Advantages of application of variational integrators on Lie groups in problems of modeling the dynamics of mechanical systems

Purpose of the research is to consider advantages of application of variational integrators on Lie groups in problems of physically correct modeling of dynamics of mechanical systems and to compare them with classical nonvariational integrators.

Methods. To demonstrate the possibilities of variational integrators on Lie groups, a mathematical model of the dynamics of a physical pendulum was developed. Methods of variational calculus and methods of Lie group theory were used to construct a mathematical model of the dynamics of a physical pendulum. The Runge-Kutta method of the 4th order was used for comparative analysis of variational and nonvariational integrators. Modeling was carried out in MATLAB software.

Results. In this research, a variational integrator algorithm on Lie groups was developed to model the dynamics of a physical pendulum. To compare the variational integrators and the 4th order Runge-Kutta method, plots were constructed to show how the angular velocity along the axes, orthogonal error, total energy, and angular momentum change over time. The graphs demonstrate that although the angular velocity is the same for both methods, the Runge-Kutta method does not preserve the geometric structure of the continuous system and does not preserve the basic constant quantities of the modeled system, namely mechanical energy and momentum.

Conclusion. Numerical modeling has shown that the preservation of symplectic properties of systems and the structure of Lie groups allows to perform physically correct computer modeling of the dynamics of mechanical systems. Variational integrators on Lie groups have significant computational advantages over classical integration methods, which do not preserve the geometric structure of the continuous system and the basic constant quantities of the system, and other variational integrators, which preserve either none or one of these properties.

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