MATHEMATICAL MODEL OF A LAGRANGIAN FRAMEWORK FOR RHYTHMIC THROWING: STABILITY AND SYNCHRONIZATION IN N-BALL CASCADE DYNAMICS
DOI:
https://doi.org/10.60787/jnamp.vol72no.659Keywords:
Juggling dynamics, Projectile motion, Timing synchronization Stability analysis, Lagrangian formalismAbstract
This study develops a unified analytical framework for rhythmic throwing and juggling involving an arbitrary number nnn of balls. Each throw is modeled as planar projectile motion under gravity, governed by fixed launch parameters and a constant rhythmic period that imposes a synchronization condition linking flight time to temporal spacing. Using a Lagrangian formulation, the equations of motion are derived systematically, and a shifted time variable enables a compact representation of multiple, phase-shifted trajectories within a single dynamical structure. The model reveals inherent symmetry and periodicity in cascade patterns and provides a scalable description applicable to nnn-ball systems. Stability analysis shows that the dynamics are robust to variations in launch speed, angle, and timing, with tolerance bounds consistent with human neuromotor variability. The inclusion of aerodynamic drag introduces small, predictable corrections that can be compensated through minor parameter adjustments. The framework integrates mechanics, timing, and stability, offering a physically grounded basis for analyzing coordinated rhythmic motion.
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