F = m A in an inertial frame, in polar coordinates, and in a rotating frame using quaternions

doug  <sweetser@world.com>

Preamble
Newton's 2nd law for an inertial reference frame in Cartesian coordinates
Newton's 2nd law in polar coordinates for a central force in a plane
Newtons's 2nd law in a nonintertial, rotating frame
Implications

Preamble

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 acheived 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  =  q[t,  x[t],  y[t],  z[t]];

R  =  (t,  (Overscript[r, ⇀]))

Operate on this once with the differential operator to get the velocity quaternion.

\(V\  == \ oneoponq[q[D, 0, 0, 0], \ R]\ .\ {1, 0, 0, 0}
\ \ 
       == {1, \(\ \(d\ x[t]\)\/\(d\ t\)\ \), \(\ \(d\ y[t]\)\/\(d\ t\)\ \), 
        \(\ \(d\ z[t]\)\/\(d\ t\)\ \)}\)

V  =  
      (((d )/(d t)), (Overscript[0, ⇀])) R = 
        (1,  ((d (Overscript[r, ⇀]))/(d t)) )

Operate on the position quaternion twice to get the classical inertial acceleration quaternion.

\(A\  == 
	
      oneoponq[q[D, 0, 0, 0], 
	oneoponq[q[D, 0, 0, 0], \ R]]\ .
        \ {1, 0, 0, 0}
\ \ 
       == {0, \(\ \(\(d\^2\) x[t]\)\/\(d\ t\^2\)\ \), \(\ 
        \(\(d\^2\) y[t]\)\/\(d\ t\^2\)\ \), \(\ 
        \(\(d\^2\) z[t]\)\/\(d\ t\^2\)\ \)}\)

A  = 
      (((((d )/(d t)), (Overscript[0, ⇀])))^2) R = 
        (0,  (((d^2) (Overscript[r, ⇀]))/(d (t^2))))

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.

(Subscript[R, plane])  =  
      q[t,  r[t] Cos[θ[t]],  r[t] Sin[θ[t]],  0];

(Subscript[R, plane])   = (t,  r Cos[θ],  r Sin[θ],  0)

The velocity in a plane.

V== oneoponq[q[D, 0, 0, 0],  (Subscript[R, plane]) ] . {1, 0, 0, 0}
   =={1,Cos[θ[t]] r'[t]-r[t] Sin[θ[t]] θ'[t],
					Sin[θ[t]] r'[t]+Cos[θ[t]] r[t] θ'[t],0}

V =(((d )/(d t)), (Overscript[0,  ⇀]) ) (Subscript[R, plane]) 
=(1,Cos[θ] r'[t]-r Sin[θ] θ',
			Sin[θ] r'+Cos[θ] r θ',0)

Acceleration in a plane.

A == \(oneoponq[q[D, 0, 0, 0], 
					
          oneoponq[q[D, 0, 0, 0], \ R\_plane\ ]]\ .\ {1, 0, 0, 0}\)
  =={0,-2 Sin[θ[t]] r'[t] θ'[t]-Cos[θ[t]] r[t] ((θ'[t])^2)+
					Cos[θ[t]] r''[t]-r[t] Sin[θ[t]] θ''[t],
					2 Cos[θ[t]] r'[t] θ'[t]-r[t] Sin[θ[t]] ((θ'[t])^2)+
					Sin[θ[t]] r''[t]+Cos[θ[t]] r[t] θ''[t],0}

A =(((((d )/(d t)), (Overscript[0,  ⇀]) ))^2) (Subscript[R, plane]) 
  =(0,-2 Sin[θ] r' θ'-Cos[θ] r ((θ')^2)+
					Cos[θ] r''-r Sin[θ] θ'',
					2 Cos[θ] r' θ'-r Sin[θ] ((θ')^2)+
				  Sin[θ] r''+Cos[θ] r θ'',0)

Not a pretty sight.  For a central force, theta' = L/mr^2, and theta" = 0.  Make these substitution and rotate the quaternion to get rid of the theta dependence.

A ==\(Simplify[
        Expand[\((
					
	
              q[Cos[θ[t]], \ 0, \ 0, \ \(-Sin[θ[t]]\)]\ .\ 


                  	oneoponq[q[D, 0, 0, 0], 
	
                    oneoponq[q[D, 0, 0, 0], \ R\_plane\ ]]\  /. \ 


                	{\(\(θ'\)'\)[t]\  -> \ 0, \ 
                  \(θ'\)[t]\  -> \ L/\ \((m\ r[t]\^2)\)})\).{1, 0, 0, 
              0}\ ]]\)
  =={0,-((L^2)/((m^2) r[t]^3))+r''[t],((2 L r'[t])/(m r[t]^2)),0}

A =(Cos[θ],  0,  0,  -Sin[θ]) 
          (((((d )/(d t)), (Overscript[0,  ⇀]) ))^2) (Subscript[R, plane])  
  =(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.

Newton's 2nd law in a noninertial, rotating frame

Consider the "noninertial" case, with the frame rotating at an angular speed omega.  The differential time operator is put into the first term of the quaternion, and the three directions for the angular speed are put in the next terms.  This quaternion is then multiplied by the position quaternion to get the velocity in a rotating reference frame. Unlike the previous examples where t did not interfer with the calcuation, this time it must be set explicitly to zero (I wonder what that means?).

\(V\  == \ 
      oneoponq[q[D, ω\_x[t], \ ω\_y[t], \ ω\_z[t]], \ R]
          \ .\ {1, 0, 0, 0}
\ \ 
       == {1 - x[t]\ ω\_x[t] - y[t]\ ω\_y[t] - 
          z[t]\ ω\_z[t], 
		\ \ 
        t\ ω\_x[t] + z[t]\ ω\_y[t] - y[t]\ ω\_z[t] + \ 
          \(d\ x[t]\)\/\(d\ t\)\ , 
		
        \(-z[t]\)\ ω\_x[t] + t\ ω\_y[t] + 
          x[t]\ ω\_z[t] + \ \(d\ y[t]\)\/\(d\ t\)\ , 
		\ \ 
        y[t]\ ω\_x[t] - x[t]\ ω\_y[t] + t\ ω\_z[t] + \(\ 
          \(d\ z[t]\)\/\(d\ t\)\ \)}\)

V = 
      (((d )/(d t)), (Overscript[ω,  ⇀]) ) 
          (0, (Overscript[r,  ⇀]) ) = 
        (-(Overscript[ω,  ⇀]) .(Overscript[r,  ⇀]) , 
          ((d (Overscript[r,  ⇀]) )/(d t))  +  
            (Overscript[ω,  ⇀])  x  (Overscript[r,  ⇀]) )

Operate on the position quaternion twice.

\(A\  == \ 
      Simplify[

        oneoponq[q[D, ω\_x[t], \ ω\_y[t], \ ω\_z[t]], 

            oneoponq[
              q[D, ω\_x[t], \ ω\_y[t], \ ω\_z[t]], \ 

              			q[0, \ x[t], \ y[t], \ z[t]]\ ]].{1, 0, 0, 0}\ ]
\ \ 
       == {\(-2\)\ ω\_x[t]\ \((\ \(d\ x[t]\)\/\(d\ t\)\ )\) - 
          2\ ω\_y[t]\ \((\ \(d\ y[t]\)\/\(d\ t\)\ )\) - 
          2\ ω\_z[t]\ \((\ \(d\ z[t]\)\/\(d\ t\)\ )\) - 
			
          x[t]\ \((\ \(d\ ω\_x[t]\)\/\(d\ t\)\ )\) - 
          y[t]\ \((\ \(d\ ω\_y[t]\)\/\(d\ t\)\ )\) - 
          z[t]\ \((\ \(d\ ω\_z[t]\)\/\(d\ t\)\ )\), 
		
        \(-x[t]\)\ 
            \((ω\_x[t]\^2 + ω\_y[t]\^2 + ω\_z[t]\^2)\)

          		 - 2\ ω\_z[t]\ \((\ \(d\ y[t]\)\/\(d\ t\)\ )\) + 
          2\ ω\_y[t]\ \((\ \(d\ z[t]\)\/\(d\ t\)\ )\)
			 + 
          z[t]\ \((\ \(d\ ω\_y[t]\)\/\(d\ t\)\ )\) - 
          y[t]\ \((\ \(d\ ω\_z[t]\)\/\(d\ t\)\ )\) + \ 
          \(\(d\^2\) x[t]\)\/\(d\ t\^2\)\ , 
		
        \(-y[t]\)\ 
            \((ω\_x[t]\^2 + ω\_y[t]\^2 + ω\_z[t]\^2)\) + 
          
			2\ ω\_z[t]\ \((\ \(d\ x[t]\)\/\(d\ t\)\ )\) - 
          2\ ω\_x[t]\ \((\ \(d\ z[t]\)\/\(d\ t\)\ )\) - 
			
          z[t]\ \((\ \(d\ ω\_x[t]\)\/\(d\ t\)\ )\) + 
          x[t]\ \((\ \(d\ ω\_z[t]\)\/\(d\ t\)\ )\) + \ 
          \(\(d\^2\) y[t]\)\/\(d\ t\^2\)\ , 
		
        \(-z[t]\)\ 
            \((ω\_x[t]\^2 + ω\_y[t]\^2 + ω\_z[t]\^2)\) - 
          
			2\ ω\_y[t]\ \((\ \(d\ x[t]\)\/\(d\ t\)\ )\) + 
          2\ ω\_x[t]\ \((\ \(d\ y[t]\)\/\(d\ t\)\ )\) + 
			
          y[t]\ \((\ \(d\ ω\_x[t]\)\/\(d\ t\)\ )\) - 
          x[t]\ \((\ \(d\ ω\_y[t]\)\/\(d\ t\)\ )\) + \(\ 
          \(\(d\^2\) z[t]\)\/\(d\ t\^2\)\ \)}\)

\(A\  = 
      \(\(\((\(d\ \)\/\(d\ t\), \(ω\& ⇀ \))\)\^2\) 
          \((0, \(r\& ⇀ \))\) = 
        \(\((\(d\ \)\/\(d\ t\), \(ω\& ⇀ \))\) 
            \((\(-\(ω\& ⇀ \).\(r\& ⇀ \)\), 
              \(d\ \(r\& ⇀ \)\)\/\(d\ t\)\  + \ 
                \(ω\& ⇀ \) x \( r\& ⇀ \))\)

          \ \  = \((
            \(\(d  ω\&⇀\)\/dt\).\(r\& ⇀ \), 
            \ \(-ω\^2\) r\&⇀\  - \ 
              2\ \(d  r\&⇀\)\/dt\ x\ ω\&⇀\ 
               + \ \(d  ω\&⇀\)\/dt\ x\ r\&⇀\ 
               + \ \(\(d\^2\) r\&⇀\)\/dt\^2)\)\)\)\)

Looks complex, but it can be simplified with vector notation.

(Subscript[(Overscript[a, ⇀]), inertial])  =  
      -(ω^2) (Overscript[r, ⇀])  -  
        2 ((d  (Overscript[r, ⇀]))/dt) x (Overscript[ω, ⇀])  +  
        ((d  (Overscript[ω, ⇀]))/dt) x (Overscript[r, ⇀])  +  
        (((d^2) (Overscript[r, ⇀]))/dt)

(Subscript[(Overscript[a, ⇀]), inertial])  =  
      -(ω^2) (Overscript[r, ⇀])  -  
        2 ((d  (Overscript[r, ⇀]))/dt) x (Overscript[ω, ⇀])  +  
        ((d  (Overscript[ω, ⇀]))/dt) x (Overscript[r, ⇀])  +  
        (((d^2) (Overscript[r, ⇀]))/dt)

In most applications,

(Overscript[ω, ⇀]) . (Overscript[r, ⇀])  =  
      0     \[Implies]     (ω^2) (Overscript[r, ⇀])  =  
        -(Overscript[ω, ⇀])  x 
          ((Overscript[ω, ⇀])  x (Overscript[r, ⇀]))

(Overscript[ω, ⇀]) . (Overscript[r, ⇀])  =  
      0     \[Implies]     (ω^2) (Overscript[r, ⇀])  =  
        -(Overscript[ω, ⇀])  x 
          ((Overscript[ω, ⇀])  x (Overscript[r, ⇀]))

The usual way the acceleration is written as

(Subscript[(Overscript[a, ⇀]), inertial])  =  
      (Overscript[ω, ⇀])  x 
          ((Overscript[ω, ⇀])  x (Overscript[r, ⇀])) +  
        2  (Overscript[ω, ⇀])  x ((d  (Overscript[r, ⇀]))/dt) +  
        ((d  (Overscript[ω, ⇀]))/dt) x (Overscript[r, ⇀])  +  
        (((d^2) (Overscript[r, ⇀]))/dt)

(Subscript[(Overscript[a, ⇀]), inertial])  =  
      (Overscript[ω, ⇀])  x 
          ((Overscript[ω, ⇀])  x (Overscript[r, ⇀])) +  
        2  (Overscript[ω, ⇀])  x ((d  (Overscript[r, ⇀]))/dt) +  
        ((d  (Overscript[ω, ⇀]))/dt) x (Overscript[r, ⇀])  +  
        (((d^2) (Overscript[r, ⇀]))/dt)

The inertial accelerations are the centrifugal, the coriolis, the azimuthal and the translational, respectively.  All four of these inertial forces are part of the vector component of the same quaternion.

The scalar term is not zero.  What this implies is not yet clear, but it may be related to the fact that the frame is not inertial.

Implications

Three forms of Newton's second law were generated by choosing appropriate operator quaternions acting on position quaternions.  The differential time operator was decoupled from any differential space operators.  This may be viewed as an operational definition of "classical" physics.