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This suggests that we interpret the Poisson bracket as derivative of the first argument with respect to the second argument. ​ Line 73: Line 68: 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: - $$\text{Commutator}\quad [\hat{f},​\hat{g}] \quad\longleftrightarrow\quad\text{Poisson bracket}\quad \{f,g\}$$ + $$\text{Commutator}\quad [\hat{f},​\hat{g}] \quad\longleftrightarrow\quad\text{Poisson bracket}\quad \{f,g\}$$ ---- ---- - **The Poisson bracket ​describes general transformations/​symmetries** ​ + We can calculate what happens when we put the canonical coordinates and momenta themselves into the Poisson bracket: - In general, the infinitesimal evolution under the transformation generated by any element $g$ of the Lie algebra of observables on our [[basic_tools:​phase_space|phase space]], (parametrized by an abstract "​angle"​ $\varphi$) is given by + \begin{align} + \{ q_i , q_j\} &=0 \notag \\ + \{ p_i , p_j \} &= 0 \notag \\ + \{ q_i , p_j \} &= \delta_{ij}\notag + \end{align} - $$\partial_\phi F = \{G,F\}.$$ + which is extremely similar to the [[formulas:​canonical_commutation_relations|canonical commutation relations]] in quantum mechanics: - This has immediately important consequences because it means that if the Poisson bracket of $F$ and $G$ vanishes, they describe an infinitesimal transformation that is a symmetry for the other. So if the Poisson bracket of an observable with the Hamiltonian vanishes it means that this observable is invariant under time translations, i.e. a constant in time. Similarly, if the Poisson bracket of an observable and the generator of rotations (= angular momentum $L^i = \epsilon^{ijk}x^jp^k$) vanishes it is invariant under rotations. + \begin{align} + [ \hat{q}_i ​, \hat{q}_j] &=0 \notag \\ + [ \hat{p}_i ​, \hat{p}_j ] &= 0 \notag \\ + [ \hat{q}_i ​, \hat{p}_j ] &= i \hbar\notag + \end{align} - More precisely, consider a "​generator" ​$\delta G$ and some quantity ​$A$.  The infinitesimal transformation,​ generated by $\delta G$ is: + In words both sets of equations state that $p_i$ generates infinitesimal translations of $q_i$. - $$A \to A+\delta A,​\quad\quad \delta A = -\{\delta G, A\}.$$ + ---- + The Poisson bracket satisfied - -->​Momentum#​ + $$\left\{ F G, H \right\} = F \left\{ G, H \right\} + G \left\{ F, H \right\} .$$ - $\delta G = \epsilon_i p_i \quad \delta A = -\epsilon_i\{p_i, A\} = \epsilon_i\frac{\partial A}{\partial q_i},$ + - $\quad \Rightarrow \quad A(p_i,​q_i)\to A(p_i,q_i)+\epsilon_i\frac{\partial A}{\partial q_i} = A(p_i,​q_i+\epsilon_i)$ + - Therefore momentum is the generator of translations. + - <-- + - -->​Angular Momentum# + which looks like the Leibniz rule in calculus. This suggests that we interpret the Poisson bracket as derivative of the first argument with respect to the second argument. ​ - $$\delta G = \epsilon_i e_{ijk} p_jq_k \quad \delta A = -\epsilon_i e_{ijk} \{p_jq_k, A\}$$ + - Here $e_{ijk}$ is the Levi-Civita symbol. Expanding yeilds ​ - $\delta A = -\epsilon_i e_{ijk} \sum_\alpha \left(\frac{\partial p_jq_k}{\partial q_\alpha}\frac{\partial A}{\partial p_\alpha}-\frac{\partial p_jq_k}{\partial p_\alpha}\frac{\partial A}{\partial q_\alpha}\right) = -\epsilon_i e_{ijk}\left(p_j\frac{\partial A}{\partial p_k}+q_j\frac{\partial A}{\partial q_k}\right)$ - $\quad \Rightarrow \quad A(p_i,​q_i)\to A(p_i,​q_i)-\epsilon_i e_{ijk}\left(p_j\frac{\partial A}{\partial p_k} + q_j\frac{\partial A}{\partial q_k}\right) = A(R_{ij}p_j,​R_{ij}q_j)$  ​ - For infinitesimal rotations $R_{ij}$ - So we conclude that angular momentum is the generator of rotations. - <-- - - -->​Energy#​ - $\delta G = \epsilon H \quad \delta A = -\epsilon\{H,​ A\} = \epsilon\frac{d A}{d t},$ - - $\quad \Rightarrow \quad A(p_i(t),​q_i(t))\to A(p_i(t),​q_i(t))+\epsilon\frac{d A}{d t} = A(p_i(t+\epsilon),​q_i(t+\epsilon))$  ​ - - Therefore energy is the generator of time evolution. - <--  ​  ​ The Poisson bracket is a tool that allows us to take the derivative of some function with respect to some other function. This becomes possible because the arena of classical mechanics is the phase space which is a symplectic manifold. A symplectic manifold is a manifold that is endowed with a [[advanced_tools:​symplectic_structure|symplectic]] form. This symplectic form can be used to map any function on the manifold to a vector field on the manifold. ​ The Poisson bracket is a tool that allows us to take the derivative of some function with respect to some other function. This becomes possible because the arena of classical mechanics is the phase space which is a symplectic manifold. A symplectic manifold is a manifold that is endowed with a [[advanced_tools:​symplectic_structure|symplectic]] form. This symplectic form can be used to map any function on the manifold to a vector field on the manifold. ​ Line 128: Line 113: ---- ---- - Any system in [[theories:​classical_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 139: Line 124:  ​  ​ - Poisson brackets are necessary to describe the time evolution of observables in the [[frameworks:​hamiltonian_formalism|Hamiltonian formulation]] of [[theories:​classical_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:classical_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.