[Paper Review] Pendulum Integration and Elliptic Functions

Pedro L. Garrido, Giovanni Gallavotti|arXiv (Cornell University)|Dec 12, 2008

Advanced Differential Equations and Dynamical Systems11 references3 citations

TL;DR

This paper provides a rigorous derivation of normal hyperbolic canonical coordinates for the classical pendulum near its unstable equilibrium using Jacobian elliptic functions, correcting prior errors and establishing a direct correspondence between hyperbolic and elliptic fixed points. The key contribution is a closed-form expression for the Hamiltonian in terms of elliptic integrals and modular parameters, enabling exact integration of pendulum motion via $ g_0(x') $, with explicit power series and product expansions for energy and action variables.

ABSTRACT

Revisiting canonical integration of the classical pendulum around its unstable equilibrium, normal hyperbolic canonical coordinates are constructed

Motivation & Objective

• To re-derive canonical normal forms for the pendulum near its unstable fixed point using the theory of elliptic functions.
• To correct inaccuracies in earlier treatments, particularly in [6, Appendix 9], regarding the construction of normal hyperbolic coordinates.
• To establish a precise correspondence between the hyperbolic dynamics near the unstable equilibrium and the elliptic dynamics near the stable equilibrium.
• To provide explicit power series and product expansions for the energy and action variables in terms of modular parameters.

Proposed method

• Uses Jacobian elliptic functions $ ext{sn}, ext{cn}, ext{dn} $ to parametrize the pendulum’s time evolution in terms of the argument $ u = gt/k $.
• Introduces the modular parameter $ x' = e^{-rac{ au}{ au'}} $, where $ au = rac{ ext{K}(h')}{ ext{K}(h)} $, to express the energy and action variables.
• Derives the Hamiltonian $ G(pq) = rac{d ilde{ ext{U}}}{d(pq)}(0) = g $, showing dependence on the product $ pq $, confirming normal hyperbolicity.
• Constructs the generating function $ S_s' $ and $ R_s' $ via infinite series involving $ p', q' $, with $ p' = \sqrt{x_s'} \cos(g_0^{(s)}t) $, $ q' = \sqrt{x_s'} \sin(g_0^{(s)}t) $.
• Applies the transformation $ (p,q) = (a_s(x_s')p', a_s(x_s')q') $ with $ a_s^2(z) = -16I \frac{d}{dz}g_0^{(s)}(z) $ to ensure canonical structure.
• Uses the product expansion $ g_0(x') = g \prod_{n=1}^\infty \left( \frac{1 + x'^n}{1 - x'^n} \right)^2 $ to express $ g_0 $ and its logarithmic derivative in terms of theta functions.

Experimental results

Research questions

• RQ1 How can the canonical normal form for the pendulum near the unstable equilibrium be rigorously derived using elliptic function theory?
• RQ2 What corrections are needed in the earlier construction of normal hyperbolic coordinates as presented in [6, Appendix 9]?
• RQ3 How is the energy $ U $ related to the modular parameter $ x' $, and what is the structure of its power series or product expansion?
• RQ4 What is the precise transformation between the hyperbolic (unstable) and elliptic (stable) cases via analytic continuation?
• RQ5 How can the action-angle variables be explicitly constructed from the elliptic function parametrization?

Key findings

• The energy $ U = 2g^2I / k^2 $ is expressed in terms of the modular parameter $ x' $, with $ g_0(x') = \frac{\pi}{2} \frac{g}{h' \text{K}(h)} $, and admits a product expansion $ g_0(x') = g \prod_{n=1}^\infty \left( \frac{1 + x'^n}{1 - x'^n} \right)^2 $.
• The energy $ U(x') $ is shown to satisfy $ \frac{d}{dx'}U(x') = g_0(x') \frac{d}{dx'}(x' a(x')^2) $, with $ a(x')^2 = 8I \frac{d}{dx'}g_0(x') $, linking energy variation to modular derivatives.
• For the stable case, the energy is given by $ U_s(x_s') = 32Ig_s^2 x_s' \prod_{n=1}^\infty \left( \frac{1 + x_s'^{2n}}{1 + x_s'^{2n-1}} \right)^8 $, with $ x_s' = -x' $, showing a dual structure to the hyperbolic case.
• The normal form Hamiltonian for the stable case is $ \mathcal{U}_s(x) = 32Ig_s^2 W\left( \frac{x}{64Ig_s} \right) $, where $ W(z) = z(1 - 2z - 4z^2 - 20z^3 - \cdots) $, matching known modular forms.
• The transformation from $ (B,\beta) $ to canonical $ (p,q) $ is constructed via $ p' = \sqrt{x_s'} \cos(g_0^{(s)}t) $, $ q' = \sqrt{x_s'} \sin(g_0^{(s)}t) $, followed by scaling with $ a_s(x_s') $ to ensure unit Jacobian.
• The logarithmic derivative of $ g_0(x') $ is $ 4 \sum_{n=1}^\infty \frac{n x'^{n-1}}{1 - x'^{2n}} $, which equals $ \frac{1}{2} \frac{d^2}{dz^2} \log \theta_4(z, x') \big|_{z=0} $, linking to theta functions.