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Consider a scalar field in one spatial dimension described by the Lagrangian

`L=(1)/(2)\theta _(p)\theta ^(\mu )\phi b-V(\phi )=(1)/(2)(\phi ^(˙)^(2)-\phi ^('2))-V(\phi )`

. (a) If

`\phi `

is a static solution of Lagrange's equation, show that

`(d)/(dx)[-(1)/(2)\phi ^('2)+V(\phi )]=0`

. (b) Suppose

`V(\phi )>=0`

and write

`V(\phi )`

in terms of a function

`W(\phi )`

in the form

`V(\phi )=(1)/(2)((d\omega )/(\sigma \phi ))^(2)`

(explain why this is always possible). If

`\phi `

is a static solution that converges to some solution of the algebraic equation

`V(\phi )=0`

as

`x->+-\infty `

, show that

`\phi `

satisfies the first-order differential equation

`(d\phi )/(dx)=+-(dW)/(d\phi )`

(c) Show that the energy

`E=\int_(-\infty )^(\infty ) d\times [(1)/(2)\phi ^(2)+V(\phi )]`

for a static solution of the type described in part (b) can be written in the form

```
E=(1)/(2)\int_(-\infty )^(\infty ) dx(\phi ^(')∓(dW)/(d\phi ))^(2)+-[W(\phi (\infty ))-W(\phi (-\infty ))]
=+-[W(\phi (\infty ))-W(\phi (-\infty ))]
```

(d) Apply this result to the example discussed in Section 11.6 and get the kink's energy without having to explicitly compute the integral in Eq. (11.136).