We can computationally model the motion of a body by determining the forces acting instantaneously on it at time t and assuming these forces to act uniformly on the body over the period [t,t+d]. The forces acting on the body may strive to fracture or deform the body and typically must be explicitly computed when determining kinematical responces so as to compare them with breaking or deformation thresholds. Assuming that the forces do not deform the body and that it remains rigid, the change in motion of the body due to them is independent of time and position and often also of body motion.
    We can think of the forces as imparting momentum to the body, and think of a body or a fluid possessing momentum by virtue of its motion. Momentum is ultimately a flow of mass. More generally, assigning any scalar property such as temperature or electric charge or baryon number to a body or to fluid particles generates a "current" of that property.

    We calculate the instantaeous forces acting on the body at t and consider them to apply uniformly over time interval [t,t+d] and model them by applying d times those forces to the body instanataneously at time t in the form of impulses individually effecting the body's motion so as to provide the desired change in momentum. We are, in effect, directly modelling the conservation of momentum.
    Gravitational forces are forces proportionate to the scalar inertial mass to which they are applied so that particulate accelerations are independant of particle mass.

Kinematics


Stances
    Suppose we have a 3D object H whose "centre" is located at a point c(t)γ and having orientation A(t)=(i,j,k) where i,j and k are t-dependant column vectors representing the "global coordinates" of the "orientation vectors" of the object. If i,j and k are always orthormal we have AA^T=1 but more general A allow for "distorted" objects. Regardless of whether A is orthonormal, the "global coordinates" of a point having coordinates p_L in the "local frame" of the object are given by p = c + Ap_L. Vector b and matrix A being expressed in global coordinates.
    The combination (c,A) representing a located (distorted) orientation is sometimes referred to as the frame of H but, as we use "frame" to describe an unpositioned set of N linearly independant ND 1-vectors , we will use the term stance for a located orientation. We will refer to a stance with an orthonormal orientation matrix as a rigid stance. Note that c is here the kinematical centre of motion which may be distinct from the dynamical centre of mass
     The conventional matrix method for representing 3D stances is to add a fourth "scaling" dimension to 3D point vectors with the understanding that r=(x,y,z,s)^T=(x/s,y/s,z/s,1)^T .
    A typical Â_4×4 stance matrix would then be:

S	=	æ				c1	ö
		ç		A		c2	÷
		ç				c3	÷
		è	0	0	0	l	ø

where A is a conventional 3x3 transformation matrix. This acts on a 4D extended vector thus  

Sr	=	æ	(Ar+c	ö	=	æ	(Ar+c)/l	ö
		è	l	ø		è	1	ø

giving a translation by c and a scaling by l^-1 in addition to the rotation and shearing effects of A.
These 4×4 transformation matrices combine according to the usual rules for matrix multiplication.

    In GHC we represent an ND stance with GHC multivector S = (½e_¥c)^↑R(-½le_¥0)^↑ = (1+½e_¥c)R( cosh(½l)- sinh(½l)e_¥0) with rotor R Î e_¥0^* = i even and satisfying RR^§=1 for a rigid stance . In the absence of a l^↓ dilation factor (-½le_¥0)^↑ or an e_¥0 negation factor,   stance S resides entirely within e_¥^* [ Recall that e_¥^* = -e_¥^* includes e_¥ while excluding e₀ ]
    When N=3, the R for a rigid stance is the standard quaternian representation of a 3D rotation.
    The GHC embedded point representaton of p is given by p_G=Sp_LS^-1 equivalent to S_§(p_L) º Sp_LS^§ for a rigid stance. Typically Se_¥S^§=e_¥ (or le_¥ if S embodies some dilation) while Se₀S^§ = c = c+e₀+½c²e_¥.
    A direction a_L in the local frame becomes scaled direction a_G = l^↓Ra_LR^§ in the global frame . In GHC we can represent this as a_Ge_¥ = Sa_Le_¥S^§ .
    The position and orientation factors of a stance rotor do not commute unless R^§cR=c.

    S is nonunitary because since e_¥^†=-2e₀ we have (½e_¥c)^↑ (½e_¥c)^↑† = (½e_¥c)^↑ (½e_¥c)^†↑ = (½e_¥c)^↑ (e₀c)^†↑    = (1+½e_¥c)(1+e₀c^†) . For Euclidean c this is 1+(½e_¥+e₀)c + ½c² - ½e_¥0c² = 1+e_mc + ½c² - ½e_¥0c² .
    This means that there is no antiHermitian logarithm for S as there is for R in a Euclidean space.

Composing Stances
    Programmers will typically represent the stance of an object relative to the stance of another "master" object. As an example consider a rotating gun turrent mounted in a bomber flying over a city at night. The stance of the gun turrent S_Q¬T º S_Q¬TQ would typically be implimented with respect to the "base frame" of the bomber. Assuming the bomber is orientated in the base frame with e₁ as the forward direction, e₂ as left, and e₃ as upward, the turret might be considered as having a centre displaced by c_TQ=(X,0,Z) from the centre of the bomber and an orientation R_TQ=(-½fe₁₂)^↑(½qe₁₃)^↑ corresponding to a roll-less turet. The notation a_Q indicates that a is expressed relative to Q ( ie. "in Q coordinates"). Thus c_TQ denotes the position of the centre of T in Q coordinate system. For a more general mobile turret, c_TQ might be variable but confined to a particular k-curve fixed within the aircraft asuch as a straight rail or a hoop.
    The stance of the bomber S_,¬Q might be expressed with respect to the stance of the Earth ,, perhaps with c_Q, = [q,f,R_,+h+a] where q and f are latitude and longitude coordinates, R_, is the sealevel radius of an assumed spherical Earth, h is a ground height above or below sea level, and a is an altitude above ground level. The stance of the Earth S_!¬, would typically be expressed with regard to the stance S_SG of Sol which in turn would be expressed with respect to the galactic centre. All stances being rigid.
    Typically we neglect galactic rotation and consider the Sun to be at rest at 0_G with identity orientation 1_G so that S_G¬!=1_G=1 where G denotes the "global" or "galactic" rest frame of the "fixed stars" of the rendered night sky. We will refer to such "base frame" coordinates as the global coordinates with "global" being used here in the sense of "universal" rather than in relation to the "Globe" of the Earth.

    The global stance of the turrent is then given by the rotor
    S_G¬T   =   S_G¬!S_!¬,S_,¬QS_Q¬T   =   S_!¬,S_,¬QS_Q¬T     since S_G¬!=1.
    Note that we compose or "cascade" relative stances from right to left rather than left to right with S_A¬C = S_A¬BS_B¬C motivating the S_B¬A º S_B¬AB notation.
    Since S_A¬B=S_B¬A^-1 is equal to S_B¬A^§ for a rigid stance, the stance of the aircraft __Acraft with respect to the city 4 is given by
    S_4¬Q = S_4¬,S_,¬Q = S_,¬4^-1S_,¬Q = S_,¬4^§S_,¬Q which, when acting on a point in bomber coordinates, returns a point in city coordinates.

    Suppose we wish to emulate gunner Gus sitting in the turrent moving his head at least, and hypothesise a camera C positioned inside Gus' head having stance S_T¬C camera. The stance of the city with respect to the camera, which we would use to render the city as seen Gus, is given by
    S_C¬4 = S_C¬TS_T¬QS_Q¬4 = S_C¬TS_Q¬T^§S_Q¬4 converting city coordinates to camera coordinates.

    For N=3, the product of any number of <0;2>-grade normalised quaternian R is another such and the product of any number of <0;2;4>-grade stance rotors is likewise just another <0;2;4>-grade rotor within Â₃^% @ Â_4,1. For N³4 however, combined stances accrue higher even grades.

    Stance rotors are applied to embedded points. If we wish to use a stance rotor S to rotate a scaled direction a, ie. to compute RaR^§ rather than SaS^§ options include evaluating (Sae_¥S^§)¿e^¥ or SaS^§ - Se₀S^§ = S(a+½a²e_¥)S^§ . The former approach extends to evaluating RxR^§ for xÎU_N as (Sxe_¥S^§)¿e^¥. However, it is usually preferable to simply reclaim R from S , ideally logically as the unexteneded U_N component of S rather than by evaluating (Se_¥)¿e^¥ (which assumes S has no e_¥0 dilation component), and then calculating RxR^§ within the unextended algebra.

Kinematics
    Let H be an N-D object moving through U^N space and rigidly rotating about a motion centre which traverses a 1-curve path b(t). Let A(t) = R(t)_§(A(0)) º R(t)(A(0))R(t)^§ be the orientation matrix of the object at time t where unit even rotor R(t) satisfies R(t)R(t)^§ = 1) . We can construct GHC stance rotor S(t)=(1+½b(t)e_¥)R(t)). [  We define matrix-multivector products columnwise, treating matrix columns as seperate 1-vectors thus b(a₁,a₂,..)=(ba₁,ba₂,..) ; (a₁,a₂,..)b=(a₁b,a₂b,.. ) . Other multivector products (¿, Ù, ×, ...) are treated likewise ]

    For brevity we will omitt the explicit (t) and  write ₀ for (0) so that A = RA₀R^§ and letting ^· denote differentiation with respect to t we have
    A^· = -w×A = -w.A = A¿w where w º -2R^·R^§ is a pure bivector known as the classical spin or spin (aka. rotational bivelocity or angular velocity bivector) with R^· = -½wR and R^·· = -½(w^· - ½w²)R .
    Writing w_L = R^§wR = -2R^§R^·R^§R = -2R^§R^· for the local spin expressed in the frame of the object, we have R^· = -½Rw_L .
[ Proof :   RR^§=1 Þ R^·R^§ + R(R^§)^· = 0 Þ R^·R^§ = -R(R^§)^· = -(R^·R^§)^§ so w=-w^§ and is even for even R, so must have grade <2;6;10;14;...>.
    a^· = (Ra₀R^§)^· = (R^·a₀R^§ + Ra₀(R^§)^·) = (R^·R^§)a - a(R^·R^§) = 2(R^·R^§)×a = -a¿2(R^·R^§) = a×w = a¿w is a 1-vector for all 1-vector a(t) so w× must either preserve the grade of or annihilate every 1-vector. Hence w has zero <6;10;...>-grade components.  .]

    For N=3 we have the conventional rotational velocity 1-vector w=w^*=we₁₂₃^-1 with a^· = -w.a = -(w^*).a = -(wÙa)i = -w×a = a×w.

    In GHC we have stance spin W º -2S^·S^§   =   (1+½e_¥b)W(1-½e_¥b)   =   (1+½e_¥b)w(1-½e_¥b) + b^·e_¥ where bivelocity W º w + b^·e_¥ embodies both the velocity b^· and the spin w with W² = w² + 2b^·Ùw generally nonscalar even for N=3 .
[ Proof :  S^· = ½e_¥b^·R + (1+½e_¥b)R^·   =   (½e_¥b^· - (1+½e_¥b)½w)R   =   ½(1+½e_¥b)((1-½e_¥b)e_¥b^· - w)R   =   -½(1+½e_¥b)(b^·e_¥ + w)R
    Þ W º -2S^·S^§     =   (1+½e_¥c)W(1-½e_¥c)  .]

Reclaiming Velocities from Spins

    The nonnull 1-velocity p_G^· = p_G^· + (p_G¿p_G^·)e_¥ perpendicular to null p_G is given by p_G^·   =   p_G¿W   + S_§(p_L^·) .
[ Proof : p_G^·   =   (Sp_LS^§)^·   =   S^·p_LS^§   + Sp_L^·S^§ + Sp_L(S^·)^§   =   -½WSp_LS^§   + Sp_L^·S^§ - ½Sp_LS^§W^§
      =   -½Wp_G   + Sp_L^·S^§ - ½p_GW^§   =   -½(Wp_G-p_GW)   + Sp_L^·S^§   =   p_G¿W   + S_§(p_L^·)  .]
    Now (p-b+e₀)¿W = b^· + (p-b)¿w + ((p-b)¿b^·)e_¥ = v + ((p-b)¿b^·)e_¥ is the instantaneous 1-velocity of p º p_G but with an additional ((p-b)¿b^·)e_¥ component . We can eliminate this by representing velocity v with 2-blade v=ve_¥ with
    ((p-b+e₀)Ùe_¥)×W = ((p-b+e₀)¿W)e_¥ = (b^· + (p-b)¿w)e_¥   providing the instamtaneous 2-blade velocity of body point p.
[ Proof :  ((p-b+e₀)Ùe_¥)×W = ((p-b)e_¥-e_¥0)×W = ((p-b+e₀)Ùe_¥)×W = ((p-b+e₀)e_¥)×W = ((p-b+e₀)×W)e_¥ = ((p-b+e₀)¿W)e_¥  .]
     Note that (p-b+e₀)Ùe_¥ = (p-b+e₀+½(p-b)²e_¥)Ùe_¥ can be regarded as p_cÙe_¥ where p_c is the GHC embedding of p-b interpreted as a point.

Iterating Stances

    Suppose we have S and W at a particular time t and we wish to evaluate S(t+d) for a small timestep d.
    We know S^· = -½WS but setting S(t+d) = S(t)-½dW(t)S(t) = (1-½dW(t))S(t) results in S(t+d)S(t+d)^§ = 1-¼d²W(t)² intead of unity.
    The correct iteration is S(t+d) = (-½dW)^↑ S , or S(t+d) = S(-½dW_L)^↑ for W_L expressed in stance local (source) coordinates. This preserves the unit roticity of S(t+d) but since W² is nonscalar it is computationally profligate and the simpler approach is to impliment velocity and spin seperately as S(t+d) = (1+½(c+dc^·)e_¥)(-½w(t))^↑R(t) with w(t)² scalar for N£3.

    To form W(t+d) given W^·(t) we can take W(t+d)=W(t) + dW^·(t) using bivector addition without subsequently violating the roticity of S but we may instead wish for acceleration control to be a precession and amplification of spin rather than simple addition of spin.
    Typically W^·(t) will be some bivector "accceleration" function Q(c,c^·,R,R^·,t) = Q(W,W^·,t) modelling the forces and torques acting upon the object at time t such as gravitational weight, air resistance, lift, engine thrust, and so forth.
    However we compute W(t+d), better _O(d²) agreement can be obtained by evaluating w(t+½d), forming S(t+d) as (1+½(c+½dc^·)e_¥)(-½w(t+½d))^↑R(t) , and then computing W(t+d).

    Thus we must iterate L timesteps multiplicatively with R(Ld) = (-½w((L-1)d)^↑ (-½w((L-2)d)...(-½w(d))^↑(-½w(0))^↑R(0) approximating R(Ld) by a particular sequence of L discrete rotations of intial state R(0). This generally differs from
     R_S º (-½(w((L-1)d)+w((L-2)d)+..+w(d)+w(0)))^↑R(0) approximating (-½ ò₀^Ld dtw(t))^↑R(0)(L-1)d)+w((L-2)d)+..+w(d)+w(0)))^↑R(0) because the w(ld) do not commute.
    R_S can be regarded as a superposition of all R reachable by any combination of the particular L subrotations, with repeated uses allowed.

U_N Spinor Representation
    We have here represented U_N^% position c as (½e_¥c)^↑_§(e₀) but provided c² has the same sign as e₁²  we also have c = Ce₁C^§ where C = (c²)^¼(½q(cÙe₁)^~)^↑ is a nonunit <0;2>-grade  rotor in U_N satisfying (CC^§)² = e₁^-2c² with C^-1 = (e₁²)^½ c^-1C^§ . Here q is the angle subtended betweem c and e₁. We  can recover c=CC^§ ; q = 2 cos^-1(C_<0>) ; (cÙe₁)^~ = C_<2>^~ .
    C is not uniquely determined by c because CA where A is any unit rotor with Ae₁A^§=e₁ also serves. For N=3, A=(fe₁i^-1) incorporates a single arbitary phase angle but for N>3 we have more general guage rotations "about" e₁.
    C^· = ½c^-1 c^·Ce₁
[ Proof :   c^· = C^·e₁C^§ + Ce₁C^§· = C^·e₁C^§ + (C^·e₁C^§)^§ = 2(C^·e₁C^§)_<0;1;4;5;..> but c^· is a 1-vector so c^· = 2C^·e₁C^§ = 2Ce₁C^·§    Þ c^·Ce₁ = 2C^·  .]

    Since c = |c²|^½ = CC^§ we have c^· = 2C^·_*C^§ .
    In GHC we have S = (1+½e_¥Ce₁C^§)R = C(c^-1+e_¥1)C^§R |e₁C^§)R = c^-1C(1+ce_¥1)C^§R   =   c^-1C(ce_¥1)^↑C^§R   =   C^~(ce_¥1)^↑C^~§R

    We can define monotonically increasing spinor time s = ò₀^t dt r^-1 regardable as an "temporally accummulated closeness measure" of c to 0 with ds/dt=c^-1 so that dC/ds º C^'   =   C^· dt/ds = cC^· = ½c^·Ce₁ .
    Whence C^'' = ½(c^··c + ½c^·2)C .
[ Proof : C^'' = cC^'· = ½c(c^·Ce₁)^·   =   ½c(c^·Ce₁)^· = ½c(c^··Ce₁ + c^·C^·e₁) = ½c(c^··Ce₁ + ½c^-1c^·2Ce₁²) = ½(cc^··Ce₁C^-1 + ½c^·2e₁²)C = ½(cc^··c^~ + ½c^·2e₁²)C = ½(c^··c + ½c^·2)C .  .]
    Now suppose that the acceleration has the form c^·· = a - m'c^-2c^~ for an attractive inverse square force plus peturbation a.
    We then have a geometric form of the Kustaanheimo-Stiefel equation
    C^'' = ½(ac - mc^-1 + ½c^·2)C which for a=0 has harmonic form C^'' = ½EC where energy E= ½c^·2 - mc^-1 is conserved .

    The key advantage to working with C(s) rather than c(t) is that solutions to the unpeturbed harmonic equation C^'' = ½EC linearly combine whareas solutions to Newtonian equation c^·· = -mc^-3c do not.
    We can reparameterise the orientation rotor R(t) as a function of spinor time R(s) with R^'(s) = cR^· .

    For E<0 corresponding to |c^·| less than escape speed (2mc^-1)^½ we have oscillatory solution C(0) cos((-½E)^½s) + C^·(0)(-½E))^-½ sin((-½E)^½) of frequency (-½E)^½ and spinor time period 2p(-½E)^-½ = 8p(-E)^-½ . Furthermore C(t) = cos( (-½E)^½ s) C₀ + sin( (-½E)^½ s) C₁ provides a solution with C(0)=C₀ and C(½p(-½E)^-½) = C₁ which is the natural constant energy trajectory from c(0) to c(T), having Cds(0) = (-½E)^½C₁ and Cds(½p(-½E)^-½) = -(-½E)^½C₀.
    The arrival time t= ò₀^½p(-½E)-½ ds r(s) is approximately ½p(-½E)^-½ r(0) for slow movement at large r and differs from the desired arrival time T.
    This suggests an alternte form of "initial conditions" in which we specify C₀=C(0), E, and one of C_±    specifying a "detemporalised" position passed through.
    A more general solution is  cosh(sbC₀ + b^-2 b  sinhsb (b) where b² = ½E . For a single N=3 particle, the angular momentum h₀ defines the orbit 2-plane so the motion is effectively 2D and it is natural to tahe b=¼|E|^½h₀^-1h₀ where h₀=|h₀²|^½ .

    Writing S(s) = (1+½e_¥Ce₁C^§)R = (1+½Ce_¥1C^§)R = R + ½Ce_¥1C^§R and noting that R^'' = (cR^·)^' = c(cR^·)^· = cc^·R^· + c²R^··
    R^' = cR^· = -½cwR.
    R^'' = cR^'· = -½cc^·wR -½ccw^·R +¼cw²R = -½c(c^·w+cw^·-½w²)

GHC spinor form

    A GHC variant of spinorising position in this way is to express a given U^N% 1-vector y=ls  dual to a hypersphere of positive squared radius s²=r²>0 as as a dilated rotatation of the unit hypersphere dual as Be₊B^§ where B=(lr)^½(½q(ye₊e₊)^~)^↑ , q being the angle subtended by s and e₊ , with cos(q) = r^-1s¿e₊ = ½r^-1(-1+c²-r²) and sÙe₊ = ce₊ - ½(1+c²-r²)e_¥0.
    Our spinor time is now s= ò₀^t dt(lr)^-1 with B^'' = ½(y^··y + ½y^·2)B .
    Now y^· = l(c^· + e_¥(cc^· - rr^·)) + l^·s     with y^·2 = lc^·2 + l^·2r² + l²(c^·¿c) , and
    y^·· = l(c^·· + e_¥(cc^·· + c^·2 - rr^·· - r^·2) + 2l^·(c^· + e_¥(cc^· - rr^·)) + l^··s     has e₀ coordinate l^·· and e_¥ coordinate l(cc^·· + c^·2 - rr^·· - r^·2) + 2l^·(cc^· - rr^·)) + ½l^··(c²-r²) .

    Taking l=1 to keep y within e⁰¿y=1 we have
    y^··y = s^··s = (c^·· + e_¥(cc^·· + c^·2 - rr^·· - r^·2)(c+e₀+ee½(c²-r²)) = c^··c + (e_¥c-1+e_¥0)(cc^·· + c^·2 - rr^·· - r^·2) + c^··e₀ + + c^··e_¥(½(c²-r²)

    y^·· = M(y)y requires l^··=M(y) and hence lc^·· + 2l^·c^· + ½l^··c = M(y)lc Þ c^··   =   ½M(y)c + 2l^-1l^·c^· .
    If we want mc^·· = m'(c)c^~ = c^-1m'(c)c we thus need M(y) = 2c^-1m'(c)  and hence l^· = ò₀^t dt   2c^-1m'(c)  = ò₀^c dc   (c^·)^-1 2c^-1m'(c)  =

    Setting l=r^-1 for unit y gives l^·=-r^-2r^· ; l^·· = 2r^-3r^· - r^-2r^·· = r^-2(2r^-1r^· - r^··)

s^··s = c^··c + c^··e₀ + ½c^··e_¥(c²-r²) +   (cc^·· + c^·2 - rr^·· - r^·2)e_¥(c+e₀).
    If r(t) is constant then s^·s = s^·Ùs is a pure 2-blade with (s^·s)² = -s^·2s² = c^·2r² .
    The radial acceleration assumption

Subbody motion

Gunturret

    Returning to our aircraft gun turret example, and assuming for generality that the turret can traverse within the aircraft as well as rotate we have W_4¬T4   =   W_4¬Q4 + (1+½e_¥c_Q4)W_Q¬T4 (1-½e_¥c_Q4)   =   W_4¬Q4 + (1+½e_¥c_T4)W_Q¬T4 (1-½e_¥c_T4)
      =   (1+½e_¥c_T4)_§( W_4¬Q4 + W_Q¬T ) .
[ Proof :  W_4¬T   =   -2S_4¬T^·S_4¬T^§   =   -2(S_4¬QS_Q¬T)^·(S_4¬QS_Q¬T)^§   =   -2(S_4¬Q^·S_Q¬T + S_4¬QS_Q¬T^·) S_Q¬T^§S_4¬Q^§
      =   W_4¬Q + S_4¬QW_Q¬TQS_4¬Q^§   =   W_4¬Q + (1+½e_¥c_Q4)R_4¬Q ((1+½e_¥c_TQ)W_Q¬TQ (1-½e_¥c_TQ) R_4¬Q^§ (1-½e_¥c_Q4)
      =   W_4¬Q + (1+½e_¥c_Q4) (1+½e_¥(c_T4-c_Q4))R_4¬Q W_Q¬TQR_4¬Q^§ (1-½e_¥(c_T4-c_Q4)) (1-½e_¥c_Q4)
      =   W_4¬Q + (1+½e_¥c_T4)R_4¬QW_Q¬TQR_4¬Q^§ (1-½e_¥c_T4)   =   W_4¬Q + (1+½e_¥c_T4)W_Q¬T4 (1-½e_¥c_T4)  .]
    The velocity of the turret is v_T4   =   v_Q4 + R_4¬Q§( v_TQ + c_TQ¿w_QQ ) while its spin is w_T4=R_4¬Q§(w_QQ+w_TQ) giving bivelocity
    W_T4 = (v_Q4 + R_4¬Q§( v_TQ + c_TQ¿w_QQ) )e_¥ + R_4¬Q§( w_QQ + w_TQ )   =   W_Q4 + R_4¬Q§( (v_TQ + c_TQ¿w_QQ )e_¥ + w_TQ ) .

    Here c_T is the position of the kinematical motion centre or attachment point of the turret rather than the centre of mass of the turret  considered in isolation. Similarly c_Q is typically a fixed reference "centre point" of the aircraft rather than its dynamical mass centre which may move with respect to c_Q due to fuel slosh, turret movement, and so forth.

    The instantaneous apparent bivelocity of the city 4 as percieved by the turret T is
    -W_TT   =   -R_T4^§_§(v_Q4)e_¥ - R_TQ^§( (v_TQ+c_TQ¿w_QQ )e_¥ + w_QQ + w_TQ )R_TQ .
[ Proof : W_TT   =   R_T4^§ W_T4 R_T4   =   (R_Q4R_TQ)^§ W_T4 (R_Q4R_TQ)   =   R_TQ^§ R_Q4^§ W_T4 R_Q4R_TQ
      =   R_TQ^§ R_Q4^§ ( v_Q4e_¥ + R_Q4§( v_TQ+c_TQ¿w_QQ )e_¥ + w_QQ + w_TQ) R_Q4R_TQ
      =   R_TQ^§R_Q4^§ v_Q4 R_Q4R_TQe_¥ + R_TQ^§ (v_TQ+c_TQ¿w_QQ )e_¥ + w_QQ + w_TQ) R_TQ  .]

Rotating Earth Frame

    Consider S_!¬,, the stance of the spinning Earth (,) with respect to the Sun (!). We will neglect the other planets and moons and assume the sun to be at rest at the centre of the solar system with the earth to orbitting it, rather than the more correct orbit of the sun-earth mass centre.
    To construct S_!¬, it is useful to form an intermediate heliocentric stance which is centered on the Earth as it orbits the sun but does not include the Earth's own daily spin. This is
    S_!¬,   =   (1+½e_¥R_,!De₁R_,!^§)R_,!   =   R_,!(1+½De_¥e₁) where D(t) is the distance from ! to ,   and R_,!=(½w_,te₁₂)^↑ where w_,=2p year^-1 . Thus the , frame has e₃ constant normal to the earth orbit plane in the solar north direction and e₁ always pointing directly away from the sun while S_,¬, = R_,¬,, is a Â₃ quaternian rotor having period one solar day.
    S_!¬,! = S_!¬,! S_,¬,, has positional velocity component as for S_!¬, and rotational velocity component R_!¬,!  =  R_!¬,!R_,¬,, with
    R_!¬,!^·   =   R_!¬,!^·R_,¬,, + R_!¬,!R_,¬,,^·   =   ½(w_!¬,!R_!¬,!R_,¬,, + R_!¬,!w_,¬,,R_,¬,,)
      =   ½(w_!¬,!R_!¬,! + R_!¬,! (R_!¬,!^§ w_,¬,! R_!¬,!) R_,¬,,)   =   ½(w_!¬,! + w_,¬,!)R_!¬,!
    Hence w_!¬,!   =   w_!¬,! + w_,¬,! as expected and
    w_!¬,,   =   w_!¬,, + w_,¬,,   =   R_!¬,!^§w_!¬,!R_!¬,! + w_,¬,, since w_,¬,, = w_,¬,, .

    w_!¬, º w_!¬,! is approximately constant providing the Earth's combined spin axis w_,! = w_!¬,e₁₂₃^-1 in the assumed inertial Sun frame. The Earth rotates about this axis with period one sidereal day. The sidereal day is shorter than the solar day (by approx 3.6 min) because the Earth spins in the same sense as it orbits the sun (clockwise as seen from below the solar south pole, looking toward the solar north pole along e_3!) , albeit about a spin axis inclined at approx 24 ^o to the orbit axis.  

Rigid Body Dynamics


Rigid Bodies
    Consider a set H of point-masses moving in Â^N, and suppose at time t the i^th particle has mass m_i, position r_i, and velocity v_i where i ranges through some indexing set I. Let m º å_im_i denote the total mass and c º m^-1 å_im_ir_i denote the instantaneous centre of mass at time t.
    The condition for these particles to form a rigid body is that v_i º r_i^· = v + (r_i-b)¿w     where 1-vector drift velocity v = b^· , 2-vector spin w, and spin centre point b may vary with t but are the same for each particle.
    In GHC we have (r_i-b+e₀)¿W = (r_i-b+e₀)×W = r_bi¿W = v_i + ((r_i-b)¿v)e_¥ where stance bivelocity W=ve_¥+w and r_bi º r_i-b + e₀ + ½(r_i-b)²e_¥ .
    Hence å_im_i(r_i-b+e₀)¿W = å_im_iv_i + m((c-b)¿v)e_¥ and we can eliminate the e_¥ compoenent by considering 2-blade velocity v_i º v_ie_¥ = v_iÙe_¥ = v_i×e_¥ = ((r_i-b+e₀)×W)×e_¥ .

Kinetic Energy

    The classical kinetic energy of a particle having velocity v_i and scalar mass m_i³0 is the scalar E_i = ½m_iv_i² , though this is actually a |v|<<c approximation to the relativistic E = m_ic²(1+(c^-1v_i)²)^½ - m_ic² = ½m_i(v_i² - ¼c^-2v_i⁴ + _O(c^-4v_i⁶)) where c denotes the speed of light.

    The net classical kinetic energy for a rigid body with drift velocity v rotating with spin w about b is thus
    E_w,v   =   å_i½m_i(v + (r_i-b)¿w)²   =   ½mv² + å_i½m_i((r_i-b)¿w)² + å_im_iv¿((r_i-b)¿w)   =   ½mv² + å_i½m_i((r_i-b)¿w)² + mv¿((c-b)¿w) .
    If w is a 2-blade with w²=-w² then the middle term is å_i½m_i(¯_w(r_i-b)w)²   =   -å_i½m_i(¯_w(r_i-b))²w²   =   ½I_w,bw² where scalar
    I_w,b º å_im_i(¯_w(r_i-b))² is the moment of inertia for the body about b "within" 2-blade w, regardable as spining about an "axis" of "direction" w passing through b . As usual, ¯_w(x) º (x¿w)w^-1 denotes projection into w.

    Thus an ND body spinning about its centre of mass (b=c) with 2-blade spin w has kinetic energy E = ½mv² - ½mI_ww² = ½m(v² ± (k_ww)²) according as w² = -/+w² ; where I_w º I_w,c and scalar radius of gyration k_w,b º (m^-1I_w,b)^½ is defined so that I_w,b = mk_w,b² .
    For N>3 and a general 2-vector spin w the å_i½m_i((r_i-b)¿w)² energy term is more complicated.

BiMomentum
    The linear momentum of a particle having position r_i Î Â^N and scalar mass m_i³0 moving with velocity v_iÎÂ^N at time t is just m_iv_i.
    Its 2-blade angular momentum about a point a, or line L={a+ln : lÎÂ} is h_a = (r_i-a)Ù(m_iv_i), or ((r_i-a)-(r_i-a)¿n)Ùm_iv_i respectively.
    The angular momentum of a system about a point or line is the sum of the angular momenta of its constituent particles about that point or line while the linear momentum of a system is the sum of the linear momenta of its constituent particles.
    We have the following two principles of Newtonian mechanics:

1. The rate of change of linear momentum of a system is equal to the sum of the external forces.
2. The rate of change of angular momentum of a system about: (a) a fixed point or line, or (b) a point or line moving with the centre of mass of the system, is equal to the total moment of external forces about that point or line.

    In GHC we have (r_aiÙe₀)×(v_ie_¥) = ((r_i-a)e₀+½(r_i-a)²e_¥0)×(v_ie_¥) = (r_i-a)Ùv_i + ((r_i-a)¿v_i)e_¥0 - ½(r_i-a)²v_ie_¥ = (r_i-a)Ùv_i + (((r_i-a)²)^· + (r_i-a)cnta^·)e_¥0 - ½(r_i-a)²v_ie_¥ where r_aiºr_i-a+e₀+½(r_i-a)²e_¥ and r_aiÙe₀ can be regarded as the bipoint {0,r_i-a}. We can eliminate the unwanted terms by forming 3-blade   rotational momentum m_i((r_aiÙe₀)×(v_ie_¥))Ùe_¥ = m_i(r_i-a)Ùv_iÙe_¥ but it is more natural to form 3-blade bimomentum m_i(r_i-a+e₀)Ùv_i = m_i((r_i-a+e₀)Ùv_i)Ùe_¥ = m_i((r_i-a)Ùv_i)e_¥ + m_iv_ie_¥0 containing both the rotational momentum m_i((r_i-a)Ùv_i)e_¥ about a and the linear momentum m_iv_ie_¥0 as 3-blades.
    A GHC body bimomentum is thus a 3-vector formed by the taking the outter product of an e₀^* 2-vector with e_¥,
    H_ia,b(W) º m_i(r_i-a+e₀)Ù((r_i-b+e₀)¿W)Ùe_¥   =   m_i( (r_i-a+e₀)Ù((r_i-b)¿w) + (r_i-a+e₀)Ùb^· )Ùe_¥   =   (h_ia,b(b^·,w)   + m_ie₀Ù((r_i-b)¿w+b^·) )Ùe_¥ embodying both angular momentum about a and linear momentum m_i((r_i-b)¿w+b^·).

  =   m_i( (r_i-c+e₀ + c-a)Ù((r_i-c+e₀ + c-b)¿W)Ùe_¥
      =   m_i( (r_i-c+e₀)Ù((r_i-c+e₀)¿W) + (c-a)Ù((r_i-c+e₀)¿W) + (r_i-c+e₀)Ù((c-b)¿W) + (c-a)Ù((c-b)¿W) )Ùe_¥
      =   m_i( (r_i-c+e₀)Ù((r_i-c)¿w) + (c-a)Ù((r_i-c)¿w) + (r_i-c+e₀)Ù((c-b)¿w) + (c-a)Ù((c-b)¿w)
    + (r_i-c+e₀)Ùb^· + (c-a)Ùb^· )Ùe_¥

Inertia Tensors


N-D Inertia 2-Multiform
    The net linear momentum of a rigid body of mass centre c spinning about b is m(v + (c-b)¿w) while its net angular momentum about a is
    h_a,b(v,w)   =   å_ih_ia,b(v,w)   =   å_ih_ia,bm_i(r_i-a)Ù((r_i-b)¿w)   =   å_i(m_i(r_i-c)²w - m_i(r_i-c)((r_i-c)¿w)) + m(c-a)Ùv + m(c-a)Ù((b-c)¿w)
      =   h_c(w) + m(c-a)Ù(v + (b-c)¿w) .
    The Inertia 2-multiform returning the angular momentum about the mass centre for the body due to a 2-vector spin w about its mass centre (a=b=c) is the linear function of w defined by
    h_c(w) º å_im_i(r_i-c)Ù((r_i-c)¿w)   =   å_im_i(r_i-c)×((r_i-c)×w)   =   å_im_i((r_i-c)²w - (r_i-c)((r_i-c)¿w)) .
    This third expression can be used to extend h_c over scalars with h_c(l) º lå_im_i((r_i-c)² .
    For 2-blade w we have h_c(w)   =   å_im_i(r_i-c)¯_w(r_i-c)w   =   (I_w +  å_im_i^_w(r_i-c)¯_w(r_i-c))w .
    For nondegenerate solid bodies with m_i>0 that extend into N₀ dimensions, linear h_c(w) is nonzero for every nonzero w and so invertible.
    h_c(w) is symmetric (self adjoint) in that c.h_c(w) = w.h_c(c) = h_c(c).w and has postive real eigenvalues for postive mass m_i>0.
    We naturally decompose w(t) = å_k=1^½N(N-1) f^k(t)f_k(t) where f_k(t) = R_§(¦_k(0)) are the unit principle spins.
    Letting D₀(c) = å_kI_k(c¿f^k(0))f_k we have h_c(c) = R_§(D₀(R_§^-1(c))) º R_§D₀R_§^-1(c) and h_c^-1(c) = R_§^-1(D₀^-1(R_§(c)))

    Because h_c is symmetric, the squared angular momentum h_c(w)² = w¿h_c²(w)   =   å_k=1^½N(N-1)  ±_k I_k² f^k2 while
    w¿h_c(w)   =   å_k=1^½N(N-1)  ±_k I_k f^k2   =   -å_im_i((r_i-c)¿w)²   =   -2E_w,0 provides the rotational kinetic energy. Here ±_k denotes the sign of f_k¿f_k .
[ Proof :  Symmetry follows from c.((r_i-c)Ù((r_i-c).w) = ((r_i-c).w).(c.(r_i-c)) = (c.(r_i-c)) . ((r_i-c).w) = ((r_i-c).c).(w.(r_i-c)) = w.((r_i-c)Ù((r_i-c).c)  .]

    The time derivative h_c^·(c)   =   h_c(w×c) - w×h_c(c) .
[ Proof : å_im_i((r_i-c)×((r_i-c)×c))^·   =   å_im_i((r_i-c)^·×((r_i-c)×c) + (r_i-c)×((r_i-c)^·×c)   =   å_im_i( ((r_i-c)×w)×((r_i-c)×c) - (r_i-c)×(c×((r_i-c)×w)))
      =   å_im_i( (r_i-c)×(w×(c×(r_i-c))) + ((r_i-c)×w)×((r_i-c)×c) - (r_i-c)×( w×(c×(r_i-c)) + c×((r_i-c)×w)) )
      =   å_im_i( (r_i-c)×( ((r_i-c)×c)×w + ((r_i-c)×c)×(w×(r_i-c)) - (r_i-c)×( w×(c×(r_i-c)) + c×((r_i-c)×w)) )
      =   å_im_i(-w×((r_i-c)×((r_i-c)×c)) + (r_i-c)×((r_i-c)×(w×c)))   =   -w×h_c(c) + h_c(w×c)  .]
    In the absence of external torques w thus satisfies Euler equation h_c^·(w) + h_c(w^·) = 0 Þ w^· = h_c^-1( h_c^·(w))
    Now w×h_c(w)   =   å_iå_j I_j fⁱ f^j f_i× f_j   =     =   å_i<j (I_j-I_i) fⁱ f^j f_i× f_j but it is more useful to exploit the conservation of h_c(w) with
    w^· = h_c^-1( -w×h₀)   =   h_c^-1( -å_i<j (I_j-I_i) fⁱ(t)f^j(0) f_i(t)×f_j(0) )
      =   -å_k I_k^-1 h_c^-1( -å_i<j (I_j-I_i) fⁱ(t)f^j(0) (f^k(t)¿ (f_i(t)×f_j(0) )) f_k
    Now (f^k(t)¿ (f_i(t)×f_j(0) ))   =   R_§(f^k(0))¿ (R_§(f_i(0))×f_j(0) ))   =   f^k(0))¿ R_§^-1(R_§(f_i(0))×f_j(0) )))   =   f^k(0))¿ R_§^-1(R_§(f_i(0)))× R_§^-1(f_j(0) )))   =   f^k(0))¿ (f_i(0)× R_§^-1(f_j(0) )))
    But w^· = -2R^··R^§ -2R^·R^·§ so w^·R = -2R^·· -2R^·R^·§R so .... ????