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A particle of mass $m$ can move in one dimension under the influence of two springs connected to fixed points a distance $a$ apart (see figure). The springs obey Hooke’s law and have zero unstretched lengths and force constants $k_1$ and $k_2$, respectively.

a) Using the position of the particle from one fixed point as the generalized coordinate $q$, find the Lagrangian and the corresponding Hamiltonian. Is the energy conserved? Is the Hamiltonian conserved?

b) Introduce a new coordinate $Q$ defined by

$$Q= q-b \;\sin(\omega t) \\ b=\frac{k_2}{k_1+k_2} a $$

What is the Lagrangian in terms of $Q$? What is the corresponding Hamiltonian? Is the energy conserved? Is the Hamiltonian conserved?

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Is this one of those "do my homework for me" questions, or is there something you do not understand and need help with? – ja72 Nov 5 '12 at 0:14

closed as too localized by Martin Beckett, Ron Maimon, Mark Eichenlaub, Qmechanic♦, mbq♦ Nov 5 '12 at 3:12

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