Variational Partitioned Runge–Kutta Methods for Lagrangians Linear in Velocities
In this paper, we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the...
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MDPI AG
2019-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/9/861 |
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| author | Tomasz M. Tyranowski Mathieu Desbrun |
| author_facet | Tomasz M. Tyranowski Mathieu Desbrun |
| author_sort | Tomasz M. Tyranowski |
| collection | DOAJ |
| description | In this paper, we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the “Hamiltonian” equations of motion can be formulated as an index-1 differential-algebraic system. We also construct variational Runge−Kutta methods and analyze their properties. The general properties of Runge−Kutta methods depend on the “velocity” part of the Lagrangian. If the “velocity” part is also linear in the position coordinate, then we show that non-partitioned variational Runge−Kutta methods are equivalent to integration of the corresponding first-order Euler−Lagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the Runge−Kutta method are retained. If the “velocity” part is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We verified our results through numerical experiments for various dynamical systems. |
| first_indexed | 2024-12-13T14:40:48Z |
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| id | doaj.art-196a5ba85fcf461d8c5b3949ee6b8894 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-13T14:40:48Z |
| publishDate | 2019-09-01 |
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| spelling | doaj.art-196a5ba85fcf461d8c5b3949ee6b88942022-12-21T23:41:36ZengMDPI AGMathematics2227-73902019-09-017986110.3390/math7090861math7090861Variational Partitioned Runge–Kutta Methods for Lagrangians Linear in VelocitiesTomasz M. Tyranowski0Mathieu Desbrun1Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, 85748 Garching, GermanyDepartment of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, USAIn this paper, we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the “Hamiltonian” equations of motion can be formulated as an index-1 differential-algebraic system. We also construct variational Runge−Kutta methods and analyze their properties. The general properties of Runge−Kutta methods depend on the “velocity” part of the Lagrangian. If the “velocity” part is also linear in the position coordinate, then we show that non-partitioned variational Runge−Kutta methods are equivalent to integration of the corresponding first-order Euler−Lagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the Runge−Kutta method are retained. If the “velocity” part is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We verified our results through numerical experiments for various dynamical systems.https://www.mdpi.com/2227-7390/7/9/861variational integratorsdegenerate LagrangiansRunge–Kutta methodsdifferential-algebraic systemssymplectic geometrydynamical systems |
| spellingShingle | Tomasz M. Tyranowski Mathieu Desbrun Variational Partitioned Runge–Kutta Methods for Lagrangians Linear in Velocities Mathematics variational integrators degenerate Lagrangians Runge–Kutta methods differential-algebraic systems symplectic geometry dynamical systems |
| title | Variational Partitioned Runge–Kutta Methods for Lagrangians Linear in Velocities |
| title_full | Variational Partitioned Runge–Kutta Methods for Lagrangians Linear in Velocities |
| title_fullStr | Variational Partitioned Runge–Kutta Methods for Lagrangians Linear in Velocities |
| title_full_unstemmed | Variational Partitioned Runge–Kutta Methods for Lagrangians Linear in Velocities |
| title_short | Variational Partitioned Runge–Kutta Methods for Lagrangians Linear in Velocities |
| title_sort | variational partitioned runge kutta methods for lagrangians linear in velocities |
| topic | variational integrators degenerate Lagrangians Runge–Kutta methods differential-algebraic systems symplectic geometry dynamical systems |
| url | https://www.mdpi.com/2227-7390/7/9/861 |
| work_keys_str_mv | AT tomaszmtyranowski variationalpartitionedrungekuttamethodsforlagrangianslinearinvelocities AT mathieudesbrun variationalpartitionedrungekuttamethodsforlagrangianslinearinvelocities |