I was reading this very nice document, to understand why symplectic geometry is the natural setting for classical mechanics. I more or less understood why there is naturally a 2-form that arises. However, I didn't really understand the argument towards the end :

[...]All that remains is to explain why ω should be closed, i.e., why dω = 0. Unfortunately I don't see how to explain that without slightly more notation, but it won't be too bad. This requirement corresponds to a slightly more subtle issue, namely that the laws of physics should not depend on time. Let Ft denote the time-t flow along the vector field corresponding to some Hamiltonian H. One natural way to get the laws of physics at time t is to look at the pullback Ft*ω. We want it to equal ω. Clearly it does so at time 0, so to check that it always does we simply differentiate with respect to t, as follows.

What does he mean by : "One natural way to get the laws of physics at time t is to look at the pullback $F_t^*\omega$"? Why is that? And why do we want it to be equal to $\omega$?


In this setting $\omega$ is "the laws of physics" (more precisely it is "the laws of classical mechanics"). It is the object that turns energy $H$ into dynamics $V$ (which is what Newton's laws do in the more usual setting). Also, in this setting $F_t$ is the diffeomorphism that "moves the clock froward by $t$ seconds", so $F_{t}^{-1}$ "moves the clock backward by $t$ seconds", and $F^*_t$ moves any physical quantity represented by a covariant tensor field (like $dH$ or $\omega$) back to what it was $t$ seconds ago. Various objects (like some other function $K$ on $M$) may change over time, so their pullback will not be the same as the object itself, but saying that the "law" does not change precisely says that $F^*_t \omega=\omega$.

  • $\begingroup$ But what exactly is $F_t$? Is it the flow such that $F_t(p)$ is the position in the phase-space after time $t$ if we started initially (at $t=0$) in the state $p$? $\endgroup$ – roi_saumon Apr 14 at 10:09
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    $\begingroup$ Yes (assuming that energy function is the given $H$). $\endgroup$ – Max Apr 14 at 10:18
  • $\begingroup$ Is there many possible choice of $\omega$ for a given problem? In my opinion there should be only one, because there is only the law of physics, no? $\endgroup$ – roi_saumon Apr 24 at 17:21
  • $\begingroup$ Maybe a better way to put it is that $\omega$ is the law(s) of mechanics as applied to a particular system. For example, if a system has configuration space $Q$ and phase space $T^*Q$ then there is a canonical symplectic form on $T^*Q$ which encodes laws of mechanics for that system. $\endgroup$ – Max Apr 25 at 7:42
  • $\begingroup$ Just one more thing : So the idea is to map the 1-fom dH to a vector field V (that models the dynamics), so basically to have a map $D:T^*M \rightarrow TM$. But what does he mean by saying, we can consider the dual approach. I.e. exhibiting a map from $F : TM \rightarrow T^*M$? Is this the same? If $A$ is a vector space, giving the map from $A$ to $A^*$ is the same as giving a map from $A^*$ to $A$? $\endgroup$ – roi_saumon May 1 at 18:14

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