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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2011

Paper 3

If $U,V$ depend only on $x=\rho+\tau$, then $\partial_\rho=\partial_\tau=\partial_x$ on them. Put $A=U-V$ and $B=U+V$. The curvature equation gives
$$
A'=-[U,V]=\tfrac12[A,B].
$$
By cyclicity of <matrix trace>,
$$
\frac d{dx}\operatorname{tr}(A^p)=p\operatorname{tr}(A^{p-1}A')=\frac p2\operatorname{tr}(A^{p-1}[A,B])=0.
$$
This is an <Isospectral Lax equation> expressed as a <Lax equation>, and it proves the assertion for every positive integer $p$.

For the given matrices and $\phi=\phi(x)$,
$$
A=i\left[\frac{\phi'}2\sigma_1+\frac{\sin\phi}{4\lambda}\sigma_2+\left(\lambda+\frac{\cos\phi}{4\lambda}\right)\sigma_3\right].
$$
Using $(a\cdot\sigma)^2=(a\cdot a)I$ gives
$$
\operatorname{tr}(A^2)=-2\left[\frac{(\phi')^2}{4}+\lambda^2+\frac{\cos\phi}{2}+\frac1{16\lambda^2}\right].
$$
Since $\lambda$ is fixed, constancy of this trace yields the <first integral>
$$
\boxed{\frac12(\phi')^2+\cos\phi=C.}
$$
Differentiating confirms $dC/dx=\phi'(\phi''-\sin\phi)=0$ for the reduced equation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 32A / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 32A / ii / Solution

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