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$\{\gamma _{i},\gamma _{j}\}=\gamma _{i}\gamma _{j}+\gamma _{j}\gamma _{i}=\beta \alpha _{i}\beta \alpha _{j}+\beta \alpha _{j}\beta \alpha _{i}=-\alpha _{i}\beta \beta \alpha _{j}-\alpha _{j}\beta \beta \alpha _{i}=-\{\alpha _{i},\alpha _{j}\}\,$

Hey cool wiki page. I just created mine so I could comment on it. I'm still wondering how $-\{\alpha _{i},\alpha _{j}\}$ is meant to lead to $2\eta$. Plus, $\alpha _{i}$ and $\alpha _{j}$ are 4-column vectors, isn't it, i.e. $\alpha _{i}=(1,\alpha _{1},\alpha _{2},\alpha _{3})$. What am I missing?

Kongguan (talk) 00:34, 5 April 2009 (UTC)

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Method of dominant balance

The method of dominant balance is used to determine the asymptotic behavior of solutions to an ODE without solving it. The process is iterative in that the result obtained by performing the method once can be used as input when the method is repeated, to obtain as many terms in the asymptotic expansion as desired.

The process is as follows:

1. Assume that the asymptotic behavior has the form

$y(x)\sim e^{S(x)}\,$.

2. Make a clever guess as to which terms in the ODE may be negligible in the limit we are interested in.

3. Drop those terms and solve the resulting ODE.

4. Check that the solution is consistent with step 2. If this is the case, then we have the controlling factor of the asymptotic behavior. Otherwise, we need to try dropping different terms in step 2.

5. Repeat the process using our result as the first term of the solution.

Example

Consider this second order ODE:

$xy''+(c-x)y'-ay=0\,$

where c and a are arbitrary constants.

This differential equation cannot be solved exactly. However, it may be useful to know how the solutions behave for large x.

We start by assuming $y\sim e^{S(x)}\,$ as $x\rightarrow \infty$. We do this with the benefit of hindsight, to make things quicker. Since we only care about the behavior of y in the large x limit, we set y equal to $e^{S(x)}\,$, and re-express the ODE in terms of S(x):

$xS''+xS'^{2}+(c-x)S'-a=0\,$, or

$S''+S'^{2}+({\frac {c}{x}}-1)S'-{\frac {a}{x}}=0\,$

where we have used the product rule and chain rule to find the derivatives of y.

Now let us suppose that a solution to this new ODE satisfies

$S'^{2}\sim S'\,$ as $x\to \infty$

$S'',{\frac {c}{x}}S',{\frac {a}{x}}=o(S'^{2}),o(S')\,$ as $x\to \infty$

We get the dominant asymptotic behaviour by setting

$S_{0}'^{2}=S_{0}'\,$

If $S_{0}$ satisfies the above asymptotic conditions, then everything is consistent. The terms we dropped will indeed have been negligible with respect to the ones we kept. $S_{0}$ is not a solution to the ODE for S, but it represents the dominant asymptotic behaviour, which is what we are interested in. Let us check that this choice for $S_{0}$ is consistent:

$S_{0}'=1\,$

$S_{0}'^{2}=1\,$

$S_{0}''=0=o(S_{0}')\,$

${\frac {c}{x}}S_{0}'={\frac {c}{x}}=o(S_{0}')\,$

${\frac {a}{x}}=o(S_{0}')\,$

Everything is indeed consistent. Thus we find the dominant asymptotic behaviour of a solution to our ODE:

$S_{0}=x\,$

$y\sim e^{x}\,$

By convention, the asymptotic series is written as:

$y\sim Ax^{p}e^{\lambda x^{r}}(1+{\frac {u_{1}}{x}}+{\frac {u_{2}}{x^{2}}}\cdots )\,$

so to get at least the first term of this series we have to do another step to see if there is a power of x out the front.

We proceed by making an ansatz that we can write

$S(x)=S_{0}(x)+C(x)\,$

and then attempt to find asymptotic solutions for C(x). Substituting into the ODE for S(x) we find

$C''+C'^{2}+C'+{\frac {c}{x}}C'+{\frac {c-a}{x}}=0\,$

Repeating the same process as before, we keep C' and (c-a)/x and find that

$C_{0}=logx^{a-c}\,$

The leading asymptotic behaviour is therefore

$y\sim x^{a-c}e^{x}\,$

Radiation force on a reflecting sphere

Electromagnetic radiation carries momentum with it. When the radiation strikes a surface To calculate the force on a reflecting sphere in a uniform radiation field, we consider the momentum delivered to a small area of the sphere by a small volume of radiation.

Poynting vector

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