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advanced_notions:poisson_bracket [2018/04/12 14:15]
bogumilvidovic ↷ Links adapted because of a move operation
advanced_notions:poisson_bracket [2018/12/18 13:00] (current)
jakobadmin
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   ​   ​
 <tabbox Concrete> ​ <tabbox Concrete> ​
-that +
  
 For two sets of canonical coordinates For two sets of canonical coordinates
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 This is one way to make the difference between quantum and classical mechanics explicit: This is one way to make the difference between quantum and classical mechanics explicit:
  
-$$  \text{Commutator}\quad [\hat{f},​\hat{g}] \quad\longleftrightarrow\quad\text{Poisson bracket}\quad \{f,g\}$$+$$ \text{Commutator}\quad [\hat{f},​\hat{g}] \quad\longleftrightarrow\quad\text{Poisson bracket}\quad \{f,g\}$$
  
 ---- ----
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 \end{align} \end{align}
  
-which is extremely similar to the [[equations:​canonical_commutation_relations|canonical commutation relations]] in quantum mechanics:+which is extremely similar to the [[formulas:​canonical_commutation_relations|canonical commutation relations]] in quantum mechanics:
  
 \begin{align} \begin{align}
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 ---- ----
  
-Any system in [[theories:newtonian_mechanics|Classical Mechanics]] can be thought of rigorously as a [[basic_tools:​phase_space|phase space]] which is more precisely formalized as a [[advanced_tools:​symplectic_structure|symplectic]] manifold $(X,ω)$, or even more precisely a Poisson Manifold. In words, this means that the algebra of all functions on our phase space $X$, is canonically equipped with a Lie bracket: the Poisson bracket. Formulated differently,​ dynamics in mechanics are modeled on the cotangent bundle $T^∗M$ which has a canonical symplectic structure.+Any system in [[theories:classical_mechanics:​newtonian|Classical Mechanics]] can be thought of rigorously as a [[basic_tools:​phase_space|phase space]] which is more precisely formalized as a [[advanced_tools:​symplectic_structure|symplectic]] manifold $(X,ω)$, or even more precisely a Poisson Manifold. In words, this means that the algebra of all functions on our phase space $X$, is canonically equipped with a Lie bracket: the Poisson bracket. Formulated differently,​ dynamics in mechanics are modeled on the cotangent bundle $T^∗M$ which has a canonical symplectic structure.
  
  
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 <tabbox Why is it interesting?> ​ <tabbox Why is it interesting?> ​
-Poisson brackets are necessary to describe the time evolution of observables in the [[formalisms:​hamiltonian_formalism|Hamiltonian formulation]] of [[theories:newtonian_mechanics|classical mechanics]]. ​ Formulated differently,​ the Poisson bracket controls the dynamics in classical mechanics. ​+Poisson brackets are necessary to describe the time evolution of observables in the [[formalisms:​hamiltonian_formalism|Hamiltonian formulation]] of [[theories:classical_mechanics:​newtonian|classical mechanics]]. ​ Formulated differently,​ the Poisson bracket controls the dynamics in classical mechanics. ​ 
  
 +Poisson brackets play more or less the same role in [[theories:​classical_mechanics:​newtonian|classical mechanics]] that [[formulas:​canonical_commutation_relations|commutators]] do in [[theories:​quantum_mechanics:​canonical|quantum mechanics]]. ​
  
-Poisson brackets ​play more or less the same role in [[theories:newtonian_mechanics|classical mechanics]] that [[equations:​canonical_commutation_relations|commutators]] do in [[theories:​quantum_mechanics|quantum ​mechanics]]+Poisson brackets ​are also important ​in thermodynamics,​ see https://​johncarlosbaez.wordpress.com/​2012/​01/​23/​classical-mechanics-versus-thermodynamics-part-2/​ and M. J. Peterson, Analogy between thermodynamics and mechanics, American Journal of Physics 47 (1979), 488–490.
  
 <tabbox FAQ> <tabbox FAQ>
advanced_notions/poisson_bracket.1523542527.txt.gz · Last modified: 2018/04/12 14:15 by bogumilvidovic