca, together with Putzi & Colombo.
Needless to say what I feel for you. Without your emotional support, I would not have
managed to hand in, nothing more to say, you were a big motivation for me to keep on
trying.
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Finally, I want to mention Peter, my friend and former neighbour who unfortunately
passed away too early, and his loving parents, Arthur and Susanne Leuzinger. Peter, I
learned a lot from you, thanks for being my friend and the time spent together at “happy
place”.
Abstract
This thesis describes the variational integration of continuous Hamiltonian systems using
discontinuous ansatz functions, i.e. the ansatz functions of the generalised coordinates
and impulses are allowed to jump at the interval boundaries. The time domain of interest
is split up into smaller intervals and the ansatz functions are required to be continuous
functions of time, for example polynomials, within these intervals. It is shown that if
the ansatz functions contain the analytical solution of the system, then the constructed
integrator finds this analytical solution and the jumps vanish. If this is not the case,
these jumps are helpful for a more accurate approximation of the analytical solution
and one is still allowed to choose relatively large time intervals. This contradicts the
Discrete Mechanics community, where usually constant ansatz functions are chosen such
that also significantly smaller time intervals are needed. It is shown at the example of the
harmonic oscillator that the presented integrator coincides with the solution of Discrete
Mechanics if constant ansatz functions are chosen for both the generalised coordinates
and the generalised impulses. The main advantage of using ansatz functions in time as
approximations for the solution is that if a higher degree of accuracy is desired, one can
simply increase the degree of the used ansatz functions and does not have to construct a
completely new integrator.
In contrary to Lagrangian mechanics, where the generalised velocities are given as the
time derivative of the generalised coordinates, the Hamiltonian approach is pursued where
the generalised coordinates and the generalised impulses can be modelled independently.
The velocities are then obtained as a consequence of Fenchel’s equality that is used to
expresstheLagrangianfunctionintermsofthegeneralisedcoordinatesandthegeneralised
impulses.
The equations of motion of a given system are obtained by Hamilton’s principle. The
time integral of the Lagrangian function is made stationary and the Euler-Lagrange equa-
tions - or Hamilton’s equations - are extracted. But instead of the discretisation of these
equations of motion, it is favourable to discretise the principle itself and imposing station-
arity with respect to the parameters of the ansatz functions. It is shown that regardless of
the choice of the ansatz functions, the conserved quantities of the original system are also
conserved by the numerical solution. Such conserved quantities may be the total energy of
the system or in the presence of cyclic coordinates the associated generalised impulses. In
a more general formulation, these impulses correspond to the Noether symmetries. This
guarantees a meaningful approximation even for long simulation times.
In the second part of the thesis, variational integration of constrained Hamiltonian
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systems is addressed. The geometric constraints are assumed to be bilateral, scleronomic
and holonomic. The geometric constraint (on position level) induces a constraint on
velocity level by derivative with respect to time. This induced constraint is incorporated
intoHamilton’sprinciplebyusingaLagrangemultiplierwhereastheconstraintonposition
level is required to be satisfied at the beginning of the time interval. The constraint on
positionlevelisincorporatedbyitsvirtualworkusingthetimederivativeoftheconstraint
impulse which represents the second Lagrange multiplier.
At the end of the thesis, the oscillatory energy behaviour of the integrator, as well as the
conservation of the generalised impulses in the sense of Noether due to symmetries in the
Lagrangian are presented at specific examples. For the latter, the problem of planetary
motion has been chosen to show the conservation of angular momentum in both Cartesian
andpolarcoordinates. ForthevariationalintegrationofconstrainedHamiltoniansystems,
three examples are given. The first one being a point mass in a gravitational field that
is confined to an inclined plane. In this case, using polynomial ansatz functions, the
integrator finds the analytical solution. The others example consists of an oscillatory
system of two point masses connected by springs. A constraint on the two point masses
is subsequently introduced. At last, the pendulum consisting of a point mass confined to
a fixed length from the origin is presented.
Zusammenfassung
DievorliegendeDissertationbehandeltdienumerischeIntegrationvonzeitkontinuierlichen
Hamilton’schen Systemen mit Ansatzfunktionen, welche Spru¨nge aufweisen k¨onnen. Zeit-
kontinuierliche Systeme meint, dass in der Theorie sowohl die Lagen als auch die Impulse
keineSpru¨ngeaufweisen,klassischeSt¨ossealsoausgeschlossensind.Dennochbew¨ahrtsich
in der Numerik eben solche Spru¨nge im Orts- und Geschwindigkeitsverlauf zuzulassen und
fu¨r den kontinuierlichen Teil des Verlaufs der entsprechenden Funktionen zum Beispiel
Polynome in der Zeit als Ansatzfunktionen zu w¨ahlen. Ein langes Zeitintervall wird in
mehrere kurze Intervalle aufgestu¨ckelt, an deren Grenzen die Ansatzfunktionen springen
k¨onnen, sonst sind sie kontinuierlich. Kann durch die erw¨ahnten Ansatzfunktionen die
(analytische) L¨osung des mechanischen Systems abgebildet werden, so liefert das numeri-
sche Integrationsschema eben diese analytische L¨osung und dementsprechend treten auch
keine Spru¨nge an den Intervallgrenzen auf. K¨onnen hingegen die Ansatzfunktionen die
analytische L¨osung nicht abbilden, so wirken die erw¨ahnten Spru¨nge korrigierend auf die
L¨osung des Integrators. Die Abweichung der numerischen von der analytischen L¨osung
wird minimiert, aber man kann noch immer relativ grosse Zeitschritte fu¨r die kontinu-
ierlichen Ansatzfunktionen w¨ahlen. Dies im Gegensatz zur Diskreten Mechanik, wo die
Ansatzfunktionen in den Lagen als konstant angenommen und die Geschwindigkeiten
u¨ber finite Differenzen der Lagen approximiert werden. Es braucht dementsprechend klei-
nere Zeitschritte fu¨r eine genu¨gend genaue Approximation der L¨osung, welche dann auch
springen muss. Es wird fu¨r den harmonischen Oszillator gezeigt, dass der beschriebene
Integrator mit dem mittlerweile verbreiteten Diskreten Mechanik-Ansatz fu¨r die Wahl von
konstanten Ansatzfunktionen der Lagen und Impulse identisch ist. Der wohl gr¨osste Vor-
teil in der Verwendung von Ansatzfunktionen liegt darin, dass fu¨r einen Integrator von
h¨oherer Genauigkeit lediglich Ans¨atze von h¨oherem Grad gew¨ahlt werden k¨onnen und
nicht ein komplett neuer Integrator konstruiert werden muss.
Statt der Mechanik basierend auf Lagrange wird der Hamilton’sche Ansatz bevorzugt.
Dies hat den Vorteil, dass die Geschwindigkeiten nicht direkt von den Lagen im Sinne der
zeitlichenAbleitungabh¨angen,sondernvondenImpulsen,welcheunabh¨angigvondenLa-
gen als eigenst¨andige Ansatzfunktionen modelliert werden du¨rfen. Die Geschwindigkeiten
erh¨alt man dann aus den Impulsen als Konsequenz der Fenchel-Gleichung.
In der Theorie erh¨alt man die Bewegungsgleichungen eines mechanischen Systems, in-
dem man das Zeitintegral der Lagrangefunktion station¨ar macht, was auch als das Prinzip
von Hamilton bekannt ist. Anstatt die Bewegungsgleichungen zu diskretisieren, ist es bes-
ser, direkt das Prinzip von Hamilton zu diskretisieren und Stationarit¨at bezu¨glich der
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Parameter der Ansatzfunktionen zu fordern. Dies hat den Vorteil, dass unabh¨angig von
der Diskretisierung die Erhaltungsgr¨ossen des mechanischen Systems auch in der Nume-
rik zumindest nahezu erhalten werden. Solche Erhaltungsgr¨ossen k¨onnen Impulserhaltung
bei zyklischen Koordinaten, allgemeiner sogenannte Noethersymmetrien, sein oder auch
die Energie.
Im zweiten Teil wird die variationelle Integration von mechanischen Systemen mit idea-
len, zweiseitigen Bindungen aufgezeigt. Dabei wird die Bindung auf Lageebene zu Beginn
des Zeitintervalls gefordert, und deren zeitliche Ableitung, also die induzierte Bindung auf
Geschwindigkeitsebene, in das Prinzip von Hamilton mit einem Lagrange-Multiplikator
einbezogen. Die Bindung auf Lageebene wird u¨ber die virtuelle Arbeit der Bindungskraft,
welche als Zeitableitung eines Impulses zu verstehen ist, mit eingebaut.
Zum Schluss der Arbeit werden Beispiele vorgestellt, welche die gefundenen Eigenschaf-
ten des Integrators aufzeigen. Diese Eigenschaften sind n¨aherungsweise Energieerhaltung
oder die Erhaltung der verallgemeinerten Impulse falls die Lagrange-Funktion Symme-
trien hat. Als Beispiel fu¨r letzteres wurde das Kepler Problem der Planetenbewegung
gew¨ahlt, um die Drallerhaltung sowohl in kartesischen Koordinaten als auch Polarkoor-
dinaten aufzuzeigen. Fu¨r die variationelle Integration von gebundenen Hamilton’schen
Systemen werden drei Beispiele vorgestellt.
Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Discrete Mechanics and Approximation by Ansatz Functions . . . . 2
1.2.2 Variational Integration of Constrained Systems . . . . . . . . . . . 3
1.3 Aim and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Some Preliminaries of Convex Analysis 7
2.1 Convex Sets and Convex Functions . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Normal Cones and Proximal Points . . . . . . . . . . . . . . . . . . . . . . 10
2.3 The Convex Conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 The Indicator and Support Function of Convex Cones . . . . . . . . 16
2.4 Some Remarks on Duality and Orthogonal Cones . . . . . . . . . . . . . . 16
3 From the Virtual Work to the Hamiltonian Formalism 19
3.1 The Concept of Virtual Work and Hamilton’s principle . . . . . . . . . . . 19
3.2 Some Properties of Lagrangian Mechanics . . . . . . . . . . . . . . . . . . 25
3.2.1 Cyclic Coordinates and Noether’s Theorem . . . . . . . . . . . . . . 25
3.2.2 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.3 Invariance of the Euler-Lagrange equations . . . . . . . . . . . . . . 30
3.3 From Hamilton’s Principle to Hamiltonian Mechanics . . . . . . . . . . . . 31
3.3.1 The Legendre Transformation of the Lagrangian . . . . . . . . . . . 32
3.3.2 A Variational Principle Leading to Hamilton’s Equations . . . . . . 34
4 Why Variational Integrators? - An Example 37
4.1 Example: The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 37
4.2 A Glance at Discrete Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 The Symplectic Euler Scheme and an Energy-like Invariant for the Har-
monic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5 Variational Integrators - Galerkin in Time 53
5.1 The Principle of Virtual Work in the Hamiltonian Context . . . . . . . . . 54
5.2 The Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
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