[Paper Review] Mathematical pendulum and its variants

Chiş Oana, Dumitru Opriş|ArXiv.org|May 27, 2009

Quantum chaos and dynamical systems9 references3 citations

TL;DR

This paper establishes a mathematical framework linking the Euler top system in R³ to the dynamics of a mathematical pendulum through conservation laws and level surface restrictions. It extends this connection to delayed, fractional, and stochastic variants of both systems, demonstrating that pendulum-like behavior emerges on invariant surfaces, with analytical solutions via elliptic functions and numerical validation using advanced integration schemes.

ABSTRACT

In this paper we show that there are applications that transform the movement of a pendulum into movements in $\mathbb{R}^3$. This can be done using Euler top system of differential equations. On the constant level surfaces, Euler top system reduces to the equation of a pendulum. Those properties are also considered in the case of system of differential equations with delay argument and in the fractional case. Another aspect presented here is stochastic Euler top system of differential equations and stochastic pendulum.

Motivation & Objective

• To establish a mathematical correspondence between the Euler top system in R³ and the mathematical pendulum using Hamilton-Poisson realizations and conservation laws.
• To extend this correspondence to systems with time delay, fractional derivatives, and stochastic perturbations.
• To derive analytical solutions for the pendulum and Euler top systems under various extensions, including elliptic and hyperbolic functions.
• To develop and validate numerical methods—such as Adams-Moulton and Milstein—for simulating fractional and stochastic systems.
• To analyze the stochastic dynamics of the Euler top and pendulum using Itô and Stratonovich calculus, including Fokker-Planck equations for probability density evolution.

Proposed method

• Utilizes the Euler top system of ODEs in R³: dx₁/dt = x₂x₃, dx₂/dt = -x₁x₃, dx₃/dt = x₁x₂, with polynomial right-hand sides.
• Applies Hamilton-Poisson realizations to identify three conservation laws (H₁, H₂, H₃) and three Casimir functions (C₁, C₂, C₃) defining invariant surfaces.
• Restricts the Euler top system to constant level surfaces of these conservation laws to recover pendulum dynamics, including the standard, delayed, fractional, and stochastic variants.
• Introduces time-delayed Euler top systems along OZ and OX axes, preserving conservation laws and yielding pendulum equations with delay.
• Applies Caputo fractional derivatives to the Euler top system, maintaining conservation laws and reducing to fractional pendulum equations on level surfaces.
• Models stochastic Euler top and pendulum systems using Itô and Stratonovich SDEs with multiplicative noise √xᵢ dWᵢ, and derives corresponding Fokker-Planck equations for probability density evolution.

Experimental results

Research questions

• RQ1 How does the restriction of the Euler top system to constant level surfaces of its conservation laws yield the mathematical pendulum equation?
• RQ2 What are the analytical solutions of the Euler top and pendulum systems in the cases of equal and unequal energy constants (H = K vs. H ≠ K)?
• RQ3 How do time delays in the Euler top system affect the emergence of pendulum-like dynamics and conservation laws?
• RQ4 Can fractional-order Euler top systems preserve conservation laws and reduce to fractional pendulum equations on invariant surfaces?
• RQ5 How do stochastic perturbations modeled via Itô and Stratonovich integrals affect the dynamics of the Euler top and pendulum, and what are the corresponding Fokker-Planck equations?

Key findings

• For H = K, the system admits exact analytical solutions in terms of hyperbolic functions: x₁(t) = ±H√2 sech(H√2 t), x₂(t) = ±H√2 tanh(H√2 t), x₃(t) = ±H√2 sech(H√2 t), representing heteroclinic orbits.
• For H ≠ K, solutions are expressed via Jacobi elliptic functions: x₂(t) = H√2 sn(H√2 t; √H/√K), x₁(t) = H√2 cn(H√2 t; √H/√K), x₃(t) = K√2 sn(H√2 t; √H/√K), with distinct periods.
• The Euler top system with delay argument retains conservation laws, and its restriction to level surfaces yields a mathematical pendulum with delay, preserving the same dynamical structure.
• Fractional Euler top systems using Caputo derivatives maintain conservation laws, and their restriction to level surfaces produces fractional pendulum equations.
• Stochastic Euler top and pendulum systems are formulated using Itô and Stratonovich SDEs with multiplicative noise √xᵢ dWᵢ, and their Fokker-Planck equations are derived for probability density evolution.
• Numerical simulations using the Milstein scheme (for SDEs) and Adams-Moulton method (for fractional systems) confirm the theoretical dynamics, with stable convergence observed in all cases.