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classical - 1 Newtons Second Law The form of Newtons second...

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1 Newton’s Second Law The form of Newton’s second law for three separate cases will be generated using quaternion operators acting on position quaternions. In classical mechanics, time and space are decoupled. One way that can be achieved algebraically is by having a time operator cat only on space, or by space operator only act on a scalar function. I call this the ”2 zero” rule: if there are two zeros in the generator of a law in physics, the law is classical. Newton’s 2nd Law for an Inertial Reference Frame in Cartesian Coordinates Define a position quaternion as a function of time. R t , R Operate on this once with the differential operator to get the velocity quaternion. V d dt , 0 t , R 1 , . R Operate on the velocity to get the classical inertial acceleration quaternion. A d dt , 0 1 , . R 0 , R This is the standard form for acceleration in Newton’s second law in an inertial reference frame. Because the reference frame is inertial, the first term is zero. Newton’s 2nd Law in Polar Coordinates for a Central Force in a Plane Repeat this process, but this time start with polar coordinates. R t , r Cos , r Sin , 0 The velocity in a plane. V d dt , 0 t , r Cos , r Sin , 0 1 , . r Cos r Sin . , . r Sin r Cos . , 0 Acceleration in a plane. A d dt , 0 1 , . r Cos r Sin . , . r Sin r Cos . , 0 0 , 2 . r Sin . r Cos . 2 r Cos r Sin , 2 . r Cos . r Sin . 2 r Sin r Cos , 0 Not a pretty sight. For a central force, . L/ 2 , and 0. Make these substitution and rotate the quaternion to get rid of the theta dependence. A Cos , 0 , 0 , Sin d dt , 0 2 t , r Cos , r Sin , 0 0 , L 2 m 2 r 3 r , 2 L . r m r 2 , 0 The second term is the acceleration in the radial direction, the third is acceleration in the theta direction for a central force in polar coordinates.
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