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Two types of variational integrators and their equivalence
Numerical Analysis · Submitted 18 Sep 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim It turns out that these variational integrators are equivalent to each other when they are used for integrating the classical mechanical system with Lagrangian function $L(q,\dot{q})=\frac{1}{2}\dot{q}^TM\dot{q}-U(q)$ ($M$ is an invertible symmetric constant matrix).
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The abstract, sentence by sentence
In this paper, we introduce two types of variational integrators, one originating from the discrete Hamilton's principle while the other from Galerkin variational approach. It turns out that these variational integrators are equivalent to each other when they are used for integrating the classical mechanical system with Lagrangian function $L(q,\dot{q})=\frac{1}{2}\dot{q}^TM\dot{q}-U(q)$ ($M$ is an invertible symmetric constant matrix). They are symplectic, symmetric, possess super-convergence order $2s$ (which depends on the degree of the approximation polynomials), and can be related to continuous-stage partitioned Runge-Kutta methods.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we introduce two types of variational integrators, one originating from the discrete Hamilton's principle while the other from Galerkin variational approach. | contribution 1.00 | 0.91 | 0.05 | asserted 1.00 |
| s1 | It turns out that these variational integrators are equivalent to each other when they are used for integrating the classical mechanical system with Lagrangian function $L(q,\dot{q})=\frac{1}{2}\dot{q}^TM\dot{q}-U(q)$ ($M$ is an invertible symmetric constant matrix). | result 1.00 | 0.86 | 0.76 | asserted 1.00 |
| s2 | They are symplectic, symmetric, possess super-convergence order $2s$ (which depends on the degree of the approximation polynomials), and can be related to continuous-stage partitioned Runge-Kutta methods. | result 0.99 | 0.82 | 0.18 | asserted 1.00 |
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Original abstract
In this paper, we introduce two types of variational integrators, one originating from the discrete Hamilton's principle while the other from Galerkin variational approach. It turns out that these variational integrators are equivalent to each other when they are used for integrating the classical mechanical system with Lagrangian function $L(q,\dot{q})=\frac{1}{2}\dot{q}^TM\dot{q}-U(q)$ ($M$ is an invertible symmetric constant matrix). They are symplectic, symmetric, possess super-convergence order $2s$ (which depends on the degree of the approximation polynomials), and can be related to continuous-stage partitioned Runge-Kutta methods.
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