My title - Departamento de Matemática da Universidade do Minho

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is constant along trajectories. The function 1 2 m | ˙q|2 is called “kynetic energy” of the system.
An alternative (and indeed useful) formulation of Newtonian mechanics is the one developed
by Lagrange. He defined the “Lagrangian” of the system as
L (q, ˙q) = 1 2 m | ˙q|2 − V (q)
and observed that Newton equations are equivalent to the (Euler)-Lagrange equations
( )
d ∂L
= ∂L
dt ∂ ˙q ∂q
The product p = m ˙q = ∂L/∂ ˙q is called “(linear) momentum”, and, since p/m is the gradient
of the kinetic energy K (p) = |p| 2 /2m, Hamilton could write Newton’s second order differential
equations as the system of first order differential equations
˙q = ∂H
∂p
ṗ = − ∂H
∂q
where H (q, p) = K (p) + V (q) is the total energy as function of q and p, nowdays called “Hamiltonian”.
It is a simple check that the energy is a constant of the motion, since
d
dt H = ∂H ∂H · ˙q +
∂q ∂p · ṗ = ∂H
∂q · ∂H
∂p − ∂H
∂p · ∂H
∂q = 0
Hamiltonian flows. The modern abstract formulation of classical mechanics is as follows. Let
(X, ω) be a symplectic manifold, i.e. a differentiable manifold X of even dimension 2n, equipped
with a smooth closed differential two-form ω such that ω n ≠ 0. Darboux theorem says that locally
one can choose “canonical” coordinates (q 1 , ..., q n , p 1 , .., p n ) such that ω = ∑ n
k=1 dp k ∧ dq k . Let
H : X → R be a smooth function, called “Hamiltonian” and thought as the “energy” of the system.
Typically, it has the form “kinetic energy+potential energy”, where the kinetic energy is a positive
definite quadratic form in the momenta p, and the potential energy is a function V depending
on the positions q and possibly on the momenta p. The Hamiltonian vector field v is defined by
the identity dH = i v ω, and the Hamiltonian flow is the flow of v. In canonical coordinates, the
equations of motion read
q˙
k = ∂H p˙
k = − ∂H
∂p k
∂q k
It happens that the Hamiltonian flow Φ preserves the energy, namely H (Φ t (x)) = H (x) for any
x ∈ X and any time t ∈ R, as follows form the fact that L v H = 0.
Geodesic flows. The simplest mechanical system, the free motion of a particle, belongs to the
class of geodesic flows. Let (M, g) be a Riemannian manifold, g beeing the Riemannian metric.
Let SM be the unit tangent bundle of M. If M is geodesically complete, to every unit vector
v ∈ SM there corresponds a unique geodesic line (i.e. a local isometry) c : R → M such that
ċ (0) = v. The geodesic flow is the action Φ : R × SM → SM, defined as Φ t (v) = ċ (t).
Particularly interesting are geodesic flows over homogeneous spaces. Apart from the rather
trivial exemple of flat spaces, a source of interesting dynamical properties is the geodesic flow
on a manifold with constant negative curvature. The proptotype is as follows. The group G =
P SL (2, R) can be seen as the orientation preserving isometry group of the Poincaré half-plane H,
equipped with the hyperbolic metric of sectional curvature −1. Its action is transitive. Since the
stabilizer of a point in the half-plane is isomorphic to the group of rotations SO (2), we can identify
SD with G. Now, let Γ be a discrete cocompact subgroup of G with no torsion. The quotient
space Σ = D/Γ is a compact Riemann surface, which comes equipped with a Riemannian metric
of sectional curvature −1, and its unit tangent bundle is diffeomorphic to G/Γ. The geodesic flow
on SΣ is then the algebraic flow Φ : R × G/Γ → G/Γ defined as Φ t (gΓ) = e t gΓ, where
( )
e
t/2
0
e t =
0 e −t/2