d'Alembert's principle

From Physics wiki

d'Alembert's principle is a generalization of the principle of virtual work in which the condition for static equilibrium, i.e.,

$\sum_i \mathbf{F}_i = 0\,$ ,

is replaced by Newton's second law,

$\sum_i ( \mathbf{F}_i - m_i \mathbf{a}_i ) = 0\,$ .

The system is therefore equivalent to a static one under the influence of an inertial force $-m_i \mathbf{a}_i\,$ . This is a condition for which the virtual work due to a virtual displacement $\delta \mathbf{r}_i\,$ , is zero:

$\delta W = \sum_i ( \mathbf{F}_i - m_i \mathbf{a}_i ) \cdot \mathbf{r}_i = 0\,$ .

We can again ignore internal forces, as these occur in pairs, and decompose the force $\mathbf{F}_i\,$ into an applied force and a (holonomic) constraint force:

$\sum_i \left( \mathbf{F}^{(a)}_i + \mathbf{F}^{(c)}_i - m_i \mathbf{a}_i \right) = 0\,$ .

Since (holonomic) constraint forces do no virtual work, we see that

$\delta W = \left(\sum_i \mathbf{F}^{(a)}_i - m_i \mathbf{a}_i\right)\cdot\delta\mathbf{r}_i = 0\,$ .

Generalized forces

If the above virtual displacements were linearly independent, we could set each factor $\mathbf{F}^{(a)}_i - m_i \mathbf{a}_i\,$ separately to zero. However, the virtual displacements are required to be consistent with zero or more constraints of the form $f(\mathbf{r}_i) = 0\,$ . Therefore, we should first parameterize these constraint surfaces by a set of $n\,$ independent generalized coordinates $q_i\,$ . Let

$\mathbf{r}_i = \mathbf{r}_i(q_1, ..., q_n, t)\,$ .

Then

$\delta \mathbf{r}_i = \sum_j \frac{\partial \mathbf{r}_i }{\partial q_j} \delta q_j\,$ ,

$\dot\mathbf{r}_i = \sum_j \frac{\partial \mathbf{r}_i }{\partial q_j} \dot q_j + \frac{\partial \mathbf{r}_i}{\partial t}\,$ ,

and, for later use,

$\frac{d}{dt}\left(\frac{\partial \mathbf{r}_i }{\partial q_j} \right) = \sum_k \frac{\partial^2 \mathbf{r}_i }{\partial q_j \partial q_k} \dot{q}_k + \frac{\partial^2 \mathbf{r}_i }{\partial t \partial q_j} = \frac{\partial \dot\mathbf{r}_i}{\partial q_j}\,$ ,

while we note that

$\frac{\partial \dot \mathbf{r}_i}{\partial \dot q_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}\,$ ,

since $\mathbf{r}_i\,$ does not depend on $\dot q_j\,$ . The virtual work due to $\mathbf{F}^{(a)}\,$ is then

$\sum_i \mathbf{F}^{(a)}_i \cdot\delta\mathbf{r}_i = \sum_{ij} \left(\mathbf{F}^{(a)}_i \cdot\frac{\partial \mathbf{r}_i }{\partial q_j}\right) \delta q_j = \sum_j Q_j \delta q_j\,$ ,

where we have defined the generalized force

$Q_j = \sum_i \mathbf{F}^{(a)}_i \cdot\frac{\partial \mathbf{r}_i }{\partial q_j}\,$ .

While it isn't necessary that $Q_j\,$ have dimensions of force nor $q_j\,$ dimensions of displacement, the product $Q_j \delta q_j\,$ must have dimensions of work.

To complete our task, we will need the product

$\sum_i m_i \ddot \mathbf{r}_i \cdot \frac{\partial \mathbf{r}_i}{\partial q_j}\,$	$= \sum_i \left[ \frac{d}{dt} \left( m_i \dot \mathbf{r}_i \cdot \frac{\partial \mathbf{r}_i}{\partial q_j} \right) - m_i \dot \mathbf{r}_i\frac{d}{dt}\left(\frac{\partial \mathbf{r}_i }{\partial q_j} \right) \right]\,$ ,
	$= \sum_i \left[ \frac{d}{dt} \left( m_i\dot \mathbf{r}_i \cdot \frac{\partial \mathbf{r}_i}{\partial q_j} \right) - m_i\dot \mathbf{r}_i \frac{\partial \dot \mathbf{r}_i }{\partial q_j}\right]\,$ ,
	$= \sum_i \left[ \frac{d}{dt} \left( m_i\dot \mathbf{r}_i \cdot \frac{\partial \dot\mathbf{r}_i}{\partial \dot q_j} \right) - m_i\dot \mathbf{r}_i \frac{\partial \dot \mathbf{r}_i }{\partial q_j}\right]\,$ ,
	$= \frac{d}{dt} \left( \frac{\partial}{\partial \dot q_j} \sum_i \frac{1}{2}m_i v_i^2 \right) - \frac{\partial}{\partial q_j} \sum_i \frac{1}{2}m_i v_i^2\,$ .

Identifying $T = \textstyle\sum_i \frac{1}{2}m_i v_i^2\,$ with the kinetic energy, d'Alembert's principle becomes

$\sum_j \left[ \left\{\frac{d}{dt} \left( \frac{\partial T}{\partial \dot q_j}\right) - \frac{\partial T}{\partial q_j} \right \} - Q_j \right] \delta q_j = 0\,$ .

By assumption, the generalized coordinates are independent, so each term in the sum must separately vanish:

$\frac{d}{dt} \left( \frac{\partial T}{\partial \dot q_j}\right) - \frac{\partial T}{\partial q_j} - Q_j = 0\,$ .

back to generalized coordinates

on to Euler-Lagrange equations

Retrieved from "http://www.physics.thetangentbundle.net/wiki/Classical_mechanics/d%27Alembert%27s_principle"

d'Alembert's principle

From Physics wiki

Generalized forces

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