Deriving the elastica 1

[StevenKatzeff]

Hi.

I would like to derive the classic Bernoulli/Newton elastica curve from first principles.

Bernoulli's/Newton's classic elastica equation is of the form:

1) curvature = dθ/ds

The RHS reforms to the well known elastica equation:

2) dy²/d²x/(1+(dy/dx)²)^3/2

But how do I get from 1) to 2)?

Here are the initial steps I took with no success:


1)ds= √dy2+dx2

2)sinθ=dy/√dy2+dx2

3)cosθ=dx/√dy2+dx2

4)1/ρ=dθ/ds

5)dsinθ/ds=cosθ.dθ/ds=cosθ/ρ


If anybody could help me that would be marvelous!

 

[dhengr]

A good starting point for you might be to read and understand such books as: “The Mathematical Theory of Elasticity,” by A.E.H. Love; “The Mathematical Theory of Elasticity,” by I.S. Sokolnikoff; “Theory of Elastic Stability,” by S.P. Timoshenko and J.M. Gere; and other Theory of Elasticity text books.

 

[BAretired]

Here is a blurb from an old math book:



BA

 

[StevenKatzeff]

Thanks Everyone.

dhengr - a brief look through Timoshenko's "Theroy of Elasticity" yielded no derivation of the elastica I am considering.
BAretired - thanks for the picture, but they also seem to jump a few crucial steps in deriving the formula that I cannot replicate.

Any more suggestions?

 

[StevenKatzeff]

With a little help from Physics Forum, Wikipedia and BAretired's picture, I finally have the solution. Excuse the bad hand-writing, I was excited...

Thanks everyone.

 

[RogerWW]

The mathematics of the elastica is thoroughly derived in a book called "The Elastica" by an author named Firsch-Fay and English publisher I believe. RogerWW