Newtonian Mechanics

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Definition 1. We consider a system of \(k\) particles with fixed masses \(m_1,\dots, m_k\), located at positions \(\mathbf x_1,\dots,\mathbf x_k\in \mathbb R^3\) at time \(t\in \mathbb R\). The \(i\)th particle is acted upon by a force \(\mathbf F_i\) that depends on the positions \(\mathbf x_1,\dots,\mathbf x_k\) and the time \(t\). We then concatenate the positions and forces into \(3k\)-vectors

\[F = (\mathbf F_1,\dots,\mathbf F_k)\in \mathbb R^{3k}\quad \text{and} \quad x = (\mathbf x_1,\dots,\mathbf x_k)\in \mathbb R^{3k},\]

and the masses into a \(3k\times 3k\) matrix

\[m = \begin{pmatrix} m_1I_3 & 0 & \dots & 0 \\ 0 & m_2I_3 & \dots & 0 \\ \vdots & \vdots & & \vdots \\ 0 & 0 & \dots & m_kI_3 \end{pmatrix}\]

where \(I_3\) is the \(3\times 3\) identity matrix. And set

\[v = \frac {dx}{dt}\quad \text{and} \quad a = \frac{dv}{dt}.\]

Principle 2 (Newton’s law). Newton’s law is a foundational physical law of dynamics of classical mechanics, which is a second-order ordinary differential equation for \(x\) when \(F\) is known:

\[F = ma.\]

Definition 3. We shall consider only autonomous and conservative forces, that is, those \(F\) that depend only on \(x\) and for which the line integral \(\int_C F\cdot dx\) vanishes for every closed curve \(C\) in \(\mathbb R^{3k}\).

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