Calculator
Formula used
For canonical coordinates \(q_i\) and momenta \(p_i\), the Poisson bracket of two phase-space functions \(F\) and \(G\) is:
This tool approximates each partial derivative using a central finite difference, then sums contributions across all degrees of freedom.
How to use this calculator
- Select the number of degrees of freedom to define qᵢ and pᵢ.
- Enter F(q,p) and G(q,p) using the supported syntax.
- Fill the phase-space point values for each variable.
- Adjust the step size; keep relative stepping on for scaling.
- Press Calculate to show {F, G} above the form.
- Use the export buttons to download CSV or PDF reports.
Example data table
| DOF | F(q,p) | G(q,p) | q1 | p1 | {F,G} |
|---|---|---|---|---|---|
| 1 | q1^2*p1 | sin(q1)+p1^2 | 2 | 0.5 | 3.6645873 |
Numbers shown are a reference; your step size can slightly change approximations.
Poisson bracket calculator guide
1) Poisson brackets in Hamiltonian mechanics
The Poisson bracket measures how two phase-space functions interact under canonical dynamics. For observables \(F(q,p)\) and \(G(q,p)\), the bracket is antisymmetric: \({F,G}=-{G,F}\). In Hamiltonian systems, time evolution is \(\dot{F}={F,H}+\partial F/\partial t\), so brackets connect directly to equations of motion and conservation laws.
2) Canonical variables and degrees of freedom
This calculator works with canonical pairs \((q_i,p_i)\) for \(i=1\ldots N\). Each degree of freedom adds one coordinate and one momentum, so the phase space has dimension \(2N\). Choose \(N\) to match your model (for example, \(N=1\) for a 1D oscillator, \(N=3\) for three translational coordinates). Units must be consistent across all inputs.
3) What the calculator computes
The tool evaluates \(F\) and \(G\) at your selected point and approximates all partial derivatives \(\partial F/\partial q_i\), \(\partial F/\partial p_i\), \(\partial G/\partial q_i\), and \(\partial G/\partial p_i\). It then sums \(N\) contributions \((\partial F/\partial q_i)(\partial G/\partial p_i)-(\partial F/\partial p_i)(\partial G/\partial q_i)\) to produce \({F,G}\).
4) Numerical differentiation and step selection
Derivatives are computed with a central difference, \(f'(x)\approx [f(x+\delta)-f(x-\delta)]/(2\delta)\). Central schemes typically reduce leading error to order \(\delta^2\) when the function is smooth. The step \(h\) controls \(\delta\). If “relative step” is enabled, \(\delta=h\,\max(1,|x|)\), which helps when variables span very different magnitudes.
5) Typical physics workflows
Common tasks include verifying canonical relations \({q_i,p_j}=\delta_{ij}\), checking whether a quantity \(C\) is conserved by testing \({C,H}=0\), and deriving equations of motion quickly. For a 1D harmonic oscillator \(H=p^2/(2m)+kq^2/2\), the expected results are \({q,H}=p/m\) and \({p,H}=-kq\). Use these as sanity checks for your expressions and step size.
6) Interpreting the term breakdown table
The breakdown table shows the two products per index \(i\) and their difference (the contribution). This is useful when \(N\) is large or when you suspect a single coordinate dominates the bracket. If one contribution is orders of magnitude larger than the others, reconsider your scaling, units, or whether the chosen point is near a singularity or steep gradient.
7) Accuracy checks and edge cases
For reliability, try multiple step sizes (for example, reduce \(h\) by a factor of 10) and see whether \({F,G}\) stabilizes. Too-large \(h\) introduces truncation error; too-small \(h\) can amplify round-off. Non-smooth expressions (absolute values, piecewise behavior) can produce unstable derivatives near kinks, so evaluate away from discontinuities when possible.
8) Exporting results for reports
Use the CSV export for spreadsheets and batch comparisons, and the PDF export for a compact shareable report. Exports include your input functions, variable values, and the computed bracket. For reproducibility, record the selected \(N\), the step \(h\), and whether relative stepping was enabled, since these numerical choices affect finite-difference estimates.
FAQs
1) What does a zero Poisson bracket mean?
Answer: If \({F,G}=0\) at a point, the functions locally commute under canonical flow there. If \({F,H}=0\) everywhere (and \(\partial F/\partial t=0\)), then \(F\) is conserved by the Hamiltonian dynamics.
2) Why do results change when I change the step size?
Answer: The calculator uses finite differences, so the derivative is an approximation. Large steps add truncation error; extremely small steps can trigger round-off error. Sweep \(h\) (for example, 1e-4 to 1e-7) and look for stable values.
3) Should I enable relative step?
Answer: Enable it when your variables vary greatly in magnitude (e.g., \(q\sim10^{-6}\) and \(p\sim10^3\)). It scales \(\delta\) with \(|x|\) to keep perturbations meaningful, improving numerical stability in many cases.
4) Which functions and operators are supported?
Answer: You can use +, −, *, /, ^, parentheses, and functions like sin, cos, tan, exp, ln, log, sqrt, abs, pow(a,b), and atan2(y,x). Variables are q1..qN and p1..pN.
5) Can I compute multiple brackets for different points?
Answer: Yes. Keep \(F\) and \(G\) fixed, change the phase-space values, and recalculate. Export each run to CSV to compare across points. For systematic scans, use consistent \(h\) and relative-step settings.
6) What if my expression returns NaN or Infinity?
Answer: That usually indicates invalid math at your point (division by zero, log of a nonpositive number, or sqrt of a negative). Adjust the phase-space point, simplify the expression, or check domain constraints before evaluating.
7) How do I verify my input is physically consistent?
Answer: Confirm units: \(q\) and \(p\) must match your Hamiltonian’s convention. Test canonical identities such as \({q_i,p_j}=\delta_{ij}\) and compare against known analytic brackets for simple systems before trusting complex models.