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Matterwave

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## Main Question or Discussion Point

So, Poisson's theorem states that if 2 variables, u and v, are constants of the motion, then one can find a third constant of the motion {u,v} where {u,v} is the Poisson bracket. This is a consequence of Jacobi's identity and the fact that:

[tex]\frac{du}{dt}=[u,H]+\frac{\partial u}{\partial t}[/tex]

Here [] are Poisson brackets (The {} symbol goes away in latex...I'm not sure how to put that in there).

And then we can find that {Lx,Ly}=Lz where L is the angular momentum. Therefore, if Lx and Ly are constants of the motion, so is Lz. This theorem seems quite general to me.

But this is very counter-intuitive for me, since it makes no sense for me why I can't just apply a torque in the z-direction. In fact, why can't I label whichever direction I apply the torque in, as the z-direction, and therefore only change Lz, but not Lx and Ly.

As a simple example, consider the pendulum. The pendulum oscillates in the x-y plane. Thus, the L vector is only in the z plane, and it oscillates from positive to negative as the pendulum bobs back and forth. In this example, only Lz is changing, Lx and Ly are identically 0 for all time.

So dLx/dt=0, dLy/dt=0, but dLz/dt=torque_z.

So it would seem contradictory with Poisson's Theorem.

I can't find the source of this contradiction, and it's bothering me. Please someone enlighten me.

[tex]\frac{du}{dt}=[u,H]+\frac{\partial u}{\partial t}[/tex]

Here [] are Poisson brackets (The {} symbol goes away in latex...I'm not sure how to put that in there).

And then we can find that {Lx,Ly}=Lz where L is the angular momentum. Therefore, if Lx and Ly are constants of the motion, so is Lz. This theorem seems quite general to me.

But this is very counter-intuitive for me, since it makes no sense for me why I can't just apply a torque in the z-direction. In fact, why can't I label whichever direction I apply the torque in, as the z-direction, and therefore only change Lz, but not Lx and Ly.

As a simple example, consider the pendulum. The pendulum oscillates in the x-y plane. Thus, the L vector is only in the z plane, and it oscillates from positive to negative as the pendulum bobs back and forth. In this example, only Lz is changing, Lx and Ly are identically 0 for all time.

So dLx/dt=0, dLy/dt=0, but dLz/dt=torque_z.

So it would seem contradictory with Poisson's Theorem.

I can't find the source of this contradiction, and it's bothering me. Please someone enlighten me.