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I'm working on a physics "potential" problem and trying to create an alternate function to describe the potential energy. I'm having trouble figuring out how to solve a nonlinear ODE, or even a limiting boundary for minimizing a drop off shape function.

I was able to reduce my problem to the following first order nonlinear ODE relationship between two real and positive valued functions. The functions [itex] f(\theta,\gamma) , s(\theta,\gamma)[/itex] are purely of a single variable [itex]\theta[/itex] and a single constant [itex]\gamma[/itex] .

[itex](f(\theta))^2 + ({ d \over {d \theta}} f( \theta ))^2 = (s(\theta))^2[/itex]

Where [itex]s:\mathbb{R} \mapsto \mathbb{R} [/itex] is usually a known shape function with physical significance.

I do not know how to solve for an 'f' given an 's'. But, with the help of numerical analysis and graphing, I was able to figure out that if I can solve for a restricted f:[itex]\theta \in [{\pi \over 2}, \pi ][/itex] that I can extend the solution to all other values of theta.

Boundary conditions for the restricted region:

I've been trying to put bounds on 'f' to narrow down the possible shapes of 'f' as it traverses from a known starting value of [itex]f({\pi \over 2})=\gamma[/itex] to an ending value of [itex]f(\pi)=\gamma^{-2}[/itex]. Where gamma is a constant: [itex]\gamma \ge 1[/itex]. (eg: Related to Lorentz contraction.)

's' is not equal to 'f' in general, but 's' does start and end with the same value as 'f'.

I know from graphing solved test cases that 'f' and 's' must both be monotonically decreasing with a few point exceptions where the slope goes to zero.

The three 'f' exceptions are at [itex]\theta[/itex] of [itex]{\pi \over 2} , { {3 \pi} \over 4}, {\pi}[/itex]

The two 's' exceptions are at [itex]\theta[/itex] of [itex] {\pi \over 2}, \pi [/itex]

Both f and s are smooth; I think that both f, s and the first and second derivatives of each are always continuous.

There are two kinds of problems I am trying to solve.

The first kind of problem is about techniques to solve for 'f' in terms of a given 's'

The second kind of problem is about extreme decay rate of the 's' functions that can be produced. The extrematization problem can ignore all zero slope inflection points except the one at [itex]\theta={\pi \over 2}[/itex]. I am just trying to find a boundary function 'f' that can produce an arbitrary 's' with a maximum rate of descent.

eg: The functions go from zero slope at [itex]s({\pi \over 2})=f({\pi \over 2})=\gamma[/itex] to whatever negative slope of 'f' produces the maximum shrinking of 's')

If 'f' descends too rapidly, the derivative squared becomes increasingly positive and makes the magnitude of s's slope smaller. eg: It's a form of negative feedback. There's a sweet spot where the curvature & slope of 'f' makes 's' decrease the quickest. I'm unsure how to formulate the problem, exactly; but intuitively it might also be discovered by something like an 'f' wich produces an 's' with minimum 's' integral over region [itex]\theta=( \pi/2 : pi )[/itex]

I was able to figure out that there are two crude function family boundaries which if 'f' is either greater in magnitude or more negative in slope, 's' will grow instead of shrink.

A crude border for maximum f magnitude is [itex]f = \gamma [/itex]

A crude border for maximum f slope is [itex]f = \gamma cos( \theta )[/itex]

Plugging either border into [itex]y=(s(\theta))^2=f^2 + f_\theta^2[/itex] gives a constant.

If the slope of 'f' is made more negative than the cosine boundary, y will be larger than it's initial value which is unacceptable. I think this means that [itex] -\sqrt {\gamma^2-f^2} \le f_\theta \le 0 [/itex]

However, I wasn't able to figure out what function 'f' produces the most steeply decaying s(). I'm unsure how to go about setting up the problem.

Any help would be appreciated.

I was able to reduce my problem to the following first order nonlinear ODE relationship between two real and positive valued functions. The functions [itex] f(\theta,\gamma) , s(\theta,\gamma)[/itex] are purely of a single variable [itex]\theta[/itex] and a single constant [itex]\gamma[/itex] .

[itex](f(\theta))^2 + ({ d \over {d \theta}} f( \theta ))^2 = (s(\theta))^2[/itex]

Where [itex]s:\mathbb{R} \mapsto \mathbb{R} [/itex] is usually a known shape function with physical significance.

I do not know how to solve for an 'f' given an 's'. But, with the help of numerical analysis and graphing, I was able to figure out that if I can solve for a restricted f:[itex]\theta \in [{\pi \over 2}, \pi ][/itex] that I can extend the solution to all other values of theta.

Boundary conditions for the restricted region:

I've been trying to put bounds on 'f' to narrow down the possible shapes of 'f' as it traverses from a known starting value of [itex]f({\pi \over 2})=\gamma[/itex] to an ending value of [itex]f(\pi)=\gamma^{-2}[/itex]. Where gamma is a constant: [itex]\gamma \ge 1[/itex]. (eg: Related to Lorentz contraction.)

's' is not equal to 'f' in general, but 's' does start and end with the same value as 'f'.

I know from graphing solved test cases that 'f' and 's' must both be monotonically decreasing with a few point exceptions where the slope goes to zero.

The three 'f' exceptions are at [itex]\theta[/itex] of [itex]{\pi \over 2} , { {3 \pi} \over 4}, {\pi}[/itex]

The two 's' exceptions are at [itex]\theta[/itex] of [itex] {\pi \over 2}, \pi [/itex]

Both f and s are smooth; I think that both f, s and the first and second derivatives of each are always continuous.

There are two kinds of problems I am trying to solve.

The first kind of problem is about techniques to solve for 'f' in terms of a given 's'

The second kind of problem is about extreme decay rate of the 's' functions that can be produced. The extrematization problem can ignore all zero slope inflection points except the one at [itex]\theta={\pi \over 2}[/itex]. I am just trying to find a boundary function 'f' that can produce an arbitrary 's' with a maximum rate of descent.

eg: The functions go from zero slope at [itex]s({\pi \over 2})=f({\pi \over 2})=\gamma[/itex] to whatever negative slope of 'f' produces the maximum shrinking of 's')

If 'f' descends too rapidly, the derivative squared becomes increasingly positive and makes the magnitude of s's slope smaller. eg: It's a form of negative feedback. There's a sweet spot where the curvature & slope of 'f' makes 's' decrease the quickest. I'm unsure how to formulate the problem, exactly; but intuitively it might also be discovered by something like an 'f' wich produces an 's' with minimum 's' integral over region [itex]\theta=( \pi/2 : pi )[/itex]

I was able to figure out that there are two crude function family boundaries which if 'f' is either greater in magnitude or more negative in slope, 's' will grow instead of shrink.

A crude border for maximum f magnitude is [itex]f = \gamma [/itex]

A crude border for maximum f slope is [itex]f = \gamma cos( \theta )[/itex]

Plugging either border into [itex]y=(s(\theta))^2=f^2 + f_\theta^2[/itex] gives a constant.

If the slope of 'f' is made more negative than the cosine boundary, y will be larger than it's initial value which is unacceptable. I think this means that [itex] -\sqrt {\gamma^2-f^2} \le f_\theta \le 0 [/itex]

However, I wasn't able to figure out what function 'f' produces the most steeply decaying s(). I'm unsure how to go about setting up the problem.

Any help would be appreciated.

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