Math 2.0

Mechanics of particles and systems Definition

Hamiltonian perturbation theory

Builds on Action-angle coordinatesCanonical transformationGenerating function of a canonical transformationFourier series on the torus

Definition

Let D⊆ℝnD \subseteq \R^n be open, let 𝕋n=ℝn/2πℤn\mathbb{T}^n = \R^n / 2\pi \Z^n, and let Hε(I,θ)=H0(I)+εH1(I,θ)H_\varepsilon(I, \theta) = H_0(I) + \varepsilon H_1(I, \theta) be a smooth Hamiltonian on D×𝕋nD \times \mathbb{T}^n in action-angle coordinates, with a real parameter ε\varepsilon. Let ω(I)=∇H0(I)\omega(I) = \nabla H_0(I) be the unperturbed frequencies, and let ⟨H1⟩(I)=(2π)−n∫𝕋nH1(I,θ) dθ\langle H_1 \rangle(I) = (2\pi)^{-n} \int_{\mathbb{T}^n} H_1(I, \theta) \, d\theta be the average over the angles. First-order Hamiltonian perturbation theory seeks a canonical transformation (I,θ)↦(J,φ)(I, \theta) \mapsto (J, \varphi) of a special form. Its type 2 generating function is F(J,θ)=J⋅θ+εS(J,θ),F(J, \theta) = J \cdot \theta + \varepsilon S(J, \theta), so that I=J+ε ∂S/∂θI = J + \varepsilon \, \partial S / \partial \theta and φ=θ+ε ∂S/∂J\varphi = \theta + \varepsilon \, \partial S / \partial J. The function SS solves the homological equation ω(J)⋅∂S∂θ(J,θ)+H1(J,θ)=⟨H1⟩(J).\omega(J) \cdot \frac{\partial S}{\partial \theta}(J, \theta) + H_1(J, \theta) = \langle H_1 \rangle(J). In the new variables the Hamiltonian is H0(J)+ε⟨H1⟩(J)+O(ε2)H_0(J) + \varepsilon \langle H_1 \rangle(J) + O(\varepsilon^2), which does not depend on the angles up to order ε2\varepsilon^2. Higher orders repeat the step with the remainder.

Remarks

The new Hamiltonian follows from a Taylor expansion: H0(J+ε∂θS)+εH1=H0(J)+ε(ω⋅∂θS+H1)+O(ε2)H_0(J + \varepsilon \partial_\theta S) + \varepsilon H_1 = H_0(J) + \varepsilon (\omega \cdot \partial_\theta S + H_1) + O(\varepsilon^2). With the Fourier series H1=∑k∈ℤnhk(J)eik⋅θH_1 = \sum_{k \in \Z^n} h_k(J) e^{i k \cdot \theta}, the solution is S(J,θ)=∑k≠0i hk(J)k⋅ω(J)eik⋅θ,S(J, \theta) = \sum_{k \ne 0} \frac{i \, h_k(J)}{k \cdot \omega(J)} e^{i k \cdot \theta}, which requires k⋅ω(J)≠0k \cdot \omega(J) \ne 0 for every kk with hk≠0h_k \ne 0; the numbers k⋅ωk \cdot \omega are the small divisors.

Example: n=1n = 1, Hε=I+εIcos⁡θH_\varepsilon = I + \varepsilon I \cos \theta, so ω=1\omega = 1 and ⟨H1⟩=0\langle H_1 \rangle = 0. The homological equation ∂θS=−Jcos⁡θ\partial_\theta S = -J \cos \theta has the solution S=−Jsin⁡θS = -J \sin \theta, and the new Hamiltonian is J+O(ε2)J + O(\varepsilon^2).

Non-example: for n=2n = 2, H0=I1−I2H_0 = I_1 - I_2 and H1=cos⁡(θ1+θ2)H_1 = \cos(\theta_1 + \theta_2), the frequency is ω=(1,−1)\omega = (1, -1) and k=(1,1)k = (1, 1) gives k⋅ω=0k \cdot \omega = 0. The homological equation then has no solution, since the left side has no ei(θ1+θ2)e^{i(\theta_1 + \theta_2)} term while H1−⟨H1⟩H_1 - \langle H_1 \rangle does; this is a resonance.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Action-angle coordinates, concept, no evidence levelAction-angle coordina…conceptCanonical coordinates, concept, no evidence levelCanonical coordinatesconceptCanonical transformation, concept, no evidence levelCanonical transformat…conceptEuclidean space, object, no evidence levelEuclidean spaceobjectFourier series on the torus, concept, no evidence levelFourier series on the…conceptFunction, concept, no evidence levelFunctionconceptHamiltonian system, concept, no evidence levelHamiltonian systemconceptPoisson bracket, concept, no evidence levelPoisson bracketconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptGenerating function of a canonical transformation, definition, no evidence levelGenerating function o…definitionDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptDiffeomorphism, concept, no evidence levelDiffeomorphismconceptDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionCanonical transformation, definition, no evidence levelCanonical transformat…definitionHamiltonian perturbation theory, definition, no evidence levelHamiltonian perturbat…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Action-angle coordinates concept
  2. Canonical coordinates concept
  3. Canonical transformation concept
  4. Euclidean space object
  5. Fourier series on the torus concept
  6. Function concept
  7. Hamiltonian system concept
  8. Poisson bracket concept
  9. Total derivative concept
  10. Function of several variables concept
  11. Generating function of a canonical transformation definition
  12. Differentiable function of several variables concept
  13. Partial derivative concept
  14. Limit of a function of several variables concept
  15. Diffeomorphism concept
  16. Differentiable function of several variables definition
  17. Jacobian matrix definition
  18. Canonical transformation definition
  19. Hamiltonian perturbation theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Perturbation_theory
About this page
TagARM5EN11YN
URIm2://ARM5EN11YN
Kinddefinition
WritingW0 stub
MSC70H09, 70H08
DefinesHamiltonian perturbation theory
UsesAction-angle coordinates, Canonical transformation, Generating function of a canonical transformation, Fourier series on the torus
Changed3 November 2026

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