- #1

Wannabe Physicist

- 17

- 3

- Homework Statement
- Consider the Lagrangian $$L = 1-\sqrt{1-\dot{q}^2}-\frac{q^2}{2}$$ of a particle executing oscillations whose amplitude is ##A## . If ##p## denotes the momentum of

the particle, then ##4p^2## is

a) ##(A^2-q^2)(4+A^2-q^2)##

b) ##(A^2+q^2)(4+A^2-q^2)##

c) ##(A^2-q^2)(4+A^2+q^2)##

d) ##(A^2+q^2)(4+A^2+q^2)##

- Relevant Equations
- 1) ##p = \frac{\partial L}{\partial \dot{q}}##

2) [not sure if this is relevant] ##H = \left(\Sigma_{i} p_i \dot{q}_i\right) - L##

The given lagrangian doesn't seem to correspond to any of the basic systems (like simple/ coupled harmonic oscillators, etc). So I calculated the momentum ##p## which is the partial derivative of ##L## with respect to generalized velocity ##\dot{q}##. Doing so I obtain

$$p = \frac{\dot{q}}{\sqrt{1-\dot{q}^2}}$$

Now, I am facing this problem: I do not know how to relate this to the amplitude of the oscillations.

In the hopes of getting a direction or line of thought to follow, I tried writing the Hamiltonian of this system, which is obtained by rewriting the above equation for ##p=p(\dot{q})## in the form ##\dot{q} = \dot{q}(p) = \displaystyle{\frac{p^2}{1+p^2}}## and substituting it in the equation (2) in the "relevant equations" listed above. I found

$$H = \sqrt{1+p^2} +\frac{q^2}{2}$$

But I am still not sure how to proceed.

After trying all of the above I looked up the answer key and the correct option is (a). But I am not sure how to get the answer.

$$p = \frac{\dot{q}}{\sqrt{1-\dot{q}^2}}$$

Now, I am facing this problem: I do not know how to relate this to the amplitude of the oscillations.

In the hopes of getting a direction or line of thought to follow, I tried writing the Hamiltonian of this system, which is obtained by rewriting the above equation for ##p=p(\dot{q})## in the form ##\dot{q} = \dot{q}(p) = \displaystyle{\frac{p^2}{1+p^2}}## and substituting it in the equation (2) in the "relevant equations" listed above. I found

$$H = \sqrt{1+p^2} +\frac{q^2}{2}$$

But I am still not sure how to proceed.

After trying all of the above I looked up the answer key and the correct option is (a). But I am not sure how to get the answer.

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