canonical one-form

Question

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?

Answer 1

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.

Answer 2

To make it even more transparent, I would do the following. Suppose $N$ is our manifold, $T^*N$ is the total space of the cotangent bundle, $\pi\colon T^*N\to N$ is the projection, and $(U,\phi)$ is a chart for $N$ with $\phi = \big(x^i\big)$. Then we get a chart $\big(\pi^{-1}(U),\tilde \phi\big)$ on $T^*N$ defined as follows. For each $\omega\in \pi^{-1}(U)$ there is a $p\in U$ so $\omega\in T^*_pN$, and $\omega$ can be written as $\omega = y_i(\omega)\,dx^i|_p$. The map $\tilde\phi\colon \pi^{-1}(U)\to \phi(U)\times\mathbb{R}^n$ is defined as $$ \omega\mapsto \big((x^1\circ\pi)(\omega),\dots,(x^n\circ\pi)(\omega),y_1(\omega),\dots,y_n(\omega)\big). $$ If we put $\widetilde x^{i} = x^i\circ\pi$, then we can write $\tilde \phi = \big(\widetilde x^{i},y_i\big)$. Moreover, for each $\omega \in \pi^{-1}(U)$, we get a basis for $T_\omega\big(T^*N\big)$ associated to the chart $\tilde\phi$: $$ {\partial\over \partial \widetilde x^{i}}\bigg|_\omega,{\partial\over \partial y_i}\bigg|_\omega, \quad i = 1,\dots,n. $$ We get a corresponding dual basis for $T_\omega^*\big(T^*N\big)$: $$ d\widetilde x^{i}\big|_\omega,d y_i\big|_\omega, \quad i = 1,\dots,n. $$ Hence if $(U,\phi) = (U,x^i)$ are local coordinates on $N$, for each $\omega\in \pi^{-1}(U)$, then the canonical $1$-form $\lambda_\omega\in T^*_\omega\big(T^*N\big)$, and $\lambda_\omega$ can be written in terms of the basis $d\widetilde x^i|_\omega,dy_i|_\omega$ as $$ \lambda_\omega = y_i(\omega)\,d\widetilde x^{i}\big|_\omega, $$ or as a local section $T^*U\to T^*\big(T^*U\big)$ as $$ \lambda = y_i\,d\widetilde x^{i}. $$