Newtonian Dynamics - Newton's Second Law in A Multidimensional Space

Newton's Second Law in A Multidimensional Space

Let's consider particles with masses in the regular three-dimensional Euclidean space. Let 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

$\frac{d\mathbf r_i}{dt}=\mathbf v_i,\qquad\frac{d\mathbf v_i}{dt}=\frac{\mathbf F_i(\mathbf r_1,\ldots,\mathbf r_N,\mathbf v_1,\ldots,\mathbf v_N,t)}{m_i},\quad i=1,\ldots,N.$

(1)

The three-dimensional radius-vectors can be built into a single -dimensional radius-vector. Similarly, three-dimensional velocity vectors can be built into a single -dimensional velocity vector:

$\mathbf r=\begin{Vmatrix} \mathbf r_1\\ \vdots\\ \mathbf r_N\end{Vmatrix},\qquad\qquad \mathbf v=\begin{Vmatrix} \mathbf v_1\\ \vdots\\ \mathbf v_N\end{Vmatrix}.$

(2)

In terms of the multidimensional vectors (2) the equations (1) are written as

$\frac{d\mathbf r}{dt}=\mathbf v,\qquad\frac{d\mathbf v}{dt}=\mathbf F(\mathbf r,\mathbf v,t),$

(3)

i. e they take the form of Newton's second law applied to a single particle with the unit mass .

Definition. The equations (3) are called the equations of a Newtonian dynamical system in a flat multidimensional Euclidean space, which is called the configuration space of this system. Its points are marked by the radius-vector . The space whose points are marked by the pair of vectors is called the phase space of the dynamical system (3).

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