This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
advanced_notions:poisson_bracket [2018/04/25 09:52] jakobadmin |
advanced_notions:poisson_bracket [2018/05/05 14:04] jakobadmin ↷ Links adapted because of a move operation |
||
---|---|---|---|
Line 113: | Line 113: | ||
---- | ---- | ||
- | 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. |
Line 124: | Line 124: | ||
<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:newtonian_mechanics|classical mechanics]] that [[equations:canonical_commutation_relations|commutators]] do in [[theories:quantum_mechanics|quantum mechanics]]. | + | Poisson brackets play more or less the same role in [[theories:classical_mechanics:newtonian|classical mechanics]] that [[equations:canonical_commutation_relations|commutators]] do in [[theories:quantum_mechanics:canonical|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. | 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. |