advanced_tools:symplectic_structure
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advanced_tools:symplectic_structure [2018/05/02 11:06] jakobadmin [Concrete] |
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| * [[http://math.mit.edu/~cohn/Thoughts/symplectic.html|Why symplectic geometry is the natural setting for classical mechanics]] by Henry Cohn | * [[http://math.mit.edu/~cohn/Thoughts/symplectic.html|Why symplectic geometry is the natural setting for classical mechanics]] by Henry Cohn | ||
| * Chapter "[[https://link.springer.com/chapter/10.1007%2F978-1-4612-0189-2_3#page-1|Phase Spaces of Mechanical Systems are Symplectic Manifolds" in the book Symmetry in Mechanics]] by Stephanie Frank Singer | * Chapter "[[https://link.springer.com/chapter/10.1007%2F978-1-4612-0189-2_3#page-1|Phase Spaces of Mechanical Systems are Symplectic Manifolds" in the book Symmetry in Mechanics]] by Stephanie Frank Singer | ||
| + | * See also Chapter 1 in Principles Of Newtonian And Quantum Mechanics by Gosson | ||
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| $$d^2 S = 0$$ | $$d^2 S = 0$$ | ||
| - | everywhere—and this gives Hamilton’s equations and the [[equations:maxwell_relations|Maxwell relations]]. | + | everywhere—and this gives Hamilton’s equations and the [[formulas:maxwell_relations|Maxwell relations]]. |
| <cite>https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/</cite> | <cite>https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/</cite> | ||
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| This is important. You might have some particles moving on a manifold like $S^3$, which is not symplectic. So the Hamiltonian mechanics point of view says that the abstract manifold that you are really interested in is something different: it must be a symplectic manifold. That's the phase space $X$. <cite>[[http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf|Lectures on Classical Mechanic]]s by J. Baez</cite></blockquote> | This is important. You might have some particles moving on a manifold like $S^3$, which is not symplectic. So the Hamiltonian mechanics point of view says that the abstract manifold that you are really interested in is something different: it must be a symplectic manifold. That's the phase space $X$. <cite>[[http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf|Lectures on Classical Mechanic]]s by J. Baez</cite></blockquote> | ||
| + | |||
| + | <blockquote>The mathematical structure underlying both classical | ||
| + | and quantum dynamical behaviour arises from symplectic geometry. It turns | ||
| + | out that, in the quantum case, the symplectic geometry is non-commutative, | ||
| + | while in the classical case, it is commutative.<cite>https://arxiv.org/pdf/1602.06071.pdf</cite></blockquote> | ||
| **Further Reading:** | **Further Reading:** | ||
advanced_tools/symplectic_structure.1525251982.txt.gz · Last modified: 2018/05/02 09:06 (external edit)
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