kill(all)$

load(lrats)$

total energy orbit in general. 'version 1.0'
peter.vlasschaert@gmail.com,16/07/2018.

kinetic energy    :

p1: E_k = 1/2*m*v^2
;

potential energy :

p2:E_p=-G*M*m/r;

total energy of orbit  :

p3:E=E_k+E_p;
p3a:E = rhs(p1)+rhs(p2);

Energy circular orbit :

How to find : only radial component .
=>   centripetal force = newton second law 
rem : - centrifugal force      = -m*v^2/r =centripetal force.
rem : newton second law  = -G*M*m/r^2

p4:F_c =-m*v^2/r ;
p4a:F_ns = -G*M*m/r^2;

To stay a circular orbit.
===================

p4b: rhs(p4)=rhs(p4a);

p4c:p4b*(-1);

orbital velocity : v= v_c  ( c: circular)
------------------------------------------------------

p5:solve(p4c,v)[2]$

p6:subst([v=v_c],p5);

total energy for circular orbit.
-------------------------------------------

p6:lhs(p3a)=subst([p5],rhs(p3a));

p6a:lratsubst([G*M = μ],p6);

Energy :  per unit mass

p6b : E=rhs(p6a)/m;

rem : r : radius of circular orbit

result : total energy of the system is negative.
general suppose : closed orbit gives negative total energy.

elliptical orbit:  a> b
a : semi minor axis
b:  semi majar axis
=================

Total Energy per unit mass of elliptical orbit.

p7:subst([r=a],p6b);

parabola, hyperbola  ,  r  =>  ∞

1e) general E : per unit mass

p8:lhs(p3a)=rhs(p3a)/m$
p8a:expand(p8);

2e )  GM = μ  into 'p8a'

p8b:subst([G*M = μ],p8a);

3e)  r => ∞

p9: E_∞ =limit(rhs(p8b),r,inf);

Total energy of a parabola ,hyperbola can't be negative.
=>  when reach infinity has still some kinetic energy : 'p9'.

4e )  v_inf  ?  

p10:solve(p9,v)[2];

p10a:([v=v_∞],p10);

result :  total Energy  E> = 0 (hyperbola,parabola)
E = 0  , parabola.
E > 0  , hyperbola.

rem : parabola  E = 0

p11:subst([E=0],p8b);

p11a:solve(p11,v)[2];

 r-> ∞  into  'p11a', v_∞ =0, 

p11b:v_∞=limit(rhs(p11a),r,inf);