Numerical Analysis for a Class of Variational Integrators

In this paper, we study a geometric framework for second-order differential systems arising in classical and relativistic mechanics. For this class of systems, we derive necessary and sufficient conditions for their Lagrangian description. The main objectives of this work are to construct efficient...

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Vydané v:Mathematics (Basel) Ročník 13; číslo 15; s. 2326
Hlavní autori: Shen, Yihan, Sun, Yajuan
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Basel MDPI AG 01.08.2025
Predmet:
Analysis
Charged particles
Differential equations
Integrators
inverse variational problem
Kepler problem
Mechanics
modified Lagrangian
Noether’s theorem
Numerical analysis
Numerical integration
Numerical methods
Partial differential equations
variational integrator
China
Germany
ISSN:2227-7390, 2227-7390
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Abstract In this paper, we study a geometric framework for second-order differential systems arising in classical and relativistic mechanics. For this class of systems, we derive necessary and sufficient conditions for their Lagrangian description. The main objectives of this work are to construct efficient structure-preserving variational integrators in a variational framework. To achieve this, we develop new variational integrators through Lagrangian splitting and prove their equivalence to composition methods. We display the superiority of the newly derived numerical methods for the Kepler problem and provide rigorous error estimates by analysing the Laplace–Runge–Lenz vector. The framework provides tools applicable to geometric numerical integration of both ordinary and partial differential equations.
AbstractList In this paper, we study a geometric framework for second-order differential systems arising in classical and relativistic mechanics. For this class of systems, we derive necessary and sufficient conditions for their Lagrangian description. The main objectives of this work are to construct efficient structure-preserving variational integrators in a variational framework. To achieve this, we develop new variational integrators through Lagrangian splitting and prove their equivalence to composition methods. We display the superiority of the newly derived numerical methods for the Kepler problem and provide rigorous error estimates by analysing the Laplace–Runge–Lenz vector. The framework provides tools applicable to geometric numerical integration of both ordinary and partial differential equations.
Audience Academic
Author Sun, Yajuan
Shen, Yihan
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StartPage 2326
SubjectTerms Analysis
Charged particles
Differential equations
Integrators
inverse variational problem
Kepler problem
Mechanics
modified Lagrangian
Noether’s theorem
Numerical analysis
Numerical integration
Numerical methods
Partial differential equations
variational integrator
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