1.3: The Microscopic Laws of Motion

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Page ID: 5096

Mark Tuckerman
New York University

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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:

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

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 \(t\).