6.3: Lagrange Equations from d’Alembert’s Principle
Bernoulli伯努利:
a change in the configuration of the system as a result of any arbitrary infinitessimal 无穷小量 instantaneous 瞬间 change of the coordinates $\delta \boldsymbol r_i$, that is consistent 一致 with the forces and constraints 约束imposed on the system at the instant $t$
Suppose the system of $n$ paticals is in equilibrium 平衡
$$ \sum_i^N\mathbf{F}_i\cdot\delta\mathbf{r}_i=0 $$
Decomposing the force, into applied force$\boldsymbol F^A_i$, and constraint forces$\boldsymbol f^C_i$
$$ \sum_i^N\mathbf{F}_i^A\cdot\delta\mathbf{r}_i+\sum_i^N\mathbf{f}_i^C\cdot\delta\mathbf{r}_i=0 $$
Bernoulli’s Principle of STATIC virtual work: constraint forces term can be ignored.
$$ \sum_i^N\mathbf{F}_i^A\cdot\delta\mathbf{r}_i=0 $$
introduce dynamics by Newton’s Law to related force and momentum
$$ \mathbf{F}_i=\dot{\mathbf{p}}_i $$
d’Alembert developed the Princple of Dynamic Virtual Work in the form
$$ \sum_i^N(\mathbf{F}_i^A-\mathbf{\dot{p}}_i)\cdot\delta\mathbf{r}_i+\sum_i^N\mathbf{f}_i^C\cdot\delta\mathbf{r}_i=0 $$
For special case where the force of constraint are zero.
Transformation to generalized coordinates
$$ \mathbf{r}_i=\mathbf{r}_i(q_1,q_2,q_3\ldots,q_n,t) $$
$$ \mathbf{v}_i\equiv\frac{d\mathbf{r}_i}{dt}=\sum_j^n\frac{\partial\mathbf{r}_i}{\partial q_j}\dot{q}_j+\frac{\partial\mathbf{r}_i}{\partial t} $$