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advanced_tools:symplectic_structure [2018/05/02 11:06] jakobadmin [Concrete] |
advanced_tools:symplectic_structure [2018/05/13 09:17] jakobadmin ↷ Links adapted because of a move operation |
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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:** |