Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

I am trying to find Hamilton's equations for a general Hamiltonian given by $$H[u]=\int_\mathbf{R} \phi(u,u_x)dx$$

Suppose $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial u}-\frac{\partial }{\partial x}\frac{\partial f}{\partial u_x}+\left(\frac{\partial }{\partial x}\right)^2\frac{\partial f}{\partial u_{xx}}+...$$ then Hamilton's equations are given by $$\frac{\partial u}{\partial t}=\{u,H[u]\}=\frac{\partial }{\partial x}\frac{\delta H[u]}{\delta u(x)}$$

But I don't know how to evaluate the last quantity. Would someone mind teaching me, please?


My guess would be

$$\frac{\partial }{\partial x}\int_\mathbf{R} \frac{\partial \phi}{\partial u}-\frac{\partial }{\partial x}\frac{\partial \phi}{\partial u_x}dx$$

Then $$\left[\frac{\partial \phi}{\partial u}-\frac{\partial }{\partial x}\frac{\partial \phi}{\partial u_x}\right]_{-\infty}^{\infty}$$ Would that be right?

share|improve this question

1 Answer 1

up vote 0 down vote accepted

What I suspect is stopping OP is different notations for the functional/variational derivative. Let there be given a functional

$$\tag{1} F[u]~=~\int \! dx ~f(x), $$

where the integrand notation $f(x)$ is a short-hand notation for the following function

$$\tag{2} f(x)~=~f(u(x), u^{\prime}(x),u^{\prime\prime}(x),\ldots;x) .$$

If the underlying variational problem (with given boundary conditions) is well-posed, then the functional derivative exists, and is usually denoted $$\tag{3} \frac{\delta F}{\delta u(x)}. $$

It is given by the Euler-Lagrange formula

$$\tag{4} \frac{\delta F}{\delta u(x)} ~=~ \frac{\partial f(x)}{\partial u(x)}-\frac{d}{dx}\left( \frac{\partial f(x)}{\partial u^{\prime}(x)}\right)+ \frac{d^2}{dx^2}\left( \frac{\partial f(x)}{\partial u^{\prime\prime}(x)}\right)+\ldots, $$

Unfortunately, another notation for the functional derivative is also often used

$$\tag{5} \frac{\delta f(x)}{\delta u(x)} ~:=~ \frac{\partial f(x)}{\partial u(x)}-\frac{d}{dx}\left( \frac{\partial f(x)}{\partial u^{\prime}(x)}\right)+ \frac{d^2}{dx^2}\left( \frac{\partial f(x)}{\partial u^{\prime\prime}(x)}\right)+\ldots. $$

Comparing eqs. (4) and (5), it becomes easy to confuse the integrand $f(x)$ with the integral $F$.

share|improve this answer
    
Aha! Thanks, Qmechanic!! –  valerie v. May 18 '13 at 10:19

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.