Question

(a) The Lagrangian for a system of one degree of freedom can be written as $$L-\frac{m}{2}\left(\phi^{2} \sin ^{2} \omega t+q q e \sin 2 \omega t+q^{2} \omega^{2}\right)$$ What is the corresponding Hamiltonian? Is it conserved? (b) Introduce a now coordinate defined by $$Q=q \sin a t$$ Find the Lagrangian in terms of the new construction the corresponding Hamill Ionian. Is $H$ conserved?

          (a) The Lagrangian for a system of one degree of freedom can be written as $$L-\frac{m}{2}\left(\phi^{2} \sin ^{2} \omega t+q q e \sin 2 \omega t+q^{2} \omega^{2}\right)$$ What is the corresponding Hamiltonian? Is it conserved? (b) Introduce a now coordinate defined by $$Q=q \sin a t$$ Find the Lagrangian in terms of the new construction the corresponding Hamill Ionian. Is $H$ conserved?

Added by Patricia M.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sri K

12/15/2022

Step 1

In this case, we have one degree of freedom, so $i = 1$. Conjugate momentum: $$p = \frac{\partial L}{\partial \dot{q}} = m\dot{q}$$ Therefore, the Hamiltonian is: $$H = p\dot{q} - L = \frac{m}{2}(\dot{q}^{2} \sin^{2} \omega t + qqe \sin 2\omega t + Show more…

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(a) The Lagrangian for a system of one degree of freedom can be written as $$L-frac{m}{2}left(phi^{2} sin ^{2} omega t+q q e sin 2 omega t+q^{2} omega^{2} ight)$$ What is the corresponding Hamiltonian? Is it conserved? (b) Introduce a now coordinate defined by $$Q=q sin a t$$ Find the Lagrangian in terms of the new construction the corresponding Hamill Ionian. Is $H$ conserved?