Minimal Function: Arc-Length c & Calc of Variations

[John_Doe]

The function of arc-length $$c$$ that minimises $$\int^b_{a}{y dx}$$ is the catenary, but why? I have tried using calculus of variations, $$\frac{\partial f}{\partial y} - \frac{d}{dx}(\frac{\partial f}{\partial \dot{y}}) + \lambda (\frac{\partial g}{\partial y}) = 0$$, however I don't know which constraint function to use.

 

[matt grime]

Surely that is the wrong integral to look at (clearly the constant function y(x)=0 for all x minimizes the integral) (edit: and equally clearly I've just said something stupid, haven't I? That integral has no minimum: assumin y(a)=y(b)=0, and why not, that a=0, b=pi then -ksin(x) can be made arbitrarily small by making k arbitrarily large).

Try instead minimizing

$$\int y \sqrt{1+y'^2}dx$$

with boundary conditions y(a)=y(b)

 

[John_Doe]

Why is the arc-length multiplied by $$y$$?

 

[matt grime]

Long answer with hand waving justification: Well, assuming a uniformly dense, unifrom cross section, the contribution of an infinitesimal bit of chain, dx, is the length of the chain times a constant (cross section times density) times the (gravitational) potential which is, ignoring constants

$$y \sqrt{1+y'^2} dx$$

adding/integrating all those gives the answer.

Short answer for those who understand mechanics: y is the gravitational potential.

 

[John_Doe]

$$y$$ is the minimal function, $$\sqrt{1 + \dot{y}^2}dx$$ is the arc-length. What is $$y \sqrt{1 + \dot{y}^2}dx$$?
Why don't I need a constraint equation?

 

[matt grime]

it is the contribution to the gravitational potential of a small part of the chain (potential times the infinitesimal length of the chain)

suppose i raise a mass of m kgs a distance of d metres, what is the increase in its gravitational potential energy? in your idea it is just d and independent of m, which is (sadly) not true.

you do not want to minimize y, you want to minimize the potential energy *of the chain*, that is the integral of y.arclength.dx

http://mathworld.wolfram.com/CalculusofVariations.html

what do you mean by constraint equation? boundary conditions? perhaps the method you've been taught differs from the one I know.

 

[John_Doe]

$$\frac{\partial f}{\partial y} - \frac{d}{dx}(\frac{\partial f}{\partial \dot{y}}) + \lambda (\frac{\partial g}{\partial y}) = 0$$

The constaint equation is $$g(y,\dot{y}, x) = 0$$

 

[matt grime]

Then you are doing something that differs from the method as I know it. It wouldn't hurt you to write more words, eg, try explaining the method you've been taught. I already understood that g from your first post was 'the constraint' equation, but had no idea what that meant. Seaching many sites on calc of variations doesn't reveal what the constraint function is either

 

[John_Doe]

I originally thought that the way to tackle the problem would be to set $$\int^b_{a}{\sqrt{1 + \dot{y}^2}dx} = c$$ as a constraint equation, and minimise $$\int^b_{a}{ydx}$$.

Can you explain geometrically $$\int^b_{a}{y \sqrt{1 + \dot{y}^2}dx$$?

 

[matt grime]

I thought I already did? length of an infinitesimal part of the chain (which is thus proportional to the mass of the infinitesimal part of the chain) times the potential that that infinitesimal mass contributes added up (integrated) over the whole length. i think that's the third time I've explained it. hopefully it is correct: I've taken this from the only source I could with enough detail in. (google is your friend.)

It appears you're trying to solve this by a combination of calculus of variation and lagrangian multipliers.

 

[John_Doe]

How is $$y$$ the gravitational potential?

 

[matt grime]

it's the 'height' of the chain, relative to the 'ground' or reference level y=0. an object of mass m and distance d above the 'ground' has gravitational potential energy m*d*g relative to the 'ground' .(Don't conflate the potential of the field with the potential energy of something in the field, something I hoped i'd managed not to do), though reading back my last post i think i failed to do so, hence the probable confusion)

g is just a constant, and the mass is a constant times the arclength, plus the lagrange equation is homogenous (minmizing L(x,y,y') is the same as minmizing kL for any constant k) so we can divide by constants hence we need to minimize y.arclength over the arc.

 

[John_Doe]

$$f = y\sqrt{1 + \dot{y}^2}$$
$$\frac{\partial{f}}{\partial{y}} = \sqrt{1 + \dot{y}^2}$$ $$\frac{\partial{f}}{\partial{\dot{y}}} = \frac{y\dot{y}}{\sqrt{1 + \dot{y}^2}}$$
$$\frac{d}{dx}(\frac{y\dot{y}}{\sqrt{1 + \dot{y}^2}}) = \frac{\dot{y}^2 + y\ddot{y} - \frac{y\dot{y}^2\ddot{y}}{\sqrt{1 + \dot{y}^2}}}{1 + \dot{y}^2} = \frac{(\dot{y}^2 + y\ddot{y})\sqrt{1 + \dot{y}^2} - y\dot{y}^2\ddot{y}}{\sqrt{1 + \dot{y}^2}^3}$$
$$\sqrt{1 + \dot{y}^2} - \frac{(\dot{y}^2 + y\ddot{y})\sqrt{1 + \dot{y}^2} - y\dot{y}^2\ddot{y}}{\sqrt{1 + \dot{y}^2}^3} = 0$$

 

[matt grime]

try this additional method for the special case when L_x=0 like this

http://planetmath.org/encyclopedia/BeltramiIdentity.html

 

[roger]

suppose i raise a mass of m kgs a distance of d metres, what is the increase in its gravitational potential energy? in your idea it is just d and independent of m, which is (sadly) not true.

Thats interesting, why not if m is just a constant ?

 

[matt grime]

Then you need to do another integral/differential equation if m varies with time, or position or something. See for instance the standard mechanics 101 exercise of the work done to launch a rocket with burning fuel making the mass change.

 

[John_Doe]

$$\frac{\partial f}{\partial x} = \dot{y} \sqrt{1 + \dot{y}^2} + \frac{y \dot{y}}{\sqrt{1 + \dot{y}^2}} = \dot{y} \frac{1 + \dot{y}^2 + y}{\sqrt{1 + \dot{y}^2}}$$