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Author	Topic: The Dome
Chaon Moderator	posted 05-21-2008 11:06 PM A Simple Violation of Determinism in Newtonian Mechanics. Read it twice. It makes my brain hurt. Please help. IP: Logged
SPQR Moderator	posted 05-22-2008 12:07 AM There is some bullshit in there, where he creates this claim that the mass rests at the apex for an arbitrary time before moving off. IP: Logged
Victor Member	posted 05-22-2008 11:04 PM I am going to have to check a text book on differential equations. I seem to remember this type of equation includes an arbitrary constant in the solution function r(t). This would correspond to an initial non-zero velocity. IP: Logged
Victor Member	posted 05-29-2008 08:43 PM It looks like the old division by zero trick. See http://www.sosmath.com/diffeq/diffeq.html for additional details to the following derivation. The equation d^2r/dt^2 = r^1/2 is a second order differential equation. Fortunately, the variable t is not part of the right side of the equation. This allows us to introduce a second variable, v = dr/dt. When we do this, we now have a set of equations. dr/dt = v dv/dt = r^1/2 The variable t can be eliminated from the second equation using the formula dv/dt = dv/dr dr/dt = dv/dr v or v dv/dr = r^1/2 or v dv = r^1/2 dr or integrating v^2/2 = 2/3 r^3/2 + C or dr/dt = v = (4/3 r^3/2 + C)^1/2 The last equation can be rearranged and integrated to solve for r in terms of t. (4/3 r^3/2 + C)^-1/2 dr = dt This is where division by zero comes in. "Rearranging" really meant dividing both sides of the equation by (4/3 r^3/2 + C)^1/2. This term is zero when r and C are both zero. As it turns out, these conditions correspond to the case where the point mass is at rest at the top of the dome. What this means is that the solution given does not apply to the point r=0, dr/dt=0. It does apply at infinitesimal distances from the top or when there is a non-zero velocity. But it does not apply when the point mass is motionless at the top of the dome. IP: Logged
Chaon Moderator	posted 05-29-2008 09:32 PM Thank you Victor! IP: Logged

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