These equations don’t make sense dimensionally. Are there supposed to be constants of proportionality that aren’t being mentioned? Are they using the convention c=1? Well, I doubt it’s relevant (scaling things shouldn’t change the result), but...
Edit: Also, perhaps I just don’t know enough differential equations, but it’s not obvious to me that a curve such as he describes exists. I expect it does; it’s easy enough to write down a differential equation for the height, which will give you a curve that makes sense when r>0, but it’s not obvious to me that everything still works when we allow x=0.
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.
Similar, for those who enjoyed discussing this problem: Did you know that Newtonian mechanics is indeterministic?
These equations don’t make sense dimensionally. Are there supposed to be constants of proportionality that aren’t being mentioned? Are they using the convention c=1? Well, I doubt it’s relevant (scaling things shouldn’t change the result), but...
Edit: Also, perhaps I just don’t know enough differential equations, but it’s not obvious to me that a curve such as he describes exists. I expect it does; it’s easy enough to write down a differential equation for the height, which will give you a curve that makes sense when r>0, but it’s not obvious to me that everything still works when we allow x=0.
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.