Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 76 / 2 / i / Solution

Apply the method of multiple scales with independent fast time $t$ and slow time $T=\epsilon t$, and write $x=x_0(t,T)+\epsilon x_1(t,T)+\cdots$. The chain rule gives $d/dt={\partial }_t+\epsilon {\partial }_T$. At leading order,At first order,A forcing component at the natural frequency produces a secular term in $x_1$. Thus the solvability condition in the method of multiple scales is orthogonality to both $\sin \theta$ and $\cos \theta$. Define the fast-phase average, holding $R,\varphi ,T$ fixed, bySince $⟨{\sin }^2\theta ⟩=⟨{\cos }^2\theta ⟩=1/2$ and $⟨\sin \theta \cos \theta ⟩=0$, the amplitude-phase equations areThey are leading averaged equations, not exact equations for a purely sinusoidal solution of a general nonlinear oscillator.

For the cubic conservative force, $f=-R^3{\cos }^3\theta$, odd symmetry gives $⟨{\cos }^3\theta \sin \theta ⟩=0$, while $⟨{\cos }^4\theta ⟩=3/8$. HenceIf $x(0)=x_{\ast }$ and $x^{\prime }(0)=v_{\ast }$, leading initial matching givesand thereforeon bounded slow time intervals for fixed bounded initial data. The positive frequency shift agrees with the increased restoring force of $x^{\prime \prime }+x+\epsilon x^3=0$. The zero initial state is the exact zero solution, with irrelevant phase. If a first waveform correction is desired, the remaining first-order forcing is $-R_0^3\cos 3\theta /4$, so $x_1=R_0^3\cos 3\theta /32$ plus a homogeneous correction chosen to match the initial data at the next order. This exhibits explicitly why the leading harmonic ansatz is approximate.