Editing Analytical mechanics (section)

===D'Alembert's principle of virtual work===
{{main |  D'Alembert's principle}}
D'Alembert's principle states that infinitesimal ''[[virtual work]]'' done by a force across reversible displacements is zero, which is the work done by a force consistent with ideal constraints of the system. The idea of a constraint is useful – since this limits what the system can do, and can provide steps to solving for the motion of the system. The equation for D'Alembert's principle is:<ref name="Torby1984">{{cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |isbn=0-03-063366-4 |chapter=Energy Methods}}</ref>{{rp|265}}
<math display="block">\delta W = \boldsymbol{\mathcal{Q}} \cdot \delta\mathbf{q} = 0 \,,</math>
where <math display="block">\boldsymbol\mathcal{Q} = (\mathcal{Q}_1, \mathcal{Q}_2, \dots, \mathcal{Q}_N)</math>
are the [[generalized forces]] (script Q instead of ordinary Q is used here to prevent conflict with canonical transformations below) and {{math|'''q'''}} are the generalized coordinates. This leads to the generalized form of [[Newton's laws]] in the language of analytical mechanics:
<math display="block">\boldsymbol\mathcal{Q} = \frac{d}{dt} \left ( \frac {\partial T}{\partial \mathbf{\dot{q}}} \right ) - \frac {\partial T}{\partial \mathbf{q}}\,,</math>

where ''T'' is the total [[kinetic energy]] of the system, and the notation
<math display="block">\frac {\partial}{\partial \mathbf{q}} = \left(\frac{\partial }{\partial q_1}, \frac{\partial }{\partial q_2}, \dots, \frac{\partial }{\partial q_N}\right)</math>
is a useful shorthand (see [[matrix calculus#Scalar-by-vector|matrix calculus]] for this notation).