The dynamical motion of mechanical systems possesses underlying geometric structures and preserving these structures in numerical integration improves the qualitative accuracy and reduces the long-time error of the simulation. For a single mechanical system, structure preservation can be achieved by adopting the variational integrator construction (Marsden, J. & West, M. (2001) Discrete mechanics and variational integrators. Acta Numer., 10, 357–514). This construction has been generalized to more complex systems involving forces or constraints as well as to the setting of Dirac mechanics (Leok, M. & Ohsawa, T. (2011) Variational and geometric structures of discrete Dirac measures. Found. Comput. Math., 11, 529–562). Forced Lagrange–Dirac systems are described by a Lagrangian and an external force pair, and two pairs of Lagrangians and external forces are said to be equivalent if they yield the same equations of motion. However, the variational discretization of a forced Lagrange–Dirac system discretizes the Lagrangian and forces separately, and will generally depend on the choice of representation. In this paper we derive a class of Dirac variational integrators with forces that yield well-defined numerical methods that are independent of the choice of representation. We present a numerical simulation to demonstrate how such equivalence-preserving discretizations avoid spurious solutions that otherwise arise from poorly chosen representations.

中文翻译：

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用力构造等价的Dirac变分积分器

机械系统的动态运动具有潜在的几何结构，并且将这些结构保留在数值积分中可以提高定性精度并减少仿真的长期误差。对于单个机械系统，结构保存可以通过采用变分积分器结构来实现（马斯登，J。＆西，M。（2001）离散力学和变积分器。ACTA NUMER。 ，10，357-514）。这种构造已经推广到涉及力或约束的更复杂系统，以及狄拉克力学的设置（Leok，M.＆Ohsawa，T.（2011）离散狄拉克测度的变分和几何结构。，11，529–562）。拉格朗日–狄拉克系统由拉格朗日和外力对来描述，如果两对拉格朗日和外力产生相同的运动方程，则它们是等效的。但是，强制拉格朗日-狄拉克系统的变分离散化将拉格朗日和力分别离散化，并且通常取决于表示的选择。在本文中，我们推导了一类Dirac变分积分器，其力产生了定义明确的数值方法，这些方法与表示的选择无关。我们提供了一个数值模拟，以证明这种保持等价的离散化如何避免否则会因选择不当的表示而引起的虚假解。