🔢 mathematics

Planar Pendulums in Complex Coordinates

This paper derives a single, simple differential equation that describes the motion of any multi-arm planar pendulum by utilizing complex coordinates and elementwise matrix products.

Original authors: Gjerrit Meinsma

Published 2026-09-15

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Gjerrit Meinsma

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The study of how objects swing and sway is a cornerstone of classical physics, a field dedicated to understanding the forces that govern motion. At its heart lies the simple pendulum: a weight hanging from a fixed point, swinging back and forth under the pull of gravity. While the motion of a single arm is predictable and easy to describe, the behavior becomes vastly more complicated when multiple arms are linked together. Imagine a chain of weights, where the second hangs from the first, the third from the second, and so on. As the number of links increases, the system becomes a tangled web of interactions. The motion of one part instantly affects all the others, creating a complex dance of forces that is notoriously difficult to calculate. For centuries, scientists have relied on standard mathematical tools to untangle these systems, but these methods often produce pages of messy equations filled with trigonometric functions that are hard to read and even harder to use for simulation.

A researcher at the University of Twente in the Netherlands has found a way to cut through this complexity. By shifting the perspective from standard angles to a different mathematical language known as complex coordinates, the author has derived a single, elegant equation that describes the motion of any planar pendulum, regardless of how many arms it has or how they are connected. This new approach does not just simplify the math; it reveals a universal structure underlying all such swinging systems. The result is a formula that is compact enough to fit on a single line of computer code, yet powerful enough to simulate everything from a simple double pendulum to a chaotic tree of dozens of swinging arms. This discovery transforms a problem that usually requires a cluttered, case-by-case analysis into a streamlined process that can be applied to any configuration instantly.

To understand why this is such a significant shift, one must first appreciate the difficulty of the traditional approach. When physicists model a multi-arm pendulum, they typically track the angle of each arm relative to a vertical line. As the arms swing, the equations governing their motion involve sines and cosines of these angles. When multiple arms are linked, these trigonometric terms multiply and combine in ways that create an explosion of algebraic terms. Every time a new arm is added to the chain, the number of terms grows rapidly, making the equations "cluttered" and prone to error. The standard method requires deriving a unique set of equations for every specific arrangement of arms, a process that is tedious and obscures the underlying physics. The author of this paper recognized that these trigonometric identities are much easier to handle when viewed through the lens of complex numbers, a mathematical system that treats points on a plane as single entities rather than separate horizontal and vertical coordinates.

The core of the new method lies in how the researcher represents the physical system. Instead of tracking angles directly, the paper describes the position of each arm's tip as a point in a complex plane. In this view, the entire structure of the pendulum is captured by a simple grid of numbers called a connectivity matrix. This matrix acts like a blueprint, recording exactly which arm is attached to which other arm. If an arm hangs from the pivot, the matrix records a connection; if a second arm hangs from the first, the matrix records that link as well. This matrix, combined with the lengths of the arms and the masses at their tips, allows the researcher to write down the position of every single tip in the system using a single, unified expression. It is a way of describing the whole tree of swinging arms as a single, coherent object rather than a collection of separate parts.

Using this framework, the author derived a single differential equation that governs the motion of the entire system. A differential equation is a mathematical rule that describes how a system changes over time. In this case, the rule calculates the acceleration of each arm based on the current positions, speeds, and the forces acting upon them. The beauty of this new equation is that it is identical for every possible pendulum configuration. Whether the system has two arms or twenty, whether they are arranged in a straight line or a branching tree, the same formula applies. The only things that change are the specific numbers plugged into the equation: the lengths, the masses, and the connectivity matrix that defines the shape of the pendulum. This universality means that a computer program written to solve this single equation can simulate any pendulum imaginable, simply by changing the input data.

The paper demonstrates this power by providing a short computer program that can simulate the motion of these systems. The code is remarkably concise, requiring only a few lines to define the lengths, masses, and connections of the pendulum. Once these values are set, the program solves the equation and outputs the motion of every arm over time. The author notes that while the resulting motion is often chaotic and unpredictable for most multi-arm pendulums, the equation itself is stable and easy to compute. This allows researchers to explore the behavior of complex systems without getting bogged down in the algebraic mess that usually accompanies such problems. The method effectively turns a problem that was once a barrier to exploration into a straightforward calculation.

Beyond simulation, the paper also uses this new equation to analyze the fundamental vibrations of these systems. When a pendulum hangs perfectly still, it can be nudged to swing in specific patterns called eigenmodes. These are the natural rhythms at which the system prefers to oscillate. The author shows that the new equation makes it possible to calculate these rhythms for any pendulum tree. By linearizing the equation—simplifying it for small movements—the researcher derives a set of values that describe how fast the system vibrates. A particularly striking finding is a general rule about the sum of the squares of these vibration frequencies. The paper proves that for any arrangement of arms with equal lengths and masses, the sum of the reciprocals of the squared frequencies is directly related to the total length of all the arms divided by gravity. This result generalizes a known fact about simple pendulums to the most complex branching structures, revealing a deep and previously hidden symmetry in how these systems behave.

The derivation of this result relies on a careful balance of forces. The author starts by considering the forces acting on each joint where the arms connect. These forces include gravity pulling down and the internal tension or compression within the arms. By expressing these forces in the complex plane, the author shows that the equations governing the motion can be separated into real and imaginary parts. The real part corresponds to the physical acceleration of the arms, while the imaginary part relates to the internal forces holding the structure together. This separation allows the author to isolate the equation of motion from the internal forces, resulting in the clean, single formula that drives the simulation. The proof is rigorous, using a method of induction to show that the relationship holds true no matter how many arms are added to the system.

The paper also addresses the practical application of these findings. While the primary focus is on the mathematical elegance and the ability to simulate motion, the author suggests that this approach could be useful for generalizing results about eigenfrequencies. The ability to quickly calculate the natural rhythms of a complex system could have implications for engineering and design, where understanding the stability of multi-part structures is crucial. However, the paper remains focused on the theoretical and computational aspects, presenting the equation as a tool for understanding the fundamental physics of swinging bodies. The work does not claim to solve the problem of chaos itself, but rather provides a clearer, more efficient way to study it.

In the end, this research offers a fresh perspective on an old problem. By abandoning the traditional angle-based approach in favor of complex coordinates and matrix algebra, the author has stripped away the unnecessary complexity that has long obscured the behavior of multi-arm pendulums. The result is a unified theory that treats every swinging system as a variation of a single, simple rule. It is a reminder that sometimes, the key to understanding a tangled problem is not to pull harder at the knots, but to change the way one looks at the whole picture. The paper stands as a testament to the power of mathematical abstraction to reveal the hidden simplicity in the physical world.

Technical Summary: Planar Pendulums in Complex Coordinates

Problem Statement
The derivation of equations of motion for multi-arm planar pendulums using standard Euler-Lagrange formalism typically results in cluttered expressions involving numerous trigonometric identities. These expressions become increasingly complex and difficult to manage as the number of arms and the complexity of their connectivity increase. The paper addresses the need for a unified, simplified mathematical framework capable of describing the dynamics of any planar pendulum tree—regardless of the number of arms or their specific connection topology—without requiring case-by-case derivation of trigonometric relations.

Methodology
The author proposes a formulation based on complex coordinates and elementwise matrix products (Hadamard products). The methodology proceeds as follows:

Complex Representation: The planar space $\mathbb{R}^2$ is identified with the complex plane $\mathbb{C}$ . The position of each arm tip $z_k$ is represented as a complex number $z_k = \ell_k e^{i\phi_k}$ , where $\ell_k$ is the arm length and $\phi_k$ is the angle relative to the positive real axis.
Connectivity Matrix ( $S$ ): A logical matrix $S \in \{0,1\}^{n \times n}$ is introduced to encode the tree structure of the pendulum. The position vector of all tips, $z$ , is expressed compactly as $z = S[\ell \odot e^{i\phi}]$ , where $\odot$ denotes the elementwise product.
Force Balance in $\mathbb{C}$ : Instead of using Lagrangian mechanics, the derivation utilizes Newton's second law applied to force balances in the complex plane. The total force on each joint includes gravitational forces and compression forces along the arms.
Duality Lemma: A key theoretical step involves proving that for any pendulum tree, the relationship between the position vector and the force vector is governed by a duality between the connectivity matrix $S$ and its inverse transpose $B = S^{-T}$ . This allows the system of force equations to be decoupled from the specific geometry of the tree.
Derivation of the Complex DE: By substituting the kinematic relations into the force balance equations and isolating the real and imaginary parts of the resulting complex equation, the author derives a single differential equation for the angular accelerations $\ddot{\phi}$ .

Key Contributions
The primary contribution is the derivation of a single, comprehensive differential equation that governs the motion of any planar pendulum tree:

$\ddot{\phi} = M_{re}^{-1}(\phi) \left[ M_{im}(\phi) \dot{\phi}^2 - g \cos(\phi) \odot \ell \odot [S^T m] \right]$

Where:

$M(\phi) = \text{diag}(e^{-i\phi} \odot \ell) S^T \text{diag}(m) S \text{diag}(\ell \odot e^{i\phi})$ is a complex mass matrix.
$M_{re}$ and $M_{im}$ are the real and imaginary parts of $M(\phi)$ .
$m$ , $\ell$ , and $\phi$ are vectors of masses, lengths, and angles, respectively.
$g$ is gravitational acceleration.

This formulation eliminates the need for explicit trigonometric expansion for each specific configuration. The paper provides a MATLAB implementation (pendusim) demonstrating that simulating any pendulum requires only defining the vectors $m, \ell$ and the matrix $S$ .

Results

Generalization: The derived equation applies to arbitrary tree structures, including standard serial chains (e.g., triple pendulum) and non-standard branching configurations.
Linearization: The paper analyzes the linearization of the system around the hanging position ( $\theta = 0$ ). It establishes that the linearized system $\ddot{\theta} = -A\theta$ possesses $n$ positive real eigenvalues and linearly independent eigenvectors, ensuring a complete set of eigenmodes.
Eigenfrequency Trace Identity: A significant analytical result is derived regarding the eigenfrequencies $\omega_k$ of the linearized system. The paper proves that the sum of the reciprocals of the squared eigenfrequencies is equal to the total length of the pendulum divided by gravity:
$\sum_{k=1}^n \frac{1}{\omega_k^2} = \frac{\ell_{tot}}{g}$
This generalizes a previous result from reference [1] (regarding standard multi-arm pendulums) to arbitrary pendulum trees.
Explicit Modes: The paper explicitly calculates and visualizes the eigenmodes for a standard double pendulum and a non-standard triple pendulum, showing how the complex formulation simplifies the identification of these modes.

Significance and Claims
The paper claims that its approach offers a "single and simple" alternative to the "cluttered expressions" of traditional methods. Its significance lies in:

Simplicity: Reducing the modeling of complex multi-body systems to the definition of a connectivity matrix and standard vector operations.
Simulation Efficiency: The formulation is described as "easy to simulate," with the provided code capable of handling any topology without modification to the core solver logic.
Analytical Insight: The complex coordinate approach facilitates the generalization of spectral properties (eigenfrequencies) that are difficult to derive using standard real-coordinate methods.

The author maintains a modest tone, noting that while the equation simplifies the derivation and simulation, the underlying dynamics of most multi-arm pendulums remain chaotic. The work is presented as a note providing a unified mathematical tool rather than a proposal for new experimental applications.

More like this