12 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGY
The cotangent bundle
This is another basic example of a symplectic manifold. The cotangent bundle T* X
carries a canonical 1-form >-can defined by
(Acan)(x,v•)(w) = v(K(w)), for WE T(x,v•)(T* X),
where 7T' : T X ---+ X is the projection. (Here x is a point in X and v E r; X.)
Then Dean = -d>-can is a symplectic form. Clearly the fibers of 11' : T * X ---+ X are
Lagrangian with respect to Dean, as is the zero section. Moreover, it is not hard to
see that:
Lemma 1.10. Let O"o: : X ---+ T* X be the section determined by the 1-form a on
X. Then O";(>-can) =a. Hence the manifold O"o:(X) is Lagrangian iff a is closed.
Exercise 1.11. (i) Take a function H on X and let fI = Ho 11'. Describe the
resulting flow on T * X.
(ii) Every diffeomorphism ¢ of X lifts to a diffeomorphism ¢ of T* X by
J(x, v) = (¢(x), (¢-^1 )v*).
Show that ¢*(>-can)= Acan·
(iii) Let <Pt be the flow on X generated by a vector field Y. If ¢t is the lift of this
flow to T X show that the Hamiltonian H : T X ---+ R that generates this flow has
the form