Elliptic functions arising from nonlinear differential equations

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Problem 8.178 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 6 "Power Series with Integral Coefficients in the Sense of Hurwitz"

The function $\varphi (z)$ , well defined by the differential equation

\left(\frac{d\varphi (z)}{dz}\right)^2 = 1-[\varphi (z)]^4

and the initial conditions $\varphi (0)=0$ , $\varphi '(0)>0$ , may be expanded in a power series with H-integral coefficients.

Problem 8.183 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 6 "Power Series with Integral Coefficients in the Sense of Hurwitz"

The coefficients $C_n$ in the series

\frac{z}{\varphi (z)} = \sum _{n=0}^\infty \frac{C_n}{(4n)!}z^{4n} ,

where $\varphi (z)$ denotes the function defined in 178, are rational numbers and moreover the denominator of $C_n$ is not divisible by any square other than 1 and is divisible only by such prime numbers as are $\equiv 1 \pmod4$ . [Expand first $z$ and then $z/\varphi (z)$ in ascending powers of $\varphi (z)$ .]

Problem 8.184 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 6 "Power Series with Integral Coefficients in the Sense of Hurwitz"

The differential equation

\left(\frac{d\wp (z)}{dz}\right)^2 = 4[\wp (z)]^3-4\wp (z)

has a well-defined solution of the form

\wp (z) = \frac{1}{z^2}+\sum _{n=1}^\infty \frac{D_n}{(4n)!}z^{4n-2} .

( $\wp (z)$ is a particular elliptic function, termed a ``lemniscate function''; its period-parallelogram is a square.) The numbers $D_n$ are rational, and moreover the denominator of $D_n$ is not divisible by any square other than 1, and is divisible only by such prime numbers as are $\equiv 1 \pmod4$ . [Show that $\wp (z)=[\varphi (z)]^{-2}$ , cf. 183. Expand first $z^2$ and then $z^2[\varphi (z)]^{-2}$ in ascending powers of $\varphi (z)$ ; $z^2$ as a function of $\varphi (z)$ satisfies a linear differential equation of second order.]

Problem 4.45 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 4 "Applications"

(i)(ii)

Consider the Hamilton function

H(p,q) = \frac{p^2}{2m} + K\frac{q^4}{4}

for the anharmonic oscillator. The equations of motion are given by

\frac{dp}{dt} = -\frac{\partial H}{\partial q} = -Kq^3, \quad \frac{dq}{dt} = \frac{\partial H}{\partial p} = \frac{p}{m}

with the initial values $p(0)$ and $q(0)$ .

(i)

Show that $H$ is a first integral.

(ii)

Solve the initial value problem using Jacobi elliptic functions

References

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.

[STH12] Steeb W., Tanski I., Hardy Y. Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications. World Scientific Publishing, 2012.