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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:**
advanced_tools/symplectic_structure.txt · Last modified: 2018/10/11 14:59 by jakobadmin