Eva Miranda
speaker
1,609 appearances
1 recordings
1 series
first heard Jan 2025
last heard Jan 2025
Eva Miranda’s voice in public audio — every appearance, attributed to the second.
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Appearances
You visualize these Li algebras as the space of velocities.
You think, oh, but velocities are matrices?
Yes, never mind.
You can put these matrices as a long vector if you want.
Okay, these are velocities of something happening on a group, which is SO3, which is the which is the group of of rotations in three dimensions.
So I was asking you a question, which is uh a tricky question, is can I have uh can this Poisson uh bracket correspond to a symplectic manifold?
And the answer is no.
Because if you look at the dimension of SO three, it cannot have a symplectic structure.
Because a symplectic manifold, and this is something I didn't say, because it looks like the cotangent bundle of something, it has to have dimension even.
So you cannot have dimension three.
It should have either dimension zero, two, four, six, eight, whatever.
Always even dimension.
So
However, there is a close connection between Poisson and contact, which is now in this this is the picture, okay, of the dual, the dual of SO3.
I can think of it as R three.
An R3, we can fill it up with uh with spheres of different radius, and these spheres of different radius have a symplectic form naturally associated to it to them, which is called the Kostan uh Suryo Kostan Kirilov.
uh symplectic structure.
And this indeed can this can be understood um physically, okay, because uh this Poisson bracket has what we call a constant of motion.
Which is the sum of x1 squared plus x2 squared plus x3 squared, this means that this is a function that is preserved.
And this is what is called a casimile of the Poisson structure.
Showing 641–660 of 1,609 · page 33 of 81
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