Skip to main content
Chemistry LibreTexts

1.3: The Microscopic Laws of Motion

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Consider a system of \(N\) classical particles. The particles a confined to a particular region of space by a "container'' of volume \(V\). The particles have a finite kinetic energy and are therefore in constant motion, driven by the forces they exert on each other (and any external forces which may be present). At a given instant in time \(t\), the Cartesian positions of the particles are \(r_1(t), \cdots , r_N(t) \)) ) . The time evolution of the positions of the particles is then given by Newton's second law of motion:

    \[ m_i \ddot {r} _i = F_i ( r_1, \cdots , r_N ) \nonumber \]

    where \(F_1, \cdots , F_N \) are the forces on each of the \(N\) particles due to all the other particles in the system. The notation \(\ddot {r} _i = \frac {d^2 r_i}{dt^2}\).

    \(N\) Newton's equations of motion constitute a set of \(3N\) coupled second order differential equations. In order to solve these, it is necessary to specify a set of appropriate initial conditions on the coordinates and their first time derivatives, \( \{r_1 (0), \cdots , r_N(0), \dot {r} _1 (0), \cdots , \dot {r} _N (0) \} \). Then, the solution of Newton's equations gives the complete set of coordinates and velocities for all time \(t\).

    This page titled 1.3: The Microscopic Laws of Motion is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mark Tuckerman.