- #1

Math100

- 756

- 201

- Homework Statement
- Consider the functional ## S[y]=\alpha y(1)^2+\int_{0}^{1}\beta y'^2dx, y(0)=0 ##, with a natural boundary condition at ## x=1 ## and subject to the constraint ## C[y]=\gamma y(1)^2+\int_{0}^{1}w(x)y^2dx=1 ##, where ## \alpha, \beta ## and ## \gamma ## are nonzero constants.

a) Show that the stationary paths of this system satisfy the Euler-Lagrange equation ## \beta\frac{d^2y}{dx^2}+\lambda w(x)y=0, y(0)=0, (\alpha-\gamma\lambda)y(1)+\beta y'(1)=0 ##, where ## \lambda ## is a Lagrange multiplier.

b) Let ## w(x)=1 ## and ## \alpha=\beta=\gamma=1 ##. Find the nontrivial stationary paths, stating clearly the eigenfunctions ## y ## (normalized so that ## C[y]=1 ##) and the values of the associated Lagrange multiplier.

- Relevant Equations
- None.

a) Proof:

Let ## \lambda ## be the Lagrange multiplier.

Then the auxiliary functional is ## \overline{S}[y]=\alpha y(1)^2+\int_{0}^{1}\beta y'^2dx-\lambda (\gamma y(1)^2+\int_{0}^{1}w(x)y^2dx-1) ##.

This gives ## \overline{S}[y+\epsilon h]=\alpha (y(1)+\epsilon h(1))^2+\int_{0}^{1}\beta (y'+\epsilon h')^2dx-\lambda (\gamma(y(1)+\epsilon h(1))^2+\int_{0}^{1}w(x)(y+\epsilon h)^2dx-1) ##, where ## y+\epsilon h ## is an admissible perturbation, so that ## h(0)=0 ##.

Note that the Gateaux differential ## \triangle\overline{S}[y, h] ## is given by ## \frac{d}{d\epsilon}\overline{S}[y+\epsilon h]\vert_{\epsilon=0} ##.

Thus ## \frac{d}{d\epsilon}\overline{S}[y+\epsilon h]\vert_{\epsilon=0}=2\alpha y(1)h(1)+2\int_{0}^{1}\beta y'h'dx-2\lambda (\gamma y(1)h(1)+\int_{0}^{1}wyhdx) ##.

From here, how should I show that the stationary paths of this system satisfy the given Euler-Lagrange equation?

b) Let ## w(x)=1 ## and ## \alpha=\beta=\gamma=1 ##.

Consider the Euler-Lagrange equation ## \beta\frac{d^2y}{dx^2}+\lambda w(x)y=0, y(0)=0, (\alpha-\gamma\lambda)y(1)+\beta y'(1)=0 ##, where ## \lambda ## is a Lagrange multiplier.

Then we have ## \frac{d^2y}{dx^2}+\lambda y=0, y(0)=0, (1-\lambda)y(1)+y'(1)=0 ##, where ## \lambda ## is a Lagrange multiplier.

This gives ## y=c_{1}sin(\sqrt{\lambda}x)+c_{2}cos(\sqrt{\lambda}x) ##.

From here, how should I find the nontrivial stationary paths?

Let ## \lambda ## be the Lagrange multiplier.

Then the auxiliary functional is ## \overline{S}[y]=\alpha y(1)^2+\int_{0}^{1}\beta y'^2dx-\lambda (\gamma y(1)^2+\int_{0}^{1}w(x)y^2dx-1) ##.

This gives ## \overline{S}[y+\epsilon h]=\alpha (y(1)+\epsilon h(1))^2+\int_{0}^{1}\beta (y'+\epsilon h')^2dx-\lambda (\gamma(y(1)+\epsilon h(1))^2+\int_{0}^{1}w(x)(y+\epsilon h)^2dx-1) ##, where ## y+\epsilon h ## is an admissible perturbation, so that ## h(0)=0 ##.

Note that the Gateaux differential ## \triangle\overline{S}[y, h] ## is given by ## \frac{d}{d\epsilon}\overline{S}[y+\epsilon h]\vert_{\epsilon=0} ##.

Thus ## \frac{d}{d\epsilon}\overline{S}[y+\epsilon h]\vert_{\epsilon=0}=2\alpha y(1)h(1)+2\int_{0}^{1}\beta y'h'dx-2\lambda (\gamma y(1)h(1)+\int_{0}^{1}wyhdx) ##.

From here, how should I show that the stationary paths of this system satisfy the given Euler-Lagrange equation?

b) Let ## w(x)=1 ## and ## \alpha=\beta=\gamma=1 ##.

Consider the Euler-Lagrange equation ## \beta\frac{d^2y}{dx^2}+\lambda w(x)y=0, y(0)=0, (\alpha-\gamma\lambda)y(1)+\beta y'(1)=0 ##, where ## \lambda ## is a Lagrange multiplier.

Then we have ## \frac{d^2y}{dx^2}+\lambda y=0, y(0)=0, (1-\lambda)y(1)+y'(1)=0 ##, where ## \lambda ## is a Lagrange multiplier.

This gives ## y=c_{1}sin(\sqrt{\lambda}x)+c_{2}cos(\sqrt{\lambda}x) ##.

From here, how should I find the nontrivial stationary paths?