- #1
Math100
- 750
- 201
- Homework Statement
- Let ## a<b ## and let ## f(x) ## be a continuously differentiable function on the interval ## [a, b] ## with ## f(x)>0 ## for all ## x\in [a, b] ##. Let ## A>0, B>0 ## be constants. Using the Jacobi equation, show that the stationary path ## y(x)=A+\beta\int_{a}^{x}\frac{dw}{\sqrt{f(w)^2-\beta^2}} ##, where ## \beta ## is a constant satisfying ## B-A=\beta\int_{a}^{b}\frac{dw}{\sqrt{f(w)^2-\beta^2}} ## gives a weak local minimum of the functional ## S[y]=\int_{a}^{b}f(x)\sqrt{1+y'^2}dx, y(a)=A, y(b)=B ##. (You are not required to solve the Jacobi equation.)
- Relevant Equations
- Jacobi equation: ## \frac{d}{dx}(P(x)\frac{du}{dx})-Q(x)u=0, u(a)=0, u'(a)=1 ##, where ## P(x)=\frac{\partial^2 F}{\partial y'^2} ## and ## Q(x)=\frac{\partial^2 F}{\partial y^2}-\frac{d}{dx}(\frac{\partial^2 F}{\partial y\partial y'}) ## vanishes at ## x=\tilde{a} ##.
For sufficiently small ## b-a ##, we have
a) if ## P(x)=\frac{\partial^2 F}{\partial y'^2}>0, a\leq x\leq b, S[y] ## has a minimum,
b) if ## P(x)=\frac{\partial^2 F}{\partial y'^2}<0, a\leq x\leq b, S[y] ## has a maximum.
Jacobi's necessary condition: If the stationary path ## y(x) ## corresponds to a minimum of the functional ## S[y]=\int_{a}^{b}F(x, y, y')dx, y(a)=A, y(b)=B ##, and if ## P(x)=\frac{\partial^2 F}{\partial y'^2}>0 ## along the path, then the open interval ## a<x<b ## does not contain points conjugate to ## a ##.
A sufficient condition: If ## y(x) ## is an admissible function for the functional ## S[y]=\int_{a}^{b}F(x, y, y')dx, y(a)=A, y(b)=B ## and satisfies the three conditions listed below, then the functional has a weak local minimum along ## y(x) ##.
a) The function ## y(x) ## satisfies the Euler-Lagrange equation, ## \frac{d}{dx}(\frac{\partial F}{\partial y'})-\frac{\partial F}{\partial y}=0 ##.
b) Along the curve ## y(x), P(x)=\frac{\partial^2 F}{\partial y'^2}>0 ## for ## a\leq x\leq b ##.
c) The closed interval ## [a, b] ## contains no points conjugate to the point ## x=a ##.
Here's my work:
Let ## F(x, y, y')=f(x)\sqrt{1+y'^2} ##.
Then ## P(x)=\frac{\partial^2 F}{\partial y'^2}=\frac{\partial}{\partial y'}(\frac{f(x)y'}{\sqrt{1+y'^2}})=\frac{\frac{\partial}{\partial y'}(f(x)y')\cdot \sqrt{1+y'^2}-(f(x)y')\cdot \frac{\partial}{\partial y'}(\sqrt{1+y'^2})}{(\sqrt{1+y'^2})^2}=\frac{f(x)\cdot \sqrt{1+y'^2}-(f(x)y')(\frac{y'}{\sqrt{1+y'^2}})}{1+y'^2}=\frac{f(x)\cdot (1+y'^2)-f(x)y'^2}{(1+y'^2)^{\frac{3}{2}}}=\frac{f(x)}{(1+y'^2)^{\frac{3}{2}}} ##.
This gives ## Q(x)=\frac{\partial^2 F}{\partial y^2}-\frac{d}{dx}(\frac{\partial^2 F}{\partial y\partial y'})=0 ##.
Thus, the Jacobi equation is ## \frac{d}{dx}(P(x)\frac{du}{dx})-Q(x)u=0\implies \frac{d}{dx}(\frac{f(x)u'}{(1+y'^2)^{\frac{3}{2}}})=0\implies \frac{f(x)u'}{(1+y'^2)^{\frac{3}{2}}}=C ##.
With this Jacobi equation found above, how should I use it and show that the given stationary path gives a weak local minimum of the given functional ## S[y] ##?
Let ## F(x, y, y')=f(x)\sqrt{1+y'^2} ##.
Then ## P(x)=\frac{\partial^2 F}{\partial y'^2}=\frac{\partial}{\partial y'}(\frac{f(x)y'}{\sqrt{1+y'^2}})=\frac{\frac{\partial}{\partial y'}(f(x)y')\cdot \sqrt{1+y'^2}-(f(x)y')\cdot \frac{\partial}{\partial y'}(\sqrt{1+y'^2})}{(\sqrt{1+y'^2})^2}=\frac{f(x)\cdot \sqrt{1+y'^2}-(f(x)y')(\frac{y'}{\sqrt{1+y'^2}})}{1+y'^2}=\frac{f(x)\cdot (1+y'^2)-f(x)y'^2}{(1+y'^2)^{\frac{3}{2}}}=\frac{f(x)}{(1+y'^2)^{\frac{3}{2}}} ##.
This gives ## Q(x)=\frac{\partial^2 F}{\partial y^2}-\frac{d}{dx}(\frac{\partial^2 F}{\partial y\partial y'})=0 ##.
Thus, the Jacobi equation is ## \frac{d}{dx}(P(x)\frac{du}{dx})-Q(x)u=0\implies \frac{d}{dx}(\frac{f(x)u'}{(1+y'^2)^{\frac{3}{2}}})=0\implies \frac{f(x)u'}{(1+y'^2)^{\frac{3}{2}}}=C ##.
With this Jacobi equation found above, how should I use it and show that the given stationary path gives a weak local minimum of the given functional ## S[y] ##?
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