kill(all)$

motion stays in one plane (angular momentum) 'version 1.0'
peter.vlasschaert@gmail.com,13/06/2018

motion stays in the plane : m2 orbit m1,m1 orbit m2
= orbital plane.
result : angular momentum must be constant.

m1 : object with mass m1
m2 : object with mass m2
Ø  : flight path angle (see : rocket,missile examples )
vector(r)  : connecting m1 to m2
vector (v) : velocity vector ( for m2) 
local horizontal : perpendicular ( = 90°= 'π/2') to vector(r)
! local horizon = local horizontal.
local vertical   : perpendicular to local horizontal.
fig :smallest angle between(r and v) = π/2+Ø.
rem : object = 'satellite,other planet,missile...)
missile : examples ( m1=earth and m2=missile=rocket)

How to find : Angular momentum :( = pecific angular motion)
vector(h)=vector(r)x vector(v) = constant ?  
x = cross product = vector product = outer product
rem : calculation cross product gives vector perpendicular to plane.
rem : calculation cross product ( = see code trajectories of missile website).

1e) How to find angle between( vector(r) and vector(v) )

general : angle between any two vectors can be find scalar product.
          (other names = dot product = inner product)

rem : angle between two vectors :θ
=> relation figure 1

rem : 'm2' point fig 1    : corresponding to 'O' point fig 2
rem : vector (v) in fig 1 : corresponding to 'vector(A)' in fig 2
!!!!! very important : 'vector (r)' must be 'negative '.
reason : to get corresponding   θ = Ø + π/2
reason : (-1)* innerproduct ' multiplication by (-1)'

p1:(-1)*innerproduct = r*v *cos(Ø+%pi/2);

p2:p1*(-1);

rem :cos(α +β) ? expand 
====================

p3:cos(α+β);
p4:''p3=trigexpand(p3);

'' : to copy  : p3

l1:α =Ø;
l2:β =%pi/2;

p5:''lhs(p4)=sublis([l1,l2],rhs(p4));

?  How to find outer product( = cross product = vector product)

!!!!!!   cos : inner product  =  sin : outer product
         sin  : inner  product  =  cos: outer product

example :  inner product  : sin(Ø) => corresponding outerproduct : cos(Ø)

p6:outerproduct = subst([sin=cos],rhs(p2));

orbital mechanics :  outerproduct =h

p7:subst([outerproduct = h],p6);

rem : calculation of 'h'
           inertial coordinate system :(coordinate system which have no acceleration)
           given : position vector ( components vector)
                         velocity vector  (components  vector)
          result : vec(h) =vec(r)xvec(v), see : missile ,rocket examples website 
          rem : 3*3 determinant  = > 3 components => pythogareon theorema (abs value)
                    use p7
           given : velocity , distance from earth , flight path angle :Ø
                    use p7

rem : calculation first : h  put  in the polar equation ,when you known :e
           (see : missile and rocket examples website)

rem : polar equation of a conic section :  ellipse,circle,hyperbola,parabola
           orbit ellipse : study

polar equation : orbit, calculus course.

p8: R=p/(1+e*cos(θ));

rem : e values :   e = 0        :circle
                                0 < e < 1 :ellipse
                                e = 1        :Parabola
                                e >  1       :Hyperbola

rem : p can be find  :θ =π/2,R=p

p9: p=h^2/μ;

p9: proof  ! not easy to understand.

see website : μ (missile,rocket examples) = constant = G*M_e
                          p (missile,rocket examples) 'missile trajectories'
G :gravitation cte (= Newton Second Law)
M_e : Mass of the earth.

p9a : [μ =M_e*G];

p10:subst([p9],p8);

subst : h from 'above' for orbit plane ( = orbit stays in one plane)

rem :  θ  = angle (FP and r)

p11:subst([p7],p10);

insert : μ 'p11'

p12:subst([p9a],p11);

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