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The Microscopic Laws of Motion

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    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:

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

    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 \(t\).

    Contributors and Attributions

    Mark Tuckerman (New York University)

    The Microscopic Laws of Motion is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.