Renormalization methods in KAM theory

It is well known that an integrable (in the sense of Arnold-Jost) Hamiltonian system gives rise to quasi-periodic motion with trajectories running on invariant tori. These tori foliate the whole phase space. If we perturb an integrable system, the Kolmogorow-Arnold-Moser (KAM) theorem states that, provided some non-degeneracy condition and that the perturbation is sufficiently small, most of the invariant tori carrying quasi-periodic motion persist, getting only slightly deformed. The measure of the persisting invariant tori is large together with the inverse of the size of the perturbation. In the first part of the thesis we shall use a Renormalization Group (RG) scheme in order to prove the classical KAM result in the case of a non analytic perturbation (the latter will only be assumed to have continuous derivatives up to a sufficiently large order). We shall proceed by solving a sequence of problems in which theperturbations are analytic approximations of the original one. We will finally show that the approximate solutions will converge to a differentiable solution of our original problem. In the second part we will use an RG scheme using continuous scales, so that instead of solving an iterative equation as in the classical RG KAM, we will end up solving a partial differential equation. This will allow us to reduce the complications of treating a sequence of iterative equations to the use of the Banach fixed point theorem in a suitable Banach space.

Homoclinic Splitting without Trees

We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, giving a simple construction of unstable KAM tori and their stable and unstable manifolds for analytic perturbations. When the coupling takes place through an even trigonometric polynomial in the angle variables, we extend analytically the solutions of the equations of motion, order by order in the perturbation parameter, to a large neighbourhood of the real line representing time. Subsequently, we devise an asymptotic expansion for the splitting (matrix) associated with a homoclinic point. This expansion consists of contributions that are manifestly exponentially small in the limit of vanishing gravity, by a shift-of-countour argument. Hence, we infer a similar upper bound for the splitting itself. In particular, the derivation of the result does not call for a tree expansion with explicit cancellation mechanisms.

Agoralla : avauksia ympäristöasiantuntijoiden vuorovaikutusprosesseista

Ympäristöasiantuntijoiden vuorovaikutusta on tutkittu agoralla (antiikin tori). Se on julkinen tila, jossa markkinat, politiikka, tiede ja yhteiskunta kohtaavat. Tutkimus kuuluu yhteiskuntatieteellisen ympäristötutkimuksen alaan, mutta siinä hyödynnetään myös tulevaisuudentutkimusta. Työn motivaationa on ollut tekijän monitieteinen koulutustausta: yhteiskuntatieteilijä ja luonnontieteilijä. Miten ja miksi vuorovaikutus eri asiantuntijoiden välillä on haasteellista ja merkityksellistä esimerkiksi metsän biodiversiteetin vähenemisen ehkäisemiseksi. Keskeisiä käsitteitä ovat asiantuntijuus, vuorovaikutus, tiedon luotettavuusja kontekstisidonnaisuus, Väitöskirja koostuu neljästä eri asiantuntijuustarinasta. Ensimmäinen (luku 2) perustuu haastatteluihin suomalaisten ja saksalaisten bio- ja yhteiskuntatietelijöiden käsityksistä luonnosta ja ympäristöstä. Tutkimusongelmana on luonnontieteilijöiden ja yhteiskuntatieteilijöiden Suomessa ja Saksassa ”kulttuurierot” luonnon ja ympäristön käsitteellistämisessä. Johtopäätöksenä on, että aistittu luonto, ympäröivä ympäristö sekä ihmisen muokkaama elinympäristö eivät tunne selkeitä tiede- eikä maanrajoja. Tämä luku toimii ponnahduslautana konstruktioiden taakse vuorovaikutuksen haasteisiin. Kirjan toinen tarina (luku 3) perustuu haastatteluihin suomalaisten metsän biodiversiteettiasiantuntijoiden vuorovaikutuksesta. Tutkimusongelman lähtökohtana on metsän biodiversiteetin väheneminen ja tästä seuraavat polittisetkin vuorovaikutustilanteet. Miten konteksti vaikuttaa eri asiantuntijoiden vuorovaikutukseen ja mitä tästä seuraa? Analyysin päätulos on implisiittisen, vahvasti kontekstisidonnaisen asiantuntijatiedon hyödyntämisen tarve ja voimavara metsän biodiversiteetin vähenemisen ennaltaehkäisemiksi. Kolmas tarina asiantuntijuudesta (luku 4) perustuu Etelä-Suomen metsien suojelutoimikunnassa (Metso) tehtyihin havainnointeihin. Tutkija on näin ollut itse eräänlaisella torilla havainnoijana. Tutkimusongelmana on ”ohipuhuminen”, tiedon luotettavuus ja implisiittien tiedon hyväksyttävyys. Johtopäätöksenä on asiantuntijuuden vahva kontekstisidonnaisuus hetkeen ja paikkaan ja yhteisen kielen (vrt. transdisiplinaarisuus) löytyminen yhteisen tavoitteen saavuttamiseksi. Merkittäviä välineitä vuorovaikutuksen onnistumiseen ovat esimerkiksi yhteinen vahva tavoitetila, interkatio, joka koskee läsnä olevia ihmisiä ei instituutioita sekä fasilitaattorin vahva rooli tulkkina ja välittäjänä. Neljäs tarina (luku 5) vie agoran konkretiaan. Tässä luvussa on kehitetty eläytymiskävely- menetelmää, jossa fasilitaattori (tutkija) johdattaa Espoon keskuksessa hallinnon, politiikan, asukkaiden ja konsultin edustajat aistimaan ja tulkitsemaan alueen sosiaalista tilaa, toiminnallisuutta ja elämyksellisyyttä. Ongelmana on aistimaailman asiantuntemuksen hyödyntämättömyys yhdyskuntasuunnittelun välineenä mm. asiantuntijoiden vuorovaikutuksen välineenä. Menetelmäkehitys on aluillaan, mutta jo tässä tapauksessa käy ilmi, että jaettu tila, jaetut aistikokemukset konkreettisella kävelyllä avaavaat vuorovaikutuksen uusiin ulottuvuuksiin, jossa implisiittiselle asiantuntemukselle annetaan sijansa vuorovaikutuksessa ja tätä kautta voidaan vaikuttaa myös tehtäviin päätöksiin, toimenpiteisiin. Johtopäätöksissä (luku 6) korostuu implisiittisen asiantuntijuuden merkitys. Onnistunut vuorovaikutteinen toiminta eri asiantuntijoiden kesken esimerkiksi erilaisia ympäristöongelmia –ja ilmiöitä ratkottaessa ja pohdittaessa vaatii vuorovaikutusosaamista. Tutkimuksen lopuksi suositellaan esimerkiksi ennakkoluulottomia avauksia agoralla. Asiantuntijuus ei ole yksi ja vain asiaatuntevuus on mahdollista. Agora on jatkuvassa liikkeessä ja juuri siinä piilee voimavara tulevaisuuden haasteisiin erilaisilla rajapinnoilla. Avainsanat: asiantuntijuus, vuorovaikutus, tieto, konteksti, agora

Helsingin Östersundomin pienvesien kartoitus

The area of Östersundom (29,1 square kilometers) was attached to Helsinki in the beginning of the year 2009. Östersundom is formed mostly from the municipality of Sipoo, and partly from the city of Vantaa. Nowadays Östersundom is still quite rural, but city planning has already started, and there are plans to develop Östersundom into a district with 45 000 inhabitants. In this study, the headwaters, streams and small lakes of Östersundom were studied to produce information as a basis for city planning. There are six main streams and five small lakes in Östersundom. The main methodology used in this study was the examination of the physical and the chemical quality of the water. The hygienic quality of the water was also studied. It was also examined whether the waters are in their natural state, or have they been treated and transformed by man. In addition, other factors affecting the waters were examined. Geographical information data was produced as a result of this work. Östersundom is the main area looked at in this study, some factors are examined in the scope of the catchment areas. Water samples were collected in three sampling periods: 31.8 4.9.2009, 3. 4.2.2010, and 10. 14.4.2010. There were 20 sampling points in Östersundom (5 in small lakes, 15 in streams). In the winter sampling period, only six samples were collected, from which one was taken from a small lake. Field measurements associated with water sampling included water temperature, oxygen concentration, pH and electoral conductivity. Water samples were analyzed in the Laboratories of Physical Geography in the University of Helsinki for the following properties: total suspended solids (TSS), total dissolved substances (TDS), organic matter, alkalinity, colour, principal anions and cations and trace elements. Metropolilab analyzed the amount of faecal coliform bacteria in the samples. The waters in Östersundom can be divided to three classes according to water quality and other characteristics: the upper course of the streams, the lower course of the streams and the small lakes. The streams in their upper course are in general acidic, and their acid neutralization capacity is low. The proportion of the organic matter is high. Also the concentrations of aluminium and iron tend to be high. The streams in the lower course have acidity closer to neutral, and the buffering capacity is good. The amounts of TSS and TDS are high, and as a result, the concentrations of many ions and trace elements are high as well. Bacteria were detected at times in the streams of the lower course. Four of the five small lakes in Östersundom are humic and acidic. TSS and TDS concentrations tend to be low, but the proportion of organic matter is often high. There were no bacteria in the small lakes. The fifth small lake (Landbonlampi) differs from the others by its water colour, which is very clear. This lake is very acidic, and its buffering capacity is extremely low. Compared to the headwaters in Finland in general, the concentrations of many ions and trace elements are higher in Östersundom. On the other hand, the characteristics of water were different according to the classification upper course streams, lower course streams, and small lakes. Generally, the best water quality was observed in the stream of Gumbölenpuro and in the lakes Storträsk, Genaträsk, Hältingträsk and Landbonlampi. Several valuable waters in their natural state were discovered from the area. The most representative example is the stream of Östersundominpuro in its lower course, where the stream flows through a broad-leaf forest area. The small lakes of Östersundom, and the biggest stream Krapuoja, with its meandering channel, are also valuable in their natural state.

Embracing Integrability in Stellar and Planetary Dynamics

Hamiltonian systems in stellar and planetary dynamics are typically near integrable. For example, Solar System planets are almost in two-body orbits, and in simulations of the Galaxy, the orbits of stars seem regular. For such systems, sophisticated numerical methods can be developed through integrable approximations. Following this theme, we discuss three distinct problems. We start by considering numerical integration techniques for planetary systems. Perturbation methods (that utilize the integrability of the two-body motion) are preferred over conventional "blind" integration schemes. We introduce perturbation methods formulated with Cartesian variables. In our numerical comparisons, these are superior to their conventional counterparts, but, by definition, lack the energy-preserving properties of symplectic integrators. However, they are exceptionally well suited for relatively short-term integrations in which moderately high positional accuracy is required. The next exercise falls into the category of stability questions in solar systems. Traditionally, the interest has been on the orbital stability of planets, which have been quantified, e.g., by Liapunov exponents. We offer a complementary aspect by considering the protective effect that massive gas giants, like Jupiter, can offer to Earth-like planets inside the habitable zone of a planetary system. Our method produces a single quantity, called the escape rate, which characterizes the system of giant planets. We obtain some interesting results by computing escape rates for the Solar System. Galaxy modelling is our third and final topic. Because of the sheer number of stars (about 10^11 in Milky Way) galaxies are often modelled as smooth potentials hosting distributions of stars. Unfortunately, only a handful of suitable potentials are integrable (harmonic oscillator, isochrone and Stäckel potential). This severely limits the possibilities of finding an integrable approximation for an observed galaxy. A solution to this problem is torus construction; a method for numerically creating a foliation of invariant phase-space tori corresponding to a given target Hamiltonian. Canonically, the invariant tori are constructed by deforming the tori of some existing integrable toy Hamiltonian. Our contribution is to demonstrate how this can be accomplished by using a Stäckel toy Hamiltonian in ellipsoidal coordinates.