Explicit energy conserving method of two classical dynamical systems

Jianqiang Sun; Jie Chen; Lijuan Zhang; Jianqiang Sun; Jie Chen; Lijuan Zhang

doi:10.3934/math.20251346

AIMS Mathematics

2025, Volume 10, Issue 12: 30683-30695. doi: 10.3934/math.20251346

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Explicit energy conserving method of two classical dynamical systems

1.
School of Mathematics and Statistics, Hainan University, Haikou, 570228, Hainan Province, China
2.
Hainan University, Key Laboratory of Engineering Modeling and Statistical Computation of Hainan Province, Haikou, 570228, Hainan Province, China

Received: 28 April 2025 Revised: 28 August 2025 Accepted: 17 September 2025 Published: 26 December 2025
MSC : 37K05, 65M06

Two classical dynamical systems, the pendulum problem and the Hénon Heiles system, are transformed into the reformulated systems by the nonlinear transformation. These new reformulated systems are transformed into ordinary differential equations on manifolds by the linear transformation. The explicit RKMK method, which is a kind of Lie group method, is applied to solve the differential equations on manifolds. The explicit energy conserving schemes of the two classical dynamical systems are obtained. Numerical simulation investigates the effectiveness of these new schemes in preserving the conservation property of these equations and well simulating dynamical behaviors.
- Hénon Heiles system,
- pendulum problem,
- RKMK method,
- scalar auxiliary variable approach
Citation: Jianqiang Sun, Jie Chen, Lijuan Zhang. Explicit energy conserving method of two classical dynamical systems[J]. AIMS Mathematics, 2025, 10(12): 30683-30695. doi: 10.3934/math.20251346

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Abstract

Two classical dynamical systems, the pendulum problem and the Hénon Heiles system, are transformed into the reformulated systems by the nonlinear transformation. These new reformulated systems are transformed into ordinary differential equations on manifolds by the linear transformation. The explicit RKMK method, which is a kind of Lie group method, is applied to solve the differential equations on manifolds. The explicit energy conserving schemes of the two classical dynamical systems are obtained. Numerical simulation investigates the effectiveness of these new schemes in preserving the conservation property of these equations and well simulating dynamical behaviors.

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