By Henry William Watson
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Extra info for A Treatise on the Application of Generalised Coordinates to the Kinetics o
J q represents the velocities in any motion, different which, when compounded with the free motion, f and this principle = = (1); necessarily positive, therefore 32 GAUSS LEAST CONSTRAINT. [17. This proposition sometimes admits of practical application be required to find the constrained motion when the free if it known. be removed by the motion is for instance, only one degree of If, constraint, then the constraint freedom may be ex pressed by making some one function of the coordinates constant in the constrained, which is not constant in the free, motion.
Result of Art. 16 is a particular case of this proposition. subjected to ^ is &c. be the velocities assumed by the system when any constraint and acted on by the same impulses, Q lf Q 2 , the kinetic energy assumed by the system, that is I Q, and the result just obtained assumes the form the same as that of Art. 16. ] the the resulting kinetic energy under given impressed impulses from rest must be increased. Substitute for the (j s i n forming the function ^ for the new system the values q lt q2 &c.
Set in one side of an The infinite plane position of the ball, position, the motion through a mass of friction- extending infinitely in all directions less incompressible fluid whole motion ball s centre #, y, z at the on and originally at rest. and therefore by our general pro is determined time t are if the coordinates of the known. Let the axis of OB be taken perpendicular to the bounding plane through any point whatever of that plane then the kinetic energy T must be a quadratic function of #, y, and #, with co efficients certain functions of #, y and z.