The canonical one-form is defined here: http://books.google.nl/books?id=uycWAu1yY2gC&lpg=PA128&dq=canonical%20one%20form%20hamiltonian&pg=PA128#v=onepage&q&f=false
My problem is this: It states that if $(x_1,\dots x_n)$ are local coordinates in $N$, a 1-form $\alpha\in T^*_x N$ is represented by $\alpha = \Sigma^n_{j=1} y_jdx_j$ It then goes on to define a special 1-form $\lambda$ on $T^*N$ by $\lambda = \Sigma^n_{j=1} y_jdx_j$.
This to me looks the same as $\alpha$ and as a 1-form on $N$ and not on $T^*N$. What am I missing here?
This sort of thing can be a bit confusing!
A local one-form on $N$ is given by $\sum_j y_j(x) dx_j$; this gives coordinates $(y_1,\ldots,y_n)$ on the fibres of $T^*N$, so we have local coordinates $(x_1,\ldots,x_n,y_1,\ldots,y_n)$ on the total space of $T^*N$. Therefore a local one-form on $T^*N$ is $\sum_j\big(\alpha_j(x,y) dx_j + \beta_j(x,y) dy_j\big)$.
The canonical one-form on $T^*N$ is given by $\alpha_j = y_j,~ \beta_j = 0$. In words, it has no components 'pointing along the fibre', and its transverse components are given by the point we're sitting at in the fibre.