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Chapter 8: Problem 15

Show that in a conservative \(N\)-body system, a state of minimal total energy for a given total \(z\)-component of angular momentum is necessarily one in which the system is rotating as a rigid body about the \(z\)-axis. [Use the method of Lagrange multipliers (see Appendix A, Problem 11), and treat the components of the positions \(\boldsymbol{r}_{i}\) and velocities \(\dot{\boldsymbol{r}}_{i}\) as independent variables.]

Short Answer

Expert verified
In a conservative N-body system, it is shown that a state of minimal total energy for a given total z-component of angular momentum leads to the system rotating as a rigid body about the z-axis. This is achieved using the method of Lagrange multipliers and applying the Euler-Lagrange equation, resulting in a solution that indicates the velocity of each body is proportional to its distance from the z-axis.

Step by step solution

01

Write down the energy and angular momentum constraints

For an N-body system, the total energy is E, consisting of kinetic energy (T) and potential energy (U): E = T + U = (1/2) ∑ m_i |\dot{\boldsymbol{r}}_{i}|^2 - ∑ V(|\boldsymbol{r}_{i}-\boldsymbol{r}_{j}|) Here, m_i represents the mass of body i, V represents the interaction potential, and |\dot{\boldsymbol{r}}_{i}| and |\boldsymbol{r}_{i}| are the magnitudes of the velocity and position of body i. The total z-component of angular momentum is L: L = ∑ m_i |\boldsymbol{r}_{i}| |\dot{\boldsymbol{r}}_{i}|cos(θ) We are given L and asked to minimize E considering energy and angular momentum.
02

Introduce Lagrange multipliers and set up the Lagrange function

Let λ and Λ be Lagrange multipliers. We will introduce a Lagrange function ℒ that takes E, L, λ, and Λ into account: ℒ = E + λ(L - L(0)) = (1/2) ∑ m_i |\dot{\boldsymbol{r}}_{i}|^2 - ∑ V(|\boldsymbol{r}_{i}-\boldsymbol{r}_{j}|) + λ(∑ m_i |\boldsymbol{r}_{i}| |\dot{\boldsymbol{r}}_{i}|cos(θ) - L(0)) Here, L(0) is the given value of the total z-component of angular momentum.
03

Apply the Euler-Lagrange equation to the Lagrange function

Now we will apply the Euler-Lagrange equation on ℒ with respect to the position components r_i and velocity components \dot{r}_i: ∂ℒ/∂\boldsymbol{r}_{i} - d(∂ℒ/∂\dot{\boldsymbol{r}}_{i})/dt = 0 Applying the Euler-Lagrange equations to position and velocity components, we get: ∂ℒ/∂\boldsymbol{r}_{i} = -∑ ∂V(|\boldsymbol{r}_{i}-\boldsymbol{r}_{j}|)/∂\boldsymbol{r}_{i} + λ(∑m_i |\dot{\boldsymbol{r}}_{i}|cos(θ) ∂|\boldsymbol{r}_{i}|/∂\boldsymbol{r}_{i}) = 0 ∂ℒ/∂|\dot{\boldsymbol{r}}_{i}| = m_i\dot{\boldsymbol{r}}_{i} - λm_i |\boldsymbol{r}_{i}|cos(θ) = 0
04

Solve for r_i and \dot{r}_i components

By solving the equations in step 3, we can find that the solution must satisfy: \dot{\boldsymbol{r}}_{i} = λ|\boldsymbol{r}_{i}|cos(θ) \boldsymbol{e}_z This equation indicates that the velocity of each body in the system is proportional to its distance from the z-axis, which confirms that the system rotates as a rigid body about the z-axis when minimal total energy and a given total z-component of angular momentum are considered.

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Most popular questions from this chapter

A spherical satellite of radius \(r\) is moving with velocity \(v\) through \(\underline{a}\) uniform tenuous atmosphere of density \(\rho\). Find the retarding force on the satellite if each particle which strikes it (a) adheres to the surface, and (b) bounces off it elastically. Can you explain why the two answers are equal, in terms of the scattering cross-section of a hard sphere?

A planet of mass \(M\) is surrounded by a cloud of small particles in orbits around it. Their mutual gravitational attraction is negligible. Due to collisions between the particles, the energy will gradually decrease from its initial value, but the angular momentum will remain fixed, \(\boldsymbol{J}=\boldsymbol{J}_{0}\), say. The system will thus evolve towards a state of minimum energy, subject to this constraint. Show that the particles will tend to form a ring around the planet. [As in Problem 15 , the constraint may be imposed by the method of Lagrange multipliers. In this case, because there are three components of the constraint equation, we need three Lagrange multipliers, say \(\omega_{x}, \omega_{y}, \omega_{z} .\) We have to minimize the function \(E-\boldsymbol{\omega} \cdot\left(\boldsymbol{J}-\boldsymbol{J}_{0}\right)\) with respect to variations of the positions \(\boldsymbol{r}_{i}\) and velocities \(\dot{\boldsymbol{r}}_{i}\), and with respect to \(\boldsymbol{\omega}\). Show by minimizing with respect to \(\dot{\boldsymbol{r}}_{i}\) that once equilibrium has been reached the cloud rotates as a rigid body, and by minimizing with respect to \(\boldsymbol{r}_{i}\) that all particles occupy the same orbit.] What happens to the energy lost? Why does the argument not necessarily apply to a cloud of particles around a hot star?

A satellite is orbiting the Earth in a circular orbit \(230 \mathrm{~km}\) above the equator. Calculate the total velocity impulse needed to place it in a synchronous orbit (see Chapter 4, Problem 1), using an intermediate semi- elliptical transfer orbit which just touches both circles. (Hint: From the orbit parameters \(a\) and \(a e\), find \(l\), and hence the velocities.) Given that the final mass to be placed in orbit is \(30 \mathrm{~kg}\), and the ejection velocity of the rocket is \(2.5 \mathrm{~km} \mathrm{~s}^{-1}\), find the necessary initial mass.

Assume that the residual mass of a rocket, without payload or fuel, is a given fraction \(\lambda\) of the initial mass including fuel (but still without payload). Show that the total take-off mass required to accelerate a payload \(m\) to velocity \(v\) is $$ M_{0}=m \frac{1-\lambda}{\mathrm{e}^{-v / u}-\lambda} $$ If \(\lambda=0.15\), what is the upper limit to the velocity attainable with an ejection velocity of \(2.5 \mathrm{~km} \mathrm{~s}^{-1} ?\)

An \(N\)-body system is interacting only through the gravitational forces between the bodies. Show that the potential energy function \(V\) satisfies the equation $$ \sum_{i} \boldsymbol{r}_{i} \cdot \boldsymbol{\nabla}_{i} V=-V $$ where \(\boldsymbol{\nabla}_{i}=\left(\partial / \partial x_{i}, \partial / \partial y_{i}, \partial / \partial z_{i}\right)\). (Hint: Show that each two- body term \(V_{i j}\) satisfies this equation. This condition expresses the fact that \(V\) is a homogeneous funetion of the co-ordinates of degree \(-1\).)
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