Editing Newtonian dynamics (section)

==Newton's second law in a multidimensional space==
Consider <math>\displaystyle N</math> particles with masses <math>\displaystyle m_1,\,\ldots,\,m_N</math> in the regular three-dimensional [[Euclidean space]]. Let <math>\displaystyle \mathbf r_1,\,\ldots,\,\mathbf r_N</math> be their radius-vectors in some [[inertial]] coordinate system. Then the motion of these particles is governed by Newton's second law applied to each of them
{{NumBlk|:|<math>
\frac{d\mathbf r_i}{dt}=\mathbf v_i,\qquad\frac{d\mathbf v_i}{dt}=\frac{\mathbf F_i(\mathbf r_1,\ldots,\mathbf r_N,\mathbf v_1,\ldots,\mathbf v_N,t)}{m_i},\quad i=1,\ldots,N. 
</math>|{{EquationRef|1}}}}
The three-dimensional radius-vectors <math>\displaystyle\mathbf r_1,\,\ldots,\,\mathbf r_N</math> can be built into a single <math>\displaystyle n=3N</math>-dimensional radius-vector. Similarly, three-dimensional velocity vectors <math>\displaystyle\mathbf v_1,\,\ldots,\,\mathbf v_N</math> can be built into a single <math>\displaystyle n=3N</math>-dimensional velocity vector:
{{NumBlk|:|<math>
\mathbf r=\begin{Vmatrix}
\mathbf r_1\\ \vdots\\ \mathbf r_N\end{Vmatrix},\qquad\qquad
\mathbf v=\begin{Vmatrix}
\mathbf v_1\\ \vdots\\ \mathbf v_N\end{Vmatrix}.
</math>|{{EquationRef|2}}}}
In terms of the multidimensional vectors ({{EquationNote|2}}) the equations ({{EquationNote|1}}) are written as 
{{NumBlk|:|<math>
\frac{d\mathbf r}{dt}=\mathbf v,\qquad\frac{d\mathbf v}{dt}=\mathbf F(\mathbf r,\mathbf v,t), 
</math>|{{EquationRef|3}}}}
i.e. they take the form of Newton's second law applied to a single particle with the unit mass <math>\displaystyle m=1</math>.

'''Definition'''. The equations ({{EquationNote|3}}) are called the
equations of a '''Newtonian [[dynamical system]]''' in a flat multidimensional [[Euclidean space]], which is called the [[Configuration space (physics)|configuration space]] of this system. Its points are marked by the radius-vector 
<math>\displaystyle\mathbf r</math>. The space whose points are marked by the pair of vectors <math>\displaystyle(\mathbf r,\mathbf v)</math> is called the [[phase space]] of the dynamical system ({{EquationNote|3}}).