Editing Newtonian dynamics (section)

==Constraint forces==

For a constrained Newtonian dynamical system the constraints described by the equations ({{EquationNote|6}}) are usually implemented by some mechanical framework. This framework produces some auxiliary forces including the force that maintains the system within its configuration manifold <math>\displaystyle M</math>. Such a maintaining force is perpendicular to <math>\displaystyle M</math>. It is called the [[normal force]]. The force <math>\displaystyle\mathbf F</math> from ({{EquationNote|6}}) is subdivided into two components
{{NumBlk|:|<math>
\mathbf F=\mathbf F_\parallel+\mathbf F_\perp</math>.|{{EquationRef|13}}}}
The first component in ({{EquationNote|13}}) is tangent to the configuration manifold <math>\displaystyle M</math>. The second component is perpendicular to <math>\displaystyle M</math>. In coincides with the [[normal force]] <math>\displaystyle\mathbf N</math>.<br>
Like the velocity vector ({{EquationNote|8}}), the tangent force 
<math>\displaystyle\mathbf F_\parallel</math> has its internal presentation
{{NumBlk|:|<math>\displaystyle\mathbf F_\parallel=\sum^n_{i=1}\frac{\partial\mathbf r}{\partial q^i}\,F^i</math>.|{{EquationRef|14}}}}
The quantities <math>F^1,\,\ldots,\,F^n</math> in ({{EquationNote|14}}) are called the internal components of the force vector.