advanced_tools:symplectic_structure
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advanced_tools:symplectic_structure [2018/04/15 11:40] ida |
advanced_tools:symplectic_structure [2018/10/11 14:59] (current) jakobadmin [Concrete] |
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| + | <blockquote>A simple way of putting it is that a two-form a way of measuring area in multivariable calculus. I believe the significance for physics boils down to the following: it turns out that a two-form is precisely what is required to translate an energy functional on phase space (a Hamiltonian) into a flow (a vector field). [See Wikipedia for how the translation goes, or read Arnold's book Mathematical Methods of Classical Mechanics, or a similar reference.] The flow describes time evolution of the system; the equations which define it are Hamilton's equations. One property these flows have is that they preserve the symplectic form; this is just a formal consequence of the recipe for going from Hamiltonian to flow using the form. So, having contemplated momentum, here we find ourselves able to describe how systems evolve using the phase space T*M, where not only is there an extremely natural extra structure (the canonical symplectic form), but also that structure happens to b preserved by the physical evolution of the system. That's pretty nice! Even better, this is a good way of expressing conservation laws. When physical evolution preserves something, that's a conservation law. So in some sense, "conservation of symplectic form" is the second most basic conservation law. (The most basic is conservation of energy, which is essentially the definition of the Hamiltonian flow.) You can use conservation of symplectic form to prove the existence of other conserved quantities when your system is invariant under symmetries (this is Noether's theorem, which can also be proved in other ways, I think, but they probably boil down to the same argument ultimately). <cite>http://qr.ae/TUTIn9</cite></blockquote> | ||
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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> | ||
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| + | <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.1523785246.txt.gz · Last modified: 2018/04/15 09:40 (external edit)
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