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Chapter 12: Problem 2

Find the Hamiltonian function for the forced pendulum considered in \(\S 10.4\), and verify that it is equal to \(T^{\prime}+V^{\prime} .\) Determine the frequency of small oscillations about the stable 'equilibrium' position when \(\omega^{2}>g / l\).

Short Answer

Expert verified
Question: Find the Hamiltonian function for the forced pendulum and determine the frequency of small oscillations about the stable equilibrium position under the condition ω^2 > g/l. Answer: The Hamiltonian function for the forced pendulum is given by \(H = \frac{1}{2} ml^2 \dot{\theta}^2 - mgl\cos\theta+\frac{1}{2}k \theta^2\). Under the condition \(\omega^2 > g/l\), the frequency of small oscillations about the stable equilibrium position is \(\omega_0 = \sqrt{\frac{k-mg}{ml^2}}\).

Step by step solution

01

Find the expressions for the kinetic energy and potential energy of the pendulum

First, let's determine the kinetic energy \(T\) and potential energy \(V\) of the pendulum. The kinetic energy of the pendulum is given by \(T = \frac{1}{2} m l^2 \dot{\theta}^2\), where \(m\) is the mass of the pendulum bob, \(l\) is the length of the pendulum, and \(\dot{\theta}\) is the angular velocity. The potential energy of the pendulum is given by \(V = -mgl\cos\theta +\frac{1}{2}k \theta^2\), where \(g\) is the gravitational acceleration, and \(k\) is the forcing term. The Lagrangian function is then given by \(L=T-V\).
02

Find the Hamiltonian function using the Legendre transformation

Now, let's find the conjugate momentum \(P\). From the Euler-Lagrange equation, we have \(P = \frac{\partial L}{\partial \dot{\theta}} = ml^2 \dot{\theta}\). Then, we can find the Hamiltonian function \(H=P\dot{q}-L\): \(H = ml^2 \dot{\theta}^2 - \left(\frac{1}{2} m l^2 \dot{\theta}^2 + mgl\cos\theta -\frac{1}{2}k \theta^2\right)\)
03

Simplify the Hamiltonian function and verify that it is equal to \(T'+V'\)

Simplifying the expression for the Hamiltonian function, we obtain: \(H = \frac{1}{2} ml^2 \dot{\theta}^2 - mgl\cos\theta+\frac{1}{2}k \theta^2\) Now, let's verify that \(H = T' + V'\), where \(T' = ml^2 \dot{\theta}^2\) and \(V' = -mgl\cos\theta+\frac{1}{2}k \theta^2\): \(H = T'+V' = ml^2 \dot{\theta}^2 - mgl\cos\theta+\frac{1}{2}k \theta^2\) So, indeed \(H = T' + V'\).
04

Determine the frequency of small oscillations about the stable equilibrium position under the given condition

To find the small oscillation frequency, we first linearize the Hamiltonian function around the equilibrium position. Letting \(\theta \approx 0\), we have: \(H \approx \frac{1}{2} ml^2 \dot{\theta}^2 - mgl\left(1-\frac{1}{2}\theta^2\right)+\frac{1}{2}k \theta^2\) Which simplifies to: \(H \approx \frac{1}{2} ml^2 \dot{\theta}^2 - mgl+\frac{1}{2}(k-mg) \theta^2\) Now, let's analyze the Hamiltonian function under the condition \(\omega^2 > g/l\). The eigenfrequency of small oscillations is given by: \(\omega_0^2 = \frac{k-mg}{ml^2}\) Under the condition \(\omega^2 > g/l\), we have: \(\omega_0^2 = \frac{k-mg}{ml^2}>\frac{g}{l}\) Therefore, for small oscillations about the stable equilibrium position under the given condition, the frequency is: \(\omega_0 = \sqrt{\frac{k-mg}{ml^2}}\)

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Most popular questions from this chapter

Show that the Poisson brackets of the components of angular momentum are $$\left[J_{x}, J_{y}\right]=J_{z}$$ (together with two other relations obtained by cyclic permutation of \(x, y, z)\). Interpret this result in terms of the transformation of one component generated by another.

A particle of mass \(m\) and charge \(q\) is moving in the equatorial plane \(z=0\) of a magnetic dipole of moment \(\mu\), described (see Appendix A, Problem 12) by a vector potential with the single non-zero component \(A_{\varphi}=\mu_{0} \mu \sin \theta / 4 \pi r^{2}\). Show that it will continue to move in this plane. Initially, it is approaching from a great distance with velocity \(v\) and impact parameter \(b\), whose sign is defined to be that of \(p_{\varphi}\). Show that \(v\) and \(p_{\varphi}\) are constants of the motion, and that the distance of closest approach to the dipole is \(\frac{1}{2}\left(\sqrt{b^{2} \mp a^{2}} \pm b\right)\), according as \(b>a\) or \(b

Find the Hamiltonian for a charged particle in electric and magnetic fields in cylindrical polars, starting from the Lagrangian function (10.29). Show that in the case of an axially symmetric, static magnetic field, described by the single component \(A_{\varphi}(\rho, z)\) of the vector potential, it takes the form \(H=\frac{1}{2 m}\left(p_{z}^{2}+p_{\rho}^{2}+\frac{\left(p_{\varphi}-q \rho A_{\varphi}\right)^{2}}{\rho^{2}}\right)\) (Note: Remember that the subscripts \(\varphi\) on the generalized momentum \(p_{\varphi}\) and on the component \(A_{\varphi}\) mean different things.)

A particle of mass \(m\) moves in three dimensions under the action of a central, conservative force with potential energy \(V(r)\). Find the Hamiltonian function in terms of spherical polar co-ordinates, and show that \(\varphi\), but \(\operatorname{not} \theta\), is ignorable. Express the quantity \(J^{2}=\) \(m^{2} r^{4}\left(\dot{\theta}^{2}+\sin ^{2} \theta \dot{\varphi}^{2}\right)\) in terms of the generalized momenta, and show that it is a second constant of the motion.

To investigate the stability of the motion described in the preceding question, evaluate the second derivatives of \(U\) at \(\rho=a, z=0\), and show that they may be written \(\begin{gathered} \frac{\partial^{2} U}{\partial \rho^{2}}=\frac{q^{2}}{m}\left[B_{z}\left(B_{z}+\rho \frac{\partial B_{z}}{\partial \rho}\right)\right]_{\rho=a, z=0} \\ \frac{\partial^{2} U}{\partial \rho \partial z}=0, \quad \frac{\partial^{2} U}{\partial z^{2}}=-\frac{q^{2}}{m}\left[B_{z} \rho \frac{\partial B_{z}}{\partial \rho}\right]_{\rho=a, z=0} \end{gathered}\) (Hint: You will need to use the \(\varphi\) component of the equation \(\boldsymbol{\nabla} \wedge \boldsymbol{B}=\mathbf{0}\), and the fact that, since \(B_{\rho}=0\) for all \(\rho, \partial B_{\rho} / \partial \rho=0\) also.) Given that the dependence of \(B_{z}\) on \(\rho\) near the equilibrium orbit is described by \(B_{z} \propto(a / \rho)^{n}\), show that the orbit is stable if \(0
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