Editing Newtonian dynamics (section)

==Constraints and internal coordinates==
In some cases the motion of the particles with the masses <math>\displaystyle m_1,\,\ldots,\,m_N</math> can be constrained. Typical [[constraint algorithm|constraints]] look like scalar equations of the form 
{{NumBlk|:|<math>\displaystyle\varphi_i(\mathbf r_1,\ldots,\mathbf r_N)=0,\quad i=1,\,\ldots,\,K</math>.|{{EquationRef|5}}}}
Constraints of the form ({{EquationNote|5}}) are called [[Holonomic constraints|holonomic]] and [[Scleronomous|scleronomic]]. In terms of the radius-vector <math>\displaystyle\mathbf r</math> of the Newtonian dynamical system ({{EquationNote|3}}) they are written as
{{NumBlk|:|<math>\displaystyle\varphi_i(\mathbf r)=0,\quad i=1,\,\ldots,\,K</math>.|{{EquationRef|6}}}}
Each such constraint reduces by one the number of degrees of freedom of the Newtonian dynamical system ({{EquationNote|3}}). Therefore, the constrained system has <math>\displaystyle n=3\,N-K</math> degrees of freedom.

'''Definition'''. The constraint equations ({{EquationNote|6}}) define an <math>\displaystyle n</math>-dimensional [[manifold]] <math>\displaystyle M</math> within the configuration space of the Newtonian dynamical system ({{EquationNote|3}}). This manifold <math>\displaystyle M</math> is called the configuration space of the constrained system. Its tangent bundle <math>\displaystyle TM</math> is called the phase space of the constrained system.

Let <math>\displaystyle q^1,\,\ldots,\,q^n</math> be the internal coordinates of a point of <math>\displaystyle M</math>. Their usage is typical for the [[Lagrangian mechanics]]. The radius-vector <math>\displaystyle\mathbf r</math> is expressed as some definite function of <math>\displaystyle q^1,\,\ldots,\,q^n</math>:
{{NumBlk|:|<math>\displaystyle\mathbf r=\mathbf r(q^1,\,\ldots,\,q^n)
</math>.|{{EquationRef|7}}}}
The vector-function ({{EquationNote|7}}) resolves the constraint equations ({{EquationNote|6}}) in the sense that upon substituting ({{EquationNote|7}}) into ({{EquationNote|6}}) the equations ({{EquationNote|6}}) are fulfilled identically in <math>\displaystyle q^1,\,\ldots,\,q^n</math>.