I have to derive the equations of motion describing a planet or comet orbiting the sun, then determine what quantity is conserved based upon those equations.
Bold denotes a vector, bold i and j are unit vectors.
Mass of comet = m
Mass of sun = M
Gravitational potential energy U = -GmM/r, G is the gravitational constant.
Position vector of comet r = rcos(α)i + rsin(α) j
Now, my plan was to find the kinetic energy, then use that to find the Lagrangian and then use the Euler-Lagrange equation to find the equations of motion.
Kinetic energy: Obviously I need to have the velocity dr/dt = ds/dt. My calculus skills are not quite up to par on that, but I was fortunate enough to have access to a formula that I believe works:
ds^2 = dr^2 + r^2dα^2
(ds/dt)^2 = (dr/dt)^2 + r^2 (dα/dt)^2
But I have two questions:
1) I'm not sure how to derive this from the position vector and the standard x = rcos(α) and y = r sin (α). If anyone can show me, that would be appreciated.
2.) When working with differentials, we can just take ds^2 = dr^2 + r^2 dα^2 and divide it all by dt^2? This is what appears to have happened.
Anyway, based upon that initial assumption:
T = (1/2)mv^2 = (1/2) m ((dr/dt)^2 + r^2 (dα/dt)^2)
Now, then therefore the Lagrangian is:
L = T - U
L = (1/2) m ((dr/dt)^2 + r^2 (dα/dt)^2) - (-GmM/r)
Now, I use the Euler-Lagrange equation on this, once for r and once for α: r' and α' are the time derivatives in this case, since I can't use dot notation here.
∂L/∂r' = mr' + 0 + 0
∂L/∂r = 0 + mr(α')^2 - GmM/r
d/dt (∂L/∂r') = mr''
md^2r/dt^2 - mr(α')^2 + GmM/r = 0
GM/r^2 = r(α')^2 - r''
∂L/∂α' = 0 + m(r^2)α' + 0
∂L/∂α = 0 + 0+ 0
d/dt(∂L/∂α') = d/dt (m(r^2)α')
d/dt (m(r^2)α') - 0 = 0
d/dt (m(r^2)α') = 0
Looking at this one, what's inside the outer parenthesis must be constant (because the derivative of it with respect to time is zero), so I am assuming that, unless m, r, α' is zero (which is useless), at least one of m, r or α' must be constant. Clearly r can change with time if the orbit isn't a perfect circle, so I am assuming that mα' is constant with respect to time. Is this not saying that angular momentum is conserved? I mean, if the "change in angular momentum with respect to time" is zero, is this not saying that it is constant with respect to time- i.e. it is conserved? Is this correct? (I would expect as much)
Anyway, are these correct? And can anyone help with the first part so this is a complete derivation?