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Chapter 14: Problem 5

For central force motion with an inverse square law force of attraction the Hamiltonian is (14.14), i.e. $$ H=\frac{p_{r}^{2}}{2 m}+\frac{p_{\theta}^{2}}{2 m r^{2}}-\frac{|k|}{r}(\equiv \text { energy } E) $$ If we fix \(p_{\theta}\), show that \(r\) is bounded \(\left(r_{1} \leq r \leq r_{2}\right)\) only when \(-k^{2} m / 2 p_{\theta}^{2} \leq E<0\) and that the energy minimum corresponds to an orbit in physical space which is a circle. Sketch the curves \(H=\) constant in the \(\left(r, p_{r}\right)\) projection of the full four-dimensional phase space for this system. Consider this projection in the light of the discussion of surfaces of section in \(\S 14.2\).

Short Answer

Expert verified
Answer: The minimum energy for an orbit with an inverse square law force of attraction is given by $$E_{\text{min}} = -\frac{k^2m}{2p_\theta^2}$$. This minimum energy corresponds to a circular orbit because at this point, the force of attraction acting along the radius is balanced, and the particle moves in a circular orbit with a fixed radius due to the balance of forces. This circle represents a stable orbit for the particle in 2D physical space.

Step by step solution

01

Hamiltonian of central force motion with an inverse square law force of attraction

In this problem, we have a given Hamiltonian for central force motion with an inverse square law force of attraction: $$ H = \frac{p_r^2}{2m} + \frac{p_\theta^2}{2mr^2} - \frac{|k|}{r} \equiv \text{energy} \ E $$
02

Identify the conditions for bounded radial motion (r1 ≤ r ≤ r2)

To find the conditions under which the radial motion is bounded, we will use the Hamiltonian. When the energy E lies in a certain range, the radial motion will be bounded. By rearranging the Hamiltonian equation, we can solve for E, and then determine the range for the energy E that results in bounded radial motion: $$ E = \frac{p_r^2}{2m} + \frac{p_\theta^2}{2mr^2} - \frac{|k|}{r} $$ When \(-k^2 m / 2p_\theta^2 \leq E<0\), the radial motion is bounded: $$ r_1 \leq r \leq r_2 $$
03

Calculate the energy minimum for bounded radial motion

We will differentiate the expression for the energy E with respect to r in order to find the minimum value of E for which the radial motion is bounded: $$ \frac{dE}{dr} = -\left(\frac{-|k|}{r^2}+\frac{p_{\theta}^2}{m r^{3}}\right) $$ Now, let's set the derivative equal to zero and solve for r: $$ 0 = \frac{p_\theta^2}{mr^3} - \frac{|k|}{r^2} $$ Upon solving for rmin, we find that: $$ r_{\text{min}} = \frac{|k| m}{p_\theta^2} $$ The energy minimum for an orbit (Emin) can be found by substituting rmin into the expression for E: $$ E_{\text{min}} = -\frac{k^2m}{2p_\theta^2} $$
04

Discuss how the energy minimum corresponds to an orbit in physical space which is a circle

The energy minimum corresponds to an orbit in the physical space which is a circle. This is because when the radial motion is at its minimum energy, the force of attraction acting along the radius is balanced. At this point, the particle moves in a circular orbit with a fixed radius, due to the balance of forces. The circle represents a stable orbit for the particle in the 2D physical space.
05

Sketch the curves H = constant in the (r, pr) projection of the full four-dimensional phase space for this system

The plot of H = constant in the (r, pr) projection of the full four-dimensional phase space for this system will consist of nested loops centered around the energy minimum (Emin) value. The loops represent bounded radial motion with the particle continuously moving between r1 and r2. The closer the loops are in the projection, the more conserved the energy is, while the wider loops indicate a greater variation in energy levels.
06

Consider the (r, pr) projection in the light of the discussion of surfaces of section in § 14.2

The (r, pr) projection can be considered in the context of the discussion of surfaces of section in § 14.2. In this context, the phase space is divided into different sections based on the surfaces of section. The projection in the (r, pr) plane provides us with the necessary information to study the dynamics, stability, and boundedness of the radial motion in the system. It helps us visualize the transitions between different regions in the phase space and gives us insights into the underlying symmetries and conservation laws governing the system. As discussed in § 14.2, studying the surfaces of section can help us understand complicated dynamical systems in a more accessible manner.

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Most popular questions from this chapter

A particle of mass \(m\) moves smoothly up and down a smooth inclined plane (inclined at an angle \(\alpha\) to the horizontal). The particle Hamiltonian is \(H=p^{2} / 2 m+m g q \sin \alpha(=E)\) where \(p=m \dot{q}\), the co-ordinate \(q \geq 0\), being measured upwards along the plane from a fixed point \(q=0\) at which the particle is perfectly elastically reflected at each impact. Show that the energy \(E\) and action \(I\) for this oscillator are related by \(E=\left[\left(9 \pi^{2} / 8\right) g^{2} m \sin ^{2} \alpha\right]^{1 / 3} I^{2 / 3}\) and find the frequency \(\omega\) of small oscillation in terms of \(g, \alpha\) and \(q_{0}\) (the amplitude of the motion). Given that the angle \(\alpha\) now decreases very slowly use the principle of adiabatic invariance to show that during the long time in which \(\alpha\) decreases from \(\pi / 3\) to \(\pi / 6\) the energy of the system decreases by about \(31 \%\) and the amplitude and period increase by about \(20 \%\) and \(44 \%\) respectively.

A particle of mass \(m\) is attached to the origin by a light elastic string of natural length \(l\), so that it is able to move freely along the \(x\)-axis if its distance from the origin is less than \(l\), but otherwise moves in a potential \(V(x)=\frac{1}{2} k(|x|-l)^{2}\) for \(|x|>l\). If the particle always moves in a straight line, sketch the potential and the phase-plane trajectories for different values of the energy \(E\). Show that \(E=\left(\sqrt{I \Omega+\beta^{2}}-\beta\right)^{2}\), where \(I\) is the action, \(\beta=\sqrt{2 k} l / \pi, \Omega=\sqrt{k / m}\). Explain briefly (without detailed calculation) how the angle variable \(\phi\) conjugate to the action may be found in the form \(\phi(x)\) and how \(x\) may be found as a function of the time \(t\). What happens in the (separate) limits of small and large energies \(E\) ?

A particle of mass \(m\) is constrained to move under the action of gravity in the vertical \((x, z)\) plane on a smooth cycloid curve given parametrically by \(x=l(\theta+\sin \theta), z=l(1-\cos \theta)\). Show that a suitable Hamiltonian is $$ H=\frac{p_{\theta}^{2}}{4 m l^{2}(1+\cos \theta)}+m g l(1-\cos \theta) $$ Use action/angle variables to show that the frequency of oscillation of the particle is independent of its amplitude, i.e. it is the same for all initial conditions with \(|\theta|<\pi\). (The substitution \(s=\sin \frac{1}{2} \theta\) is useful. This tautochrone property of the cycloid was known to Huygens in the seventeenth century and, in principle at least, it leads to some quite accurate clock mechanisms. Contrast the tautochrone property with the brachistochrone property of Chapter 3 , Problem 15.)

For the billiard in an elliptical enclosure (Fig. 14.12) use the result of Appendix B, Problem 2 to show that the product \(\Lambda\) of the angular momenta of the ball measured about the two foci of the ellipse is preserved through each bounce, so that it is conserved and therefore a constant of the motion. Using elliptical co-ordinates (see Chapter 3, Problem 24), show that \(\Lambda=\left(\cosh ^{2} \lambda-\cos ^{2} \theta\right)^{-1}\left(\sinh ^{2} \lambda p_{\theta}^{2}-\sin ^{2} \theta p_{\lambda}^{2}\right)\) and that \(H=\) \(\left[2 m c^{2}\left(\cosh ^{2} \lambda-\cos ^{2} \theta\right)\right]^{-1}\left(p_{\lambda}^{2}+p_{\theta}^{2}\right)\). Hence show that the reflected trajectory from the boundary \(x^{2} / a^{2}+y^{2} / b^{2}=1\) is necessarily tangent (when \(\Lambda>0\) ) to the ellipse \(\lambda=\operatorname{arcsinh} \sqrt{\Lambda / 2 m c^{2} H}\) [Closure for such a trajectory implies closure for all trajectories tangent to the same inner ellipse - an example of a general result due to Poncelet (1822).]

A particle of mass \(m\) moves in a one-dimensional potential \(V(q)=\) \(\frac{1}{2}\left(k q^{2}+\lambda / q^{2}\right)\), where \(k, \lambda, q>0\). Sketch the potential and the phase portrait. Show that the energy \(E\) and action \(I\) are related by $$ E=\sqrt{k \lambda}+2 I \sqrt{k / m} $$ and that the period is then independent of amplitude. Discuss how the dependence of \(q\) on the angle variable \(\phi\) may be found and then the dependence of \(q\) on the time \(t\). (This one-dimensional problem models the purely radial part of the motion of the isotropic harmonic oscillator of \(\$ 4.1\). The integral $$ \int_{x_{1}}^{x_{2}} \sqrt{\left(x_{2}-x\right)\left(x-x_{1}\right)} \frac{\mathrm{d} x}{x}=\frac{1}{2} \pi\left(x_{1}+x_{2}\right)-\pi \sqrt{x_{1} x_{2}} $$ where \(x_{2}>x_{1}>0\) and \(x=q^{2}\) will prove useful!)
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