- #1
MadRocketSci2
- 47
- 1
I have a question about calculus of variations that is driving me absolutely nuts right now:
I have followed the standard derivation of differential equations from the extrimization of a functional S = ∫(F(x,dx/dt,t)dt)
By doing some manipulation involving an arbitrary perturbation to your function, you end up with the folowing differential equation which has to be zero
dF/dx - d/dt(dF/d(dx/dt)) = 0;
I can sort of follow this, but I want to be able to derive this from the local properties of the function being integrated.
If the path taken is independent of the interval over which the functional is integrated (and it better be!) then there should be a way to arrive at the differential equation from nothing but the local gradients in F.
I have been trying to do this, and have encountered no end of trouble. My current line of reasoning is as follows:
dS/dt = F; We want to extremize dS/dt at every point t in order to extremize S overall.
d (dS/dt) over a specific interval dt = 0;
d (dS/dt) = dF/dx*dx + dF/d(dx/dt) * d(dx/dt) = 0;
The task then is choosing d(dx/dt) over the interval such that d (dS/dt) = 0;
dx over the interval must be dx/dt*dt or some combination of the derivatives of x in time.
d(dx/dt) is the acceleration d^2x/dt^2 * dt;
divide the time interval out.
dF/dx * xdot = dF/dxdot * xdotdot;
The strange thing about this is that it works for some specific problems, but does not yield equivalent results in general. I am getting the same answers for the principle of least action and the brachistochrone problem that I should be getting. But in other more general problems, the hypothetically minimized functions are slightly off for some reason.
Also, I am getting mysterious sign changes in certain terms (such as in the brachistochrone problem derivation).
Does anyone know where I might be going wrong? Do you see what I am trying to do?
I have followed the standard derivation of differential equations from the extrimization of a functional S = ∫(F(x,dx/dt,t)dt)
By doing some manipulation involving an arbitrary perturbation to your function, you end up with the folowing differential equation which has to be zero
dF/dx - d/dt(dF/d(dx/dt)) = 0;
I can sort of follow this, but I want to be able to derive this from the local properties of the function being integrated.
If the path taken is independent of the interval over which the functional is integrated (and it better be!) then there should be a way to arrive at the differential equation from nothing but the local gradients in F.
I have been trying to do this, and have encountered no end of trouble. My current line of reasoning is as follows:
dS/dt = F; We want to extremize dS/dt at every point t in order to extremize S overall.
d (dS/dt) over a specific interval dt = 0;
d (dS/dt) = dF/dx*dx + dF/d(dx/dt) * d(dx/dt) = 0;
The task then is choosing d(dx/dt) over the interval such that d (dS/dt) = 0;
dx over the interval must be dx/dt*dt or some combination of the derivatives of x in time.
d(dx/dt) is the acceleration d^2x/dt^2 * dt;
divide the time interval out.
dF/dx * xdot = dF/dxdot * xdotdot;
The strange thing about this is that it works for some specific problems, but does not yield equivalent results in general. I am getting the same answers for the principle of least action and the brachistochrone problem that I should be getting. But in other more general problems, the hypothetically minimized functions are slightly off for some reason.
Also, I am getting mysterious sign changes in certain terms (such as in the brachistochrone problem derivation).
Does anyone know where I might be going wrong? Do you see what I am trying to do?