This chapter shows how the Newtonian law of motion of a particle subject to a gradient force derived from a ‘potential energy’ can always be obtained from an extremal principle, or ‘principle of least action’. According to Newton’s first law, the trajectory representing the motion of a free particle between two points p1 and p2 is a straight line. In other words, out of all the possible paths between p1 and p2, the trajectory effectively followed by a free particle is the one that minimizes the length. However, even though the use of the principle of extremal length of the paths between two points gives the straight line joining the points, this does not mean that the straight-line path is traced with constant velocity in an inertial frame. Moreover, the trajectory describing the motion of a particle subject to a force is not uniform and rectilinear.
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