These equations don’t make sense dimensionally. Are there supposed to be constants of proportionality that aren’t being mentioned?
That is my guess. The simplest way IMO would be replace the g in eq. 1 by a constant c with units of distance^(-1/2). The differential equation becomes r″ = g c r^1/2, which works dimensionally. The nontrivial solution (eq. 4) is correct with an added (c g)^2 in front.
it’s not obvious to me that a curve such as he describes exists.
I’m not sure what you mean here. What could be wrong in principle with a curve h = c r^3/2 describing the shape of a dome, even at r = 0?
Notice “r” here does not mean the usual radial distance but rather the radial distance along the curve itself. I don’t see any obvious barrier to such a thing and I expect it exists, but that it actually does isn’t obvious to me; the resulting differential equation seems to have a problem at h=0, and not knowing much about differential equations I have no idea if it’s removable or not (since getting an explicit solution is not so easy).
You piqued my curiosity, so I sat to play a little with the equation. If R is the usual radial coordinate, I got:
dh/dR = c y^(1/3) / [(1 - c^2 y^(2/3))^(1/2)]
with y = 3h/(2c), using the definition of c in my previous comment. (I got this by switching from dh/dr to dh/dR with the relation between sin and tan, and replaciong r by r(h). Feel free to check my math, I might have made a mistake.) This was easily integrated by Mathematica, giving a result that is too long to write here, but has no particular problem at h = 0, other than dR/Dh being infinite there. That is expected, it just means dh/dR = 0 so the peak of the dome is a smooth maximum.
That is my guess. The simplest way IMO would be replace the g in eq. 1 by a constant c with units of distance^(-1/2). The differential equation becomes r″ = g c r^1/2, which works dimensionally. The nontrivial solution (eq. 4) is correct with an added (c g)^2 in front.
I’m not sure what you mean here. What could be wrong in principle with a curve h = c r^3/2 describing the shape of a dome, even at r = 0?
Notice “r” here does not mean the usual radial distance but rather the radial distance along the curve itself. I don’t see any obvious barrier to such a thing and I expect it exists, but that it actually does isn’t obvious to me; the resulting differential equation seems to have a problem at h=0, and not knowing much about differential equations I have no idea if it’s removable or not (since getting an explicit solution is not so easy).
You piqued my curiosity, so I sat to play a little with the equation. If R is the usual radial coordinate, I got:
dh/dR = c y^(1/3) / [(1 - c^2 y^(2/3))^(1/2)]
with y = 3h/(2c), using the definition of c in my previous comment. (I got this by switching from dh/dr to dh/dR with the relation between sin and tan, and replaciong r by r(h). Feel free to check my math, I might have made a mistake.) This was easily integrated by Mathematica, giving a result that is too long to write here, but has no particular problem at h = 0, other than dR/Dh being infinite there. That is expected, it just means dh/dR = 0 so the peak of the dome is a smooth maximum.