My title - Departamento de Matemática da Universidade do Minho

6 FLOWS 39
6 Flows
“... forse stima che la filosofia sia un libro e una fantasia d’un uomo, come l’Iliade e
l’Orlando furioso, libri ne’ quali la meno importante cosa è che quello che vi è scritto
sia vero. Signor Sarsi, la cosa non istà così. La filosofia è scritta in questo grandissimo
libro che continuamente ci sta aperto innanzi agli occhi (io dico l’universo), ma non si
può intendere se prima non s’impara a intender la lingua, e conoscer i caratteri, ne’
quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi,
ed altre figure geometriche, senza i quali mezi è impossibile a intenderne umanamente
parola; senza questi è un aggirarsi vanamente per un oscuro laberinto.”
6.1 Structure of physical models
Galileo Galilei, Il saggiatore, 1623.
Flows of vector fields. The main way in which dynamical systems enter in physics is through
differential equations. Let X be a differentiable manifold, and let v be a vector field on X. If we
assume that the autonomous differential equation
ẋ = v(x)
with any given initial condition x (0) = x, has solutions t ↦→ x (t) which exist for any time t ∈ R
(as is the case when v is smooth and X is compact), then the flow of v is the action Φ : R×X → X
given by Φ t (x) = x (t).
From flows to maps. Given a flow Φ : R×X → X, one could specialize to discrete time looking
at the system at multiples integers nτ of a given time-unit τ > 0, and this amounts to iterate the
transformation f = Φ τ .
Also, given a submanifold U ⊂ X of codimension 1 which is transversal to the flow (i.e. the
tangent space T x U is transversal to Rv(x) for any x ∈ U), one could define a first return map
f : U → U sending a point x ∈ U into x(t) if t is the smallest positive time t > 0 such that
Φ t (x) ∈ U.
Newtonian mechanics. According to greeks, the “velocity” ˙q = d dtq of a planet, where
q ∈ R 3 is its position in our euclidean space and t is time, was determined by gods or whatever
forced planets to move around circles. Then came Galileo, and showed that gods could at most
determine the “acceleration” ¨q = d2
dt
q, since the laws of physics should be written in the same
2
way by an observer in any reference system at uniform rectilinear motion with respect to the fixed
stars. Finally came Newton, who decided that what gods determined was to be called “force”,
and discovered that the trajectories of planets, fulfilling Kepler’s experimental three laws 10 , were
solutions of his famous (second order differential) equation
m¨q = F
where m is the mass of the planet, and where the attractive force F between the planet and the
Sun is proportional to the product of their masses and inverse proportional to the square of their
distance.
Later, somebody noticed that most observed forces were “conservative”, could be written as
F = −∇V , for some real valued function V (q) called “potential energy”. There follows that
Newton equations can be written as m¨q = −∇V , and that the “total energy”
E = 1 2 m | ˙q|2 + V (q)
10 In Astronomia nova, 1609, and Harmonices mundi, 1619, Johannes Kepler published his three laws of planetary
motions:
i) planets moves in ellipses with focus at the Sun,
ii) the radius vector describes equal areas in equal times,
iii) the squares of the periods are to each other as the cubes of the mean distance from the Sun.
It was with the purpose to derive Kepler laws from a second order differential equation m¨q = F that Isaac Newton
realized that the force of gravitational attraction between the Sun and a planet (hence between any two bodies!)
should be proportional to m/ρ 2 (Philosophiae naturalis principia mathematica, 1687).