The Poincaré integral invariant of a closed curve 
in the phase space of a Hamiltonian
system is the integral of the canonical one-form 
,
 |
(1)
|
If 
is the phase flow generated by the Hamiltonian, then
 |
(2)
|
Thus the integral is unchanged when every point of 
is transported by the phase flow (Arnold 1989, pp. 237-239).
The corresponding symplectic form is
 |
(3)
|
Since the phase flow preserves 
, it also preserves its exterior
powers. For 
and a transported 
-dimensional region 
,
the absolute integral invariants are therefore
 |
(4)
|
For 
,
this is the sum of the signed areas of the projections
of 
onto the coordinate planes 
. For 
, it is the phase space volume
 |
(5)
|
whose invariance is Liouville's phase space theorem.
On extended phase space, the Poincaré-Cartan integral invariant includes the Hamiltonian,
 |
(6)
|
It is unchanged when 
is transported by the corresponding phase
flow.
