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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Bibliographic Details
Published in:Mathematics (Basel) Vol. 13; no. 15; p. 2326
Main Authors: Shen, Yihan, Sun, Yajuan
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.08.2025
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math13152326