kill(all)$

How to understand     :                motion equation 3D ,'version 1.0'
(initial reference system)
peter.vlasschaert@gmail.com,04/06/2018

rem :initial reference system (=frame ) ≠ rotating system( =frame)
======================================================

How get this : taylor expansion about  ( x=0 )     'ref'
Menu: Calculus:Get series
Expression : f(x)
Variable       : x
Point             : 0
Depth            :4

rem : 2=2!,6=3! ......

p1:'taylor(f(x), x, 0, 4) =taylor(f(x), x, 0, 4);

ref  => ' see above ', Point  : a  =>  (' Taylor expansion around "a" ')

p1a:'taylor(f(x), x, a, 4) =taylor(f(x), x, a, 4);

differentation f(x) with respect  'x'
===========================
Mechanics : 'differentation f(x)  to ' t ' :( t = time , x=space (1D = length))'
===========================================================

p2:ff(x,Δx,n) := sum(diff(f(x),x,i)/i!*(Δx)^i,i,0,n);

g(x,Δx,n) = f(x+Δx,n) =f(x)+diff(f(x),x,1)/1!*Δx+..........+diff(f(x),x,n)/n!*(Δx)^n
rem : representation 'p1' 

p3:ff(x,Δx,4);

p4:expand(p3);

Mechanics : construct other function , time involved.
============================================

pm(x,t,q,n):=sum(diff(f(x,t),t,i)/i!*(q)^i,i,0,n);

We have truncate : taylor expansion ,why ?
-----------------------------------------------------------------

p5:pm(x,t,q,2);

Answer  : see 'p5' f=f(x,t)
 f=f(x,t)             = position            (units ,m)
 df/dt                = velocity             (units,m/s)      
 d^2(f)/dt^2    = acceleration     (units,m/s^2)

acceleration :  symbol  'a'
=====================

p6:part(p5,1);

p6a:coeff(p5,diff(f(x,t),t,2));

p6b:a =1/p6a*p6;

velocity      :  symbol  'v'
====================

p7:part(p5,2);

p7a:coeff(p5,diff(f(x,t),t,1));

p7b:v=1/p7a*p7;

position    : symbol 's0', initial position.
================================

s : will be some ' other position in 'space'  '

p8:part(p5,3);

p8a:coeff(p5,diff(f(x,t),t,0));

p8b:s0=1/p8a*p8;

result : position = s
=================

p9:s=p6a*lhs(p6b)+p7a*lhs(p7b)+p8a*lhs(p8b);

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rem  : q*v = units 'm'  .  q*m/s = m
q: must be '  time = t '
---------------------------------------------------

p10:subst([q=time],p9);

p11:subst([q=t],p9);

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assume  3D  : s=s(sx,sy,sz),v=v(vx,vy,vz),a=a(ax,ay,az)
                           s0=s0(s0x,s0y,s0z)
==============================================

x compenents :
-----------------------

p12a :sublis([s=sx,s0=s0x,v=vx,a=ax],p11);

y compenents :
-----------------------

p12b :sublis([s=sy,s0=s0y,v=vy,a=ay],p11);

z compenents :
-----------------------

p12c :sublis([s=sz,s0=s0z,v=vz,a=az],p11);

xy plane , z axis above : gravitation 'downwards'  to 'xy  plane'.
**********************************************************************************************
only acceleration : gravity field earth ( negative sign in front)
                                                                       ----------------------------------
a = a(ax,ay,az) =a(0,0,-g)

rem : when no external force exit.
                      ===============

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x,y,z components,gravity field of the earth
===================================

p13a:sublis([ax=0],p12a);
p13b:sublis([ay=0],p12b);
p13c:sublis([az=-g],p12c);

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