Analytical Investigation of Human Finger Dynamics using Lie Symmetry Analysis

Abstract

This study explores the dynamics of human finger movement through the lens of a nonlinear differential equation. By employing Lie symmetry analysis, we derive the exact form of the torque parameter  and obtain an analytical solution to the governing equation. The results are visualized for various initial conditions, showcasing the utility of Lie symmetry methods in biomechanical modelling. This work not only advances the understanding of human finger dynamics but also provides a mathematical foundation for applications in robotics and physiology.

Introduction: The human finger is a complex biomechanical system whose dynamics can be modeled using nonlinear differential equations. Understanding these dynamics is crucial for applications in robotics, prosthetics, and rehabilitation medicine.

Objectives: This study aims to: (1) derive the exact form of the torque parameter  governing finger movement, (2) obtain an analytical solution to the nonlinear differential equation describing finger dynamics, and (3) validate the solution through graphical representation under various initial conditions.

Methods: We employ Lie symmetry analysis to systematically determine the form of  and solve the governing equation. This mathematical approach identifies invariant transformations that reduce the equation's complexity, enabling the derivation of an exact solution. The methodology includes symmetry generator identification, equation reduction, and analytical solution derivation.

Results: The analysis yields a quadratic form for the torque parameter,, and provides the exact solution  . Graphical representations demonstrate the solution's behavior for different initial conditions, showing excellent agreement with expected physical behaviour.

Conclusion: This work demonstrates that Lie symmetry analysis is a powerful tool for solving nonlinear biomechanical systems. The derived analytical solution provides a rigorous benchmark for numerical simulations and experimental studies of human finger dynamics, with potential applications in robotic hand design and rehabilitation engineering.