A particle of mass m can move in one dimension under the influence of two springs connecte...

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 k1 and k2, respectively.

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

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(b) Introduce a new coordinate Q defined by

 

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