## Question

###### Strongly nonlinear oscillators 130 marks] Most of the nonlinear oscillators YOu studlied in lectures are weakly nonlinear. That is the restoring force is linear to leading order plus small nonlinear correction. (The relaxation oscillator, which Can serve a5 model of the human heart , is an exception:) In this question; we consider strongly nonlinear oscillators where the restoring force is something other than linear to leading order, i.e. the nonlinearity is Iot just sInall correction: Conside

Strongly nonlinear oscillators 130 marks] Most of the nonlinear oscillators YOu studlied in lectures are weakly nonlinear. That is the restoring force is linear to leading order plus small nonlinear correction. (The relaxation oscillator, which Can serve a5 model of the human heart , is an exception:) In this question; we consider strongly nonlinear oscillators where the restoring force is something other than linear to leading order, i.e. the nonlinearity is Iot just sInall correction: Consider one-dimensional oscillator of mass m described by generalised coordinate q(t) (units: m) ald generalised momentum p(t) (units: kg In 1). Suppose that the restoring force has magnitude klqla - where 0 is not necessarily an integer What are the SI units of k? Starting from the Lagrangian, show that thc oscillator's Hamiltonian takes thc form H(g,p) 2mn Suppose that the oscillator is isolated from its envirOlment, so that it vibrates with constant mnechanical energy; E Sketch its phase-space trajectory on the q P plane Label the intercepts_ Explain wby the phase space trajectory is closed for all positive values of You may find it useful to illustrate yOur answer with diagram canonical transformation cxists which converts q(t) and p(t) to action and angle variables J(t) and 0(t) respectively. Show that_ when E is constant, one obtains 1/ a J = (8rmnE)1/2 38+} where I(z) symbolises the Euler gamma functiol; with the properties I(1) = ad C(I+l) = IC(c): You may find the integrals at the end of the question helpful: Express the transformed Hamiltonian K(J,0) H[q( J,0) , p( J,8)] purely in terms of J, ie inde- pendent of 0. Write down Hamnilton $ equations for J(t) and 0(t) . It can be shown that 0(t) increases by unity during one period, of duration T_ i6 0(t+T)- 0(t) = 1 for all t. By solving Hamilton $ equations_ show that the period is given by (a +2)J 2aE with J given by (2)- By the way; You may be tempted to think that 0(t) should increase by 2v during one period; befits any decent angular coordinate! In fact; though the increment is unity. We do not prove this result here. If you are curious and would like to try proving it "for fun" (highly optional; zero marksl ) . OIlC good approach is to relate 0(+T) _ 0() = } da &q4 to the generating function for the canonical transformation that yields the action-angle variables. Now let' $ calculate the period in more traditional way. Write down Hamnilton $ equations for the original Hamniltoniall, H(g,p).