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<p>A multiple pendulum is defined as a system composed of <i>n</i> pendulums connected in series. In the small-oscillation regime, a closed-form expression for the characteristic polynomial is obtained assuming equal masses and lengths. For a fixed value of <i>n</i>, this formula provides the polynomial whose roots correspond to the normal frequencies of the system. The closed-form expression is shown to be proportional to the Laguerre polynomial of order <i>n</i>, in agreement with previously established results. Furthermore, several algebraic properties of these frequencies are derived. Although these properties are closely related to known identities satisfied by the zeros of Laguerre polynomials, their formulation in terms of the normal frequencies of the linearized multiple pendulum does not appear to have been explicitly reported in the physics literature.</p>

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Closed-form expression for the characteristic polynomial of the multiple pendulum

  • 期刊论文

  • 发表日期：

    2026年6月15日

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  • D. A. Pulido-Caviedes,
  • J. H. Muñoz

摘要

A multiple pendulum is defined as a system composed of n pendulums connected in series. In the small-oscillation regime, a closed-form expression for the characteristic polynomial is obtained assuming equal masses and lengths. For a fixed value of n, this formula provides the polynomial whose roots correspond to the normal frequencies of the system. The closed-form expression is shown to be proportional to the Laguerre polynomial of order n, in agreement with previously established results. Furthermore, several algebraic properties of these frequencies are derived. Although these properties are closely related to known identities satisfied by the zeros of Laguerre polynomials, their formulation in terms of the normal frequencies of the linearized multiple pendulum does not appear to have been explicitly reported in the physics literature.