Examples of the Search for Multipliers. Attraction of a Point by a Fixed Centre in a Resisting Medium and in Empty Space

Abstract

In order to show the applicability of the theory of multipliers, we shall first consider a case in which, deviating from all other examples to which these investigations relate, X _i , Y _i , Z _i will be functions not merely of the coordinates, but will also of the velocities, so that M is not a constant. This case is that of a planet which moves around the sun in a resisting medium. Without taking into account the resistance, it is well known that the equations for the motion of a planet are the following:

$$\frac{{{d^2}x}} {{d{t^2}}} = - {k^2}\frac{x} {{{r^3}}},\;\frac{{{d^2}y}} {{d{t^2}}} = - {k^2}\frac{y} {{{r^3}}},\;\frac{{{d^2}z}} {{d{t^2}}} = - {k^2}\frac{z} {{{r^3}}}$$

, where x, y, z are the heliocentric coordinates of the planet, r its distance from the sun and k² the attraction which the sun exerts at unit distance. If $v = \sqrt {{{x'}^2} + {{y'}^2} + {{z'}^2}}$ is the velocity of the planet in the direction of the tangent to its trajectory and V the resistance in the same direction, then the components of the resistance along the axes x, y, z are respectively

$$\frac{{Vx'}} {v},\;\frac{{Vy'}} {v},\;\frac{{Vz'}} {v}$$

. These quantities are to be added to the right sides of the differential equations, with the same sign as those the terms based on the attraction have. The equations of motion then become

$$\frac{{{d^2}x}} {{d{t^2}}} = - {k^2}\frac{x} {{{r^3}}} - \frac{{Vx'}} {v},\;\frac{{{d^2}y}} {{d{t^2}}} = - {k^2}\frac{y} {{{r^3}}} - \frac{{Vy'}} {v},\;\frac{{{d^2}z}} {{d{t^2}}} = - {k^2}\frac{z} {{{r^3}}} - \frac{{Vz'}} {v}$$

. If we take the resistance proportional to the nth power of the velocity

$$V = f{v^n}$$

, where f is a constant, one has the following differential equations

$$\begin{gathered} \frac{{{d^2}x}} {{d{t^2}}} = - {k^2}\frac{x} {{{r^3}}} - f{v^{n - 1}}x' = A, \hfill \\ \frac{{{d^2}y}} {{d{t^2}}} = - {k^2}\frac{y} {{{r^3}}} - f{v^{n - 1}}y' = B, \hfill \\ \frac{{{d^2}z}} {{d{t^2}}} = - {k^2}\frac{z} {{{r^3}}} - f{v^{n - 1}}z' = C \hfill \\ \end{gathered}$$

((16.1))

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