HAMILTONIAN DYNAMICS, FALL 2015
PROBLEM SET 5
FEDERICA PASQUOTTO AND OLIVER FABERT
Exercise 1
1
Let E denote the Hilbert space of loops H 2 (S 1 ; R2n ) and consider the functional
Φ : E → R defined by
Z 1
1
1
Φ(x) = kx+ k2 − kx− k2 −
H(x).
2
2
0
i) Prove that ẋ = −∇Φ(x) is Lipschitz continuous and hence defines a a global
flow
φ : E × R → R, (t, x) 7→ φt (x).
ii) Suppose that the sequence {xj } ⊂ E satisfies ∇Φ(xj ) −−−→ 0. Prove that
j→∞
1
{xj } is bounded (with respect to the H 2 -norm).
iii) Prove that if every Palais-Smale sequence is bounded and ∇Φ is of the form
L + K, where L is boundedly invertible and K is a compact operator, then
Φ satisfies the Palais-Smale condition.
iv) Let Σ ⊂ E be defined by
Σ = {x = x− + x0 + se+ kx− + x0 k ≤ τ, 0 ≤ s ≤ τ },
where e+ (t) = e2πJt e1 , and S ⊂ E + by
S = {x ∈ E + |kxk = α }.
Show that φt (Σ) ∩ S 6= 0 if and only if x + B(t, x) = 0 and x ∈ Σ, where
B(t, x) = (e−t P − + P 0 )K(t, x) + P + {(kΦt (x)k − α)e+ − x}
and K is defined by the following decomposition for the flow:
Φt (x) = et x− + x0 + (P − + P 0 )K(t, x).
Exercise 2
Consider the Hilbert space
X
`2 = {x = (x1 , x2 , . . .) kxk2`2 =
x2k < +∞}.
k
The mapping


s
X
f (x1 , x2 , . . .) =  1 −
x2k , x1 , x2 , . . .
k
maps B ∞ = {kxk2`2 ≤ 1} to S ∞ = ∂B ∞ .
i) Prove that the mapping f does not have any fixed point in B ∞ .
ii) Prove that any two mappings S ∞ → S ∞ are homotopic. In particular, S ∞
is contractible.