LECTURE 1. PROBLEMS, BASIC CONCEPTS AND OVERVIEW 39l. Given a closed hypersurface S in IR^2 n and Hamiltonians H and K having
S as a regular energy surface, the orbits on S of XH and XK coincide upto reparametrisation., i.e. if x = XH(x) and iJ = XK(Y) with x, y E S
and x (to) = y(t 1 ), then there exists a diffeomorphism {3: IR~ IR such that
{3(to) = t1 and x = yo {3. Hence XH has a periodic orbit on S iff X K has
one.- Call a compact hypersurface S in IR^2 n star-shaped if it bounds a domain,
which is star-shaped with respect to some point x 0. That means that there
is a diffeomorphism
(2) s;n-^1 (xo) := {x E IR^2 nl Ix - xol = c} ~ s: x ~ Xo + t(x)(x - xo)
for a suitable smooth map t : s;n-^1 (xo) ~ (0, oo). Given a Hamiltonian
H having a starshaped regular level set S there exists a periodic orbit onS. (Hint: Construct a new Hamiltonian XK having Sas a regular level set,
say S = K-^1 ( 1), which has the following additional property. There exists
co > 0 such that St: = K-^1 (1 + c) contains a periodic orbit for c E [-co, co]
iff So = S does. Then apply theorem 1.5.Before we continue let us state the general Weinstein conjecture.Conjecture 1.8. (Weinstein) Assume that H is an autonomous Ha miltonian on a
symplectic manifold (W,w). Let I;= H -^1 (E) be a closed regular level surface for
H. Suppose there exists a 1-form >..on I; such that d>.. = wlI; and >..(z)(XH(z)) -:/:- 0
for every z E I;. Then there exists a periodic orbit on I;.
Weinstein assumed in addition that HJR(L;) = 0 (deRham co homology). How-
ever, the results so far indicate that this restriction is presumably not necessary.
Exercise 1.9. Prove Viterbo's result by completing the following steps:
l. With the notation used in the Weinstein conjecture construct a 1-form T
defined on an open neighbourhood of I; which induces >.. and satisfies nearI; the identity dT = w.
2. Define a vector field T/ near I; by iryw = T. Show that 'f/ is transversal to I;.
- Using T/ construct a Hamiltonian near I; having I; as a regular energy sur-
face, such that the periodic orbits on nearby energy surfaces are in bijective
correspondence to those of I;. - Extend the previously constructed Hamiltonian to all of IR^2 n so that it sat-
isfies H(z ) ~ oo for lzl ~ oo.
- Apply theorem 1.5.
For the following we focus on the three-dimensional case. If n is a volume form
on S^3 and X is a nowhere vanishing vector field preserving the volume form we
find that
0 = Lxn = (dix + ixd)D = d(ixD).
Since H'JR(S^3 ) = 0 there exists a one-form >.. satisfying
ixD = d>...Observe that there is the freedom of adding a dh to >.., where his a smooth map
on S^3.