分析力学更新于 2026-07-12

达朗贝尔原理与第二类拉格朗日方程

从牛顿第二定律引入惯性力，推导达朗贝尔原理和第二类拉格朗日方程。

#达朗贝尔原理#拉格朗日方程#动能

达朗贝尔原理

牛顿第二定律写作

\mathbf F_i+\mathbf F_{Ni}=m_i\mathbf a_i.

移项后把 $-m_i\mathbf a_i$ 看作惯性力。对任意允许虚位移作虚功，并利用理想约束力虚功为零：

\sum_i(\mathbf F_i-m_i\mathbf a_i)\cdot\delta\mathbf r_i=0.

动能

T=\frac12\sum_i m_i v_i^2.

笔记推导使用

\frac{\partial \mathbf v_i}{\partial \dot q_\sigma} =\frac{\partial \mathbf r_i}{\partial q_\sigma}, \qquad \frac{\partial \mathbf v_i}{\partial q_\sigma} =\frac{d}{dt}\left( \frac{\partial \mathbf r_i}{\partial q_\sigma} \right).

于是

\sum_i m_i\mathbf a_i\cdot \frac{\partial \mathbf r_i}{\partial q_\sigma} = \frac{d}{dt}\left(\frac{\partial T}{\partial \dot q_\sigma}\right) -\frac{\partial T}{\partial q_\sigma}.

代回达朗贝尔原理，得到

\frac{d}{dt}\left(\frac{\partial T}{\partial \dot q_\sigma}\right) -\frac{\partial T}{\partial q_\sigma} =Q_\sigma.

若主动力有势， $Q_\sigma=-\frac{\partial V}{\partial q_\sigma}$ ，令 $L=T-V$ ：

\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_\sigma}\right) -\frac{\partial L}{\partial q_\sigma}=0.