Generating Function

Twisted Product Form

For $(M_1,{\omega }_1)$ and $(M_2,{\omega }_2)$ as two 2n-dimensional symplectic manifolds. Consider ${\text{pr}}_1:M_1\times M_2\to M_1$ and ${\text{pr}}_2:M_1\times M_2\to M_2$. Then $\omega =({\text{pr}}_1{)}^{\ast }{\omega }_1+({\text{pr}}_2{)}^{\ast }{\omega }_2$ is closed and symplectic. More generally, if ${\lambda }_1,{\lambda }_2\in \mathbb{R}\setminus \{0\}$, then ${\lambda }_1({\text{pr}}_1{)}^{\ast }{\omega }_1+{\lambda }_2({\text{pr}}_2{)}^{\ast }{\omega }_2$ is also a symplectic form on $M_1\times M_2$. Take ${\lambda }_1=1$ and ${\lambda }_2=-1$ to obtain the twisted product form on $M_1\times M_2$: $\tilde{\omega }=({\text{pr}}_1{)}^{\ast }{\omega }_1-({\text{pr}}_2{)}^{\ast }{\omega }_2.$

Symplectomorphisms as Lagrangians

Method of Generating Function

For any $f\in C^{\infty }(X_1\times X_2)$, $df$ is a closed 1-form on $X_1\times X_2$. The lagrangian submanifold generated by $f$ is $Y_f:=\{((x,y),(df)(x,y))|(x,y)\in X_1\times X_2\}$ $d_{x}f:=(df{)}_{(x,y)}\text{ projected to }{T}_x^{\ast }{X}_1\times \{0\},$ $d_{y}f:=(df{)}_{(x,y)}\text{ projected to }\{0\}\times T_y^{\ast }{X}_2,$ which enables us to write $Y_f=\{(x,y,d_{x}f,d_{y}f)\mid (x,y)\in X_1\times X_2\}$ and $Y_f^{\sigma }=\{(x,y,d_{x}f,-d_{y}f)\mid (x,y)\in X_1\times X_2\}.$ When $Y_f^{\sigma }$ is in fact the graph of a diffeomorphism $\phi :M_1\to M_2$, we can call $\phi$ the symplectomorphism generated by $f$, and call $f$ the generating function of $\phi :M_1\to M_2$. To find the image $(y,\eta )=\phi (x,\xi )$, we must solve the “Hamilton” equations: ${\xi }_i=\frac{\partial f}{\partial x_i}(x,y)(\ast )$ ${\eta }_i=-\frac{\partial f}{\partial y_i}(x,y)(\ast \ast )$ By implicit function theorem, necessary local condition for $f$ to generate a symplectomorphism $\phi$ is: $\det {\left[\frac{\partial }{\partial y_j}\left(\frac{\partial f}{\partial x_i}\right)\right]}_{i,j=1}^n\ne 0.$

Remark

We think $f$ as the function telling us given end and beginning position, how particle would move, and thus $\phi :M_1\to M_2$ corresponds to the rule of how a system($M_1=T^{\ast }{X}_1$) “transfer” to another system($M_2=T^{\ast }{X}_2$), as $T^{\ast }{X}_1$ is just the phase space $(x,\xi )$ with $x$ position and $\xi$ momentum.

Recurrence

Proposition

There is a one-to-one correspondence between the fixed points of $\phi$ and the critical point of $\psi :X\to \mathbb{R}$, $\psi (x)=f(x,x)$.