Poisson Bracket

Prerequisites: Hamiltonian Formalism. Full path (13 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ

Intuitive

The Poisson bracket is a mathematical tool that allows us to calculate the time evolution in classical mechanics.

In the Hamiltonian picture the state of a system is a point in phase space, the space of all positions and momenta. Every quantity we can compute from the state, such as energy, momentum or angular momentum, is a function on this space.

Each such quantity also tells us how to move through phase space. The energy moves the system forward in time. The momentum shifts it in space, and the angular momentum rotates it. The Poisson bracket of two quantities answers one question: how fast does the first quantity change when we move the system along the motion generated by the second? When the second quantity is the energy, this is the rate of change of the first quantity in time, which is why the Poisson bracket gives the equations of motion.

This one tool also connects symmetries and conservation laws. If the bracket of some quantity with the energy vanishes, and the quantity does not depend on time explicitly, it does not change in time, so it is conserved. The bracket only changes sign when the two quantities are swapped, so the same statement says that the energy does not change when we move the system along the motion generated by that quantity. So a quantity is conserved exactly when the motion it generates is a symmetry of the energy, which is the Hamiltonian version of Noether's theorem.

In quantum mechanics the commutator of two operators plays the same role, and the rule of replacing Poisson brackets by commutators (divided by Planck's constant times the imaginary unit) is the starting point of canonical quantization.

Concrete

For two sets of canonical coordinates $q_1,q_2, … q_N$ and momenta $p_1, p_2, … p_N$ the Poisson bracket is defined by

$$\left \{F,G \right \} = \sum_{n = 1}^{N} \left( \frac{\partial F}{\partial q_n} \frac{\partial G}{\partial p_n} - \frac{\partial F}{\partial p_n} \frac{\partial G}{\partial q_n} \right)$$ for any two functions $F$ and $G$ of the canonical coordinates and momenta.

The Poisson bracket of two observables can be thought of as the rate of change of the first along the flow given by the second. The most famous example is the time evolution. To get the time evolution of some observable $O$ all we have to do is calculate the Poisson bracket of $O$ with the Hamiltonian function.

The Poisson Bracket describe time evolution

In general, the time derivative of a function $F$ that is a function of generalized position and momentum coordinates, but not a direct function of time (which is often accurate) is

$$\frac{dF}{dt}=\frac{\partial F}{\partial q}\frac{dq}{dt}+\frac{\partial F}{\partial p}\frac{dp}{dt}$$

Hamilton's equations of motion are $$\frac{dq}{dt}=\frac{\partial H}{\partial p}$$ and $$\frac{dp}{dt}=-\frac{\partial H}{\partial q}.$$

Putting these equations into our general time derivative from above yields

$$\frac{dF}{dt}=\frac{\partial F}{\partial q}\frac{\partial H}{\partial p}-\frac{\partial F}{\partial p}\frac{\partial H}{\partial q}\equiv \{F,H\},$$

which shows that indeed Poisson brackets describe the time evolution of observables. If the function $F$ explicitly depends on the time we get instead $$\frac{\mathrm{d}F}{\mathrm{d}t} = \{F,H\} + \frac{\partial F}{\partial t}.$$ Take note how similar this equation is to the Heisenberg equation which describes the time evolution in quantum mechanics $$\frac{\mathrm{d}\hat F}{\mathrm{d}t} = -\frac{i}{\hbar}[\hat F,\hat H] + \frac{\partial \hat F}{\partial t}.$$ The only difference is that the Poisson brackets have been replaced with the commutator.

This is one way to make the difference between quantum and classical mechanics explicit:

$$\text{Commutator}\quad [\hat{f},\hat{g}] \quad\longleftrightarrow\quad\text{Poisson bracket}\quad \{f,g\}$$

We can calculate what happens when we put the canonical coordinates and momenta themselves into the Poisson bracket:

$$\begin{align} \{ q_i , q_j\} &=0 \\ \{ p_i , p_j \} &= 0 \\ \{ q_i , p_j \} &= \delta_{ij} \end{align}$$

which is extremely similar to the canonical commutation relations in quantum mechanics:

$$\begin{align} [ \hat{q}_i , \hat{q}_j] &=0 \\ [ \hat{p}_i , \hat{p}_j ] &= 0 \\ [ \hat{q}_i , \hat{p}_j ] &= i \hbar \delta_{ij} \end{align}$$

In words both sets of equations state that $p_i$ generates infinitesimal translations of $q_i$.

The Poisson bracket satisfied

$$\left\{ F G, H \right\} = F \left\{ G, H \right\} + G \left\{ F, H \right\} .$$

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.

• No-Nonsense Classical Mechanics by Jakob Schwichtenberg, section "Hamilton's General Equation"

Abstract

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 symplectic form. This symplectic form can be used to map any function on the manifold to a vector field on the manifold.

Each vector field can be used to define (integral) curves. Starting with one point on the manifold, we can use the vector field to get a curve.

Now, the Poisson bracket tells us how we can differentiate a function along one such curve. We do this by comparing the value of the function at one point to its value at another point on the curve which lies infinitesimally close.

In this sense $\{F,G\}$ is a short-hand notation for using the symplectic form to map $G$ into a vector field and then taking the derivative of $F$ along the curves that are described by this vector field.

Any system in Classical Mechanics can be thought of rigorously as a phase space which is more precisely formalized as a 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.

The Poisson bracket is like a ”Lie derivative” found in differential geometry. The Poisson bracket of two observables can be thought of as as the rate of change of the first along the flow given by the second. This is formally expressed as a Lie derivative:$$\{A, B\} = \mathcal{L}_{X_B} A .$$ In canonical coordinates this takes the form:$$\{A, B\} = \sum_{i=1}^N \frac{\partial A}{\partial q_i} \frac{\partial B}{\partial p_i} - \frac{\partial A}{\partial p_i} \frac{\partial B}{\partial q_i}$$

The Poisson bracket Lie algebra $\mathscr{P}(X,ω)$, which is the algebra of classical observables, is an extension of the Lie algebra of Hamiltonian vector fields $\mathscr{H}_{\mathcal{v}}(X)$ on our phase space $X$ by the "line Lie algebra $\mathbb{R}$" $$\mathbb{R} \longrightarrow \mathscr{P}(X,ω) \longrightarrow \mathscr{H}_{\mathcal{v}}(X).$$

Why is it interesting?

Poisson brackets are necessary to describe the time evolution of observables in the Hamiltonian formulation of classical mechanics. Formulated differently, the Poisson bracket controls the dynamics in classical mechanics.

Poisson brackets play more or less the same role in classical mechanics that commutators do in quantum mechanics.

Poisson brackets are also important in thermodynamics, see Classical Mechanics versus Thermodynamics (Part 2) | Azimuth and M. J. Peterson, Analogy between thermodynamics and mechanics, American Journal of Physics 47 (1979), 488–490.

FAQ

Path to this page

1. Calculus
2. Linear Algebra
3. Functional
4. Legendre Transformation
5. Newton's Second Law
6. Vector Calculus
7. Newtonian Mechanics
8. Variational Calculus
9. Configuration Space
10. Euler-Lagrange Equations
11. Lagrangian Formalism
12. Phase Space
13. Hamiltonian Formalism

Leads to: Canonical Commutation Relations, Phase Space Quantum Mechanics.