Nonlinear springs with orbits in radicals: a solvability criterion
Abstract
An odd polynomial oscillator has a periodic orbit given by a quadrature. When the zeros of the energy polynomial are all simple or all of even multiplicity, that quadrature is elementary exactly when the polynomial is a perfect square, and the time is a sum of inverse tangents. Recovering the displacement inverts that relation, algebraic when their coefficients are commensurate, making a closed form a Galois question. This paper identifies the group as the monodromy group of a rational map, computes its branch data from the parameters, and shows the branch value at the phase-angle singularity never vanishes, so every decomposition pairs the amplitudes reciprocally. In the commensurate family with distinct real factors, the orbit is radical exactly when the primitive inversion degree is at most four, or when reciprocal pairs leave a reduced spring of that degree. Radical orbits occur at the fifth, ninth, thirteenth and seventeenth force orders, the last only for reciprocal members, an expression recorded for each. The multiplicity hypothesis cannot be dropped. Factoring the energy polynomial into square and squarefree parts turns the quadrature into a differential of the third kind on a hyperelliptic curve, where elementarity is a torsion condition on the divisor of its poles with a vanishing holomorphic remainder. The first nonsquare elementary member appears at the ninth force order, and the power pullback inverts in closed form at every such order, giving a radical orbit for a confining spring of arbitrarily high degree with a sufficient condition for positive coefficients.
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BibTeX
@misc{pillai2026nonlinear,
title = {{Nonlinear springs with orbits in radicals: a solvability criterion}},
author = {Unnikrishna Pillai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37730},
eprint = {2610.37730},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37730},
}
RIS
TY - UNPB
TI - Nonlinear springs with orbits in radicals: a solvability criterion
AU - Unnikrishna Pillai
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37730
UR - https://arxiv.org/abs/2610.37730
AB - An odd polynomial oscillator has a periodic orbit given by a quadrature. When the zeros of the energy polynomial are all simple or all of even multiplicity, that quadrature is elementary exactly when the polynomial is a perfect square, and the time is a sum of inverse tangents. Recovering the displacement inverts that relation, algebraic when their coefficients are commensurate, making a closed form a Galois question. This paper identifies the group as the monodromy group of a rational map, computes its branch data from the parameters, and shows the branch value at the phase-angle singularity never vanishes, so every decomposition pairs the amplitudes reciprocally. In the commensurate family with distinct real factors, the orbit is radical exactly when the primitive inversion degree is at most four, or when reciprocal pairs leave a reduced spring of that degree. Radical orbits occur at the fifth, ninth, thirteenth and seventeenth force orders, the last only for reciprocal members, an expression recorded for each. The multiplicity hypothesis cannot be dropped. Factoring the energy polynomial into square and squarefree parts turns the quadrature into a differential of the third kind on a hyperelliptic curve, where elementarity is a torsion condition on the divisor of its poles with a vanishing holomorphic remainder. The first nonsquare elementary member appears at the ninth force order, and the power pullback inverts in closed form at every such order, giving a radical orbit for a confining spring of arbitrarily high degree with a sufficient condition for positive coefficients.
ER -