The Solution to the Differential Equation with Linear Damping Describing a Physical Systems Governed by a Cubic Energy Potential

Kim Johannessen

doi:doi:10.11648/j.acm.20251406.17

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The Solution to the Differential Equation with Linear Damping Describing a Physical Systems Governed by a Cubic Energy Potential

Kim Johannessen^*

Published in Applied and Computational Mathematics (Volume 14, Issue 6)

Received: 3 December 2025 Accepted: 16 December 2025 Published: 31 December 2025

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Abstract

An analytical solution to the nonlinear differential equation describing the equation of motion of a particle moving in an unforced physical system with linear damping, governed by a cubic potential well, is presented in terms of the Jacobi elliptic functions. In the attractive region of the potential the system becomes an anharmonic damped oscillator, however with asymmetric displacement. An expression for the period of oscillation is derived, which for a nonlinear damped system is time dependent, and in particular it contains a quartic root of an exponentially decaying term in the denominator. Initially the period is longer as compared to that of a linear oscillator, however gradually it decreases to that of a linear damped oscillator. Transforming the undamped nonlinear differential equation into the differential equation describing orbital motion of planets, the perihelion advance of Mercury can be estimated to 42.98 arcseconds/century, close to present day observations of 43.1±0.5 arcseconds/century. Some familiarity with the Jacobi elliptic functions is required, in particular with respect to the differential behavior of these functions, however, they are standard functions of advanced mathematical computer algebra tools. The expression derived for the solution to the nonlinear physical system, and in particular the expression for the period of oscillation, is useful for an accurate evaluation of experiments in introductory and advanced physics labs, but also of interest for specialists working with nonlinear phenomena governed by the cubic potential well.

Published in	Applied and Computational Mathematics (Volume 14, Issue 6)
DOI	10.11648/j.acm.20251406.17
Page(s)	367-377
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Linear Damping, Cubic Potential Well, Jacobi Elliptic Functions, Period of Oscillation, Asymmetric Oscillations, Perihelion Precession

References

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Johannessen, K. (2025). The Solution to the Differential Equation with Linear Damping Describing a Physical Systems Governed by a Cubic Energy Potential. Applied and Computational Mathematics, 14(6), 367-377. https://doi.org/10.11648/j.acm.20251406.17

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ACS Style

Johannessen, K. The Solution to the Differential Equation with Linear Damping Describing a Physical Systems Governed by a Cubic Energy Potential. Appl. Comput. Math. 2025, 14(6), 367-377. doi: 10.11648/j.acm.20251406.17

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AMA Style

Johannessen K. The Solution to the Differential Equation with Linear Damping Describing a Physical Systems Governed by a Cubic Energy Potential. Appl Comput Math. 2025;14(6):367-377. doi: 10.11648/j.acm.20251406.17

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@article{10.11648/j.acm.20251406.17,
  author = {Kim Johannessen},
  title = {The Solution to the Differential Equation with Linear Damping Describing a Physical Systems Governed by a Cubic Energy Potential},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {6},
  pages = {367-377},
  doi = {10.11648/j.acm.20251406.17},
  url = {https://doi.org/10.11648/j.acm.20251406.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251406.17},
  abstract = {An analytical solution to the nonlinear differential equation describing the equation of motion of a particle moving in an unforced physical system with linear damping, governed by a cubic potential well, is presented in terms of the Jacobi elliptic functions. In the attractive region of the potential the system becomes an anharmonic damped oscillator, however with asymmetric displacement. An expression for the period of oscillation is derived, which for a nonlinear damped system is time dependent, and in particular it contains a quartic root of an exponentially decaying term in the denominator. Initially the period is longer as compared to that of a linear oscillator, however gradually it decreases to that of a linear damped oscillator. Transforming the undamped nonlinear differential equation into the differential equation describing orbital motion of planets, the perihelion advance of Mercury can be estimated to 42.98 arcseconds/century, close to present day observations of 43.1±0.5 arcseconds/century. Some familiarity with the Jacobi elliptic functions is required, in particular with respect to the differential behavior of these functions, however, they are standard functions of advanced mathematical computer algebra tools. The expression derived for the solution to the nonlinear physical system, and in particular the expression for the period of oscillation, is useful for an accurate evaluation of experiments in introductory and advanced physics labs, but also of interest for specialists working with nonlinear phenomena governed by the cubic potential well.},
 year = {2025}
}

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Author Information

Kim Johannessen

Department of Mathematics, Vucfyn, Odense, Denmark

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http://orcid.org/0000-0002-0270-3485

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Johannessen, K. (2025). The Solution to the Differential Equation with Linear Damping Describing a Physical Systems Governed by a Cubic Energy Potential. Applied and Computational Mathematics, 14(6), 367-377. https://doi.org/10.11648/j.acm.20251406.17

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ACS Style

Johannessen, K. The Solution to the Differential Equation with Linear Damping Describing a Physical Systems Governed by a Cubic Energy Potential. Appl. Comput. Math. 2025, 14(6), 367-377. doi: 10.11648/j.acm.20251406.17

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AMA Style

Johannessen K. The Solution to the Differential Equation with Linear Damping Describing a Physical Systems Governed by a Cubic Energy Potential. Appl Comput Math. 2025;14(6):367-377. doi: 10.11648/j.acm.20251406.17

Copy | Download

@article{10.11648/j.acm.20251406.17,
  author = {Kim Johannessen},
  title = {The Solution to the Differential Equation with Linear Damping Describing a Physical Systems Governed by a Cubic Energy Potential},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {6},
  pages = {367-377},
  doi = {10.11648/j.acm.20251406.17},
  url = {https://doi.org/10.11648/j.acm.20251406.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251406.17},
  abstract = {An analytical solution to the nonlinear differential equation describing the equation of motion of a particle moving in an unforced physical system with linear damping, governed by a cubic potential well, is presented in terms of the Jacobi elliptic functions. In the attractive region of the potential the system becomes an anharmonic damped oscillator, however with asymmetric displacement. An expression for the period of oscillation is derived, which for a nonlinear damped system is time dependent, and in particular it contains a quartic root of an exponentially decaying term in the denominator. Initially the period is longer as compared to that of a linear oscillator, however gradually it decreases to that of a linear damped oscillator. Transforming the undamped nonlinear differential equation into the differential equation describing orbital motion of planets, the perihelion advance of Mercury can be estimated to 42.98 arcseconds/century, close to present day observations of 43.1±0.5 arcseconds/century. Some familiarity with the Jacobi elliptic functions is required, in particular with respect to the differential behavior of these functions, however, they are standard functions of advanced mathematical computer algebra tools. The expression derived for the solution to the nonlinear physical system, and in particular the expression for the period of oscillation, is useful for an accurate evaluation of experiments in introductory and advanced physics labs, but also of interest for specialists working with nonlinear phenomena governed by the cubic potential well.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - The Solution to the Differential Equation with Linear Damping Describing a Physical Systems Governed by a Cubic Energy Potential
AU  - Kim Johannessen
Y1  - 2025/12/31
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251406.17
DO  - 10.11648/j.acm.20251406.17
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 367
EP  - 377
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251406.17
AB  - An analytical solution to the nonlinear differential equation describing the equation of motion of a particle moving in an unforced physical system with linear damping, governed by a cubic potential well, is presented in terms of the Jacobi elliptic functions. In the attractive region of the potential the system becomes an anharmonic damped oscillator, however with asymmetric displacement. An expression for the period of oscillation is derived, which for a nonlinear damped system is time dependent, and in particular it contains a quartic root of an exponentially decaying term in the denominator. Initially the period is longer as compared to that of a linear oscillator, however gradually it decreases to that of a linear damped oscillator. Transforming the undamped nonlinear differential equation into the differential equation describing orbital motion of planets, the perihelion advance of Mercury can be estimated to 42.98 arcseconds/century, close to present day observations of 43.1±0.5 arcseconds/century. Some familiarity with the Jacobi elliptic functions is required, in particular with respect to the differential behavior of these functions, however, they are standard functions of advanced mathematical computer algebra tools. The expression derived for the solution to the nonlinear physical system, and in particular the expression for the period of oscillation, is useful for an accurate evaluation of experiments in introductory and advanced physics labs, but also of interest for specialists working with nonlinear phenomena governed by the cubic potential well.
VL  - 14
IS  - 6
ER  -

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