The principle of least action is a variational principle that states an object will always take the path of least action as compared to any other conceivable path. This principle can be used to derive the equations of motion of many systems, and therefore provides a unifying equation that has been applied in many fields of physics and mathematics. Hamilton’s formulation of the principle of least action typically only accounts for conservative forces, but can be reformulated to include non-conservative forces such as friction by using the approach suggested by Wang and co-workers (Lin and Wang, 2014; Wang, 2015). However, it turns out that this modified action is dependent upon the nature and amount of damping in the system and the optimality argument fails to hold for sufficiently large values of the damping coefficient. In this paper, we specifically investigate the modified action principle for the cases of a linearly and cubically damped, nonlinear pendulum.

最小作用量原则是一个变分原则，它表明与任何其他可想象的路径相比，对象将始终采用最小作用量的路径。这个原理可以用来推导出许多系统的运动方程，因此提供了一个统一的方程，已经应用于物理和数学的许多领域。Hamilton 对最小作用量原理的表述通常只考虑保守力，但可以通过使用 Wang 及其同事建议的方法重新表述为包括非保守力，例如摩擦力（Lin 和 Wang，2014 年；Wang，2015 年） . 然而，事实证明，这种修改后的动作取决于系统中阻尼的性质和数量，并且对于足够大的阻尼系数值，最优性论证无法成立。在本文中，