so i want to derive the O(1) algorithm of earth going around the sun(i mean a formula to calculate a smol object orbiting a large object,and only the smol object can get gravity from the large object,and i dont mean something like x²+y²=r² for circle orbits(it has to do with gravity etc,and i want to derive it not graph it and find the pattern(circle)))
so gravity is a function of position(and some constants like gravity constant and big object mass and smol object mass)
gravity sums up(∫) to velocity
velocity sums up to position
but there is a problem!(ik how to do ∫∫s)i see that the pre-integral function has something to do with its integral's integral!
i thought that for a day and did that ∫∫ so i ended up with(i used complex numbers not vectors cuz the |v| function is undifferentiable)
gravity constant*mass(smol object)mass(big object)(real part of position of smol object to the third power+im.p.o.p.o.s.o to the 3rd power times i)*square root of(re.p.o.p.o.s.o to the 4th power+im.p.o.p.o.s.o to the 4th power),and this should end up with re.poposo+im.poposoi
the problem is that i have no idea with how to root this(it has a degree of 3+3+4+4=14!!!)

edit:DIDNT U LISTEN TO PHYSICS CLASS THE GRAVITY FORMULA GIVES U A FORCE NOT AN ACCELERATION YOU 1/8BYTECH
---physics teacher
edit edit:dividing by mass is just multiplying a constant so its no difference

i have all the gravity equations in that spaceship project you've seen
f=G(m_1 * m_2)/d
f is force, d is distance
f=ma
a*m_1=G(m_1 * m_2)/d
a=G(m_2)/d
acceleration = G * mass of other object / distance

....and then take its 2nd integral and solve for position(wait it should be a=Gm/r ^2)
so a=Gm/(x^2+y^2)
factor out constants
GmS(x^2+y^2)^-1
take the reciprocal rule(and addition rule)
Gmln(Sx^2+Sy^2)
power rule
Gmln(1/3(x^3+y^3))=a
integrate again
GmSln(1/3(x^3+y^3))
ln rule xln(x)-x(too much xs so take that out)
a=Gm(ulnu-u)
u=1/3Sx^3+1/3Sy^3
power rule
u=1/12(x^4+y^4)
put u back
a=Gm(1/12(x^4+y^4)ln(1/12(x^4+y^4))-1/12(x^4+y^4))
constant of integration
a=Gm(1/12(x^4+y^4)ln(1/12(x^4+y^4))-1/12(x^4+y^4))+C
$$a=1/12Gm((x^4+y^4)ln(1/12(x^4+y^4))-1/12(x^4+y^4))+C$$

i don't understand anything you mean by whatevers after factor out constants
those aren't equations anymore and it's not clear what S is supposed to be, how any of the things you're putting there are useful, or what your goal is with the end of that

ill be honest all i could tell from your original post is that it's a massive block of text with something about gravity, i can't understand any of it

Then simplify it and move (x,y)to the left
x^4+y^4=(x²+√2xy+y²)(x²-√2xy+y²)
a=Gm(1/12(x²+√2xy+y²)(x²-√2xy+y²)ln((x²+√2xy+y²)(x²-√2xy+y²)))-1/12(x²+√2xy+y²)(x²-√2xy+y²))
(the constant of integral here means the absolute position of the sun and the starting position of the earth so its irrelavent)

i can do calculus just fine, i've done it for physics before. i just can't get through massive blocks of text. i can't tell what's where and get lost
id probably do better with images of math equations made with latex or something

i've also just never dealt with gravity equations before that one space project so there's nothing really familiar to look at

k
$$a=Gm(1/12(x²+√2xy+y²)(x²-√2xy+y²)ln((x²+√2xy+y²)(x²-√2xy+y²)))-1/12(x²+√2xy+y²)(x²-√2xy+y²))$$
it doesnt change much aside some 1/12s,cuz its not simplifyed at all

well this is basically the same as my answer,but organized so the ks are out and not duplicate :)(the ?s are weird and not intended)
wait u have lost a Gm
the last 1/12 k should get multiplied by Gm

i'll probably try and read this over for that space ship project because i never did the deltatime calculations for gravity and it's bugging me quite a bit