By Phillip A Griffiths, Mathematiker USA
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Extra resources for Topics in transcendental algebraic geometry : (a seminar; Princeton - N.J., 1981-1982)
Things are different when we consider the relative velocity of one object with respect to another. For example, the velocity with which a runner approaches the ribbon at the finish line is a mechanical quantity, since it does not depend on the inertial reference frame in which we consider runner and ribbon. Whether our coordinate system is linked to the earth or to the fixed stars (in which case both runner and ribbon move with tremendous, "cosmic," velocities), the relative velocity of the runner, which depends on how fast the distance between runner and ribbon is decreasing, is the same.
For such points it makes sense to define the special distance (6) In fact, if the abscissas of A(x,y) and AI(xl'YI) coincide (XI = x), then a motion (1) takes these points to points A'(x',y') and Ai(xi,yl), with x'=xi =x+a and y'=vx+y+b, Yi =VX+YI +b. Hence yi-y'=(vx+YI + b)-(vx+y+b)=YI-Y· Thus the difference YI - Y is unchanged by a motion, and so has geometric significance in the Galilean plane. On the other hand, if the distance dAA I = X I - x between A and A I is not zero, then the difference Y I - Y of their ordinates is not preserved by a motion, for, in that case, Yi-y'=(vx l +YI +b)-(vx+y+b)=YI-y+v(xl-x)*YI-Y.
It follows that if VI and V2 are the velocities of two objects (for example runner and ribbon) in one reference frame, and vi, V; are their velocities in another reference frame, then vI=vi+a and V2=V;+a, so that VI -V2 =vi -V;. Things are much the same if VI and V2 are the velocities of an object at two instants tl and t2. , the vector a is assumed constant, it follows, just as before, that VI =vi +a, V2=V;+a, and VI- V2=vi -V;. lIICf. Galileo's "Dialogues" , pp. l7l-172. 23 I. What is mechanics?
Topics in transcendental algebraic geometry : (a seminar; Princeton - N.J., 1981-1982) by Phillip A Griffiths, Mathematiker USA