Questions and Answers of Mechanics

Show that there are no stable circular orbits of a particle in the Schwarzschild geometry with a radius less than \(6 G M / c^{2}\).
Show from the effective potential corresponding to the Schwarzschild metric that if \(U_{\text {eff }}\) can be used for arbitrarily small radii, there are actually two radii at which a particle can
Kepler's second law for classical orbits states that planets sweep out equal areas in equal times. Is that still true in Schwarzschild spacetime, assuming orbital radii \(r>2 G M / c^{2}\) ? (a)
Earth's orbit has a semimajor axis \(a=1.496 \times 10^{8} \mathrm{~km}\) and eccentricity \(\epsilon=0.017\). Find the general relativistic precession of the earth's perihelion in seconds of arc per
Sometimes more than one coordinate system can usefully describe the same spacetime geometry. This is true in particular for the Schwarzschild geometry surrounding a spherically symmetric mass \(M\).
The geometry on the surface of a sphere is noneuclidean, so the circumference \(C\) and radius \(R\) of a circle drawn on the sphere do not obey \(C=2 \pi R\), where for example the circumference is
Before the age of relativity, some people calculated that light would be deflected by the sun in a classical model in which light consists of particles of tiny mass \(m\) moving at speed (c\), pulled
(a) Find the escape velocity \(d r / d \tau\) of a particle of mass \(m\) starting from rest at radius \(r_{0}=4 G M / c^{2}\) in a Schwarzschild spacetime of mass \(M\), where \(\tau\) is read on
Tachyons are hypothetical particles (never observed, at least so far) that always travel faster than light. Therefore in general relativistic spacetimes they would follow spacelike (rather than
Consider two concentric coplanar circles in the Schwarzschild metric surrounding the sun, with measured circumferences \(C_{1}\) and \(C_{2}\). In terms of \(C_{1}\) and \(C_{2}\), find an expression
The Robertson-Walker metrics\[d s^{2}=-c^{2} d t^{2}+a(t)^{2}\left[\frac{d r^{2}}{1-k(r / R)^{2}}+r^{2} d \theta^{2}+r^{2} \sin ^{2} \theta d \varphi^{2}\right]\]are applicable to universes that are
Inspired from equations 10.77, write gravitational field vectors describing a gravitational wave of angular frequency \(\omega\) propagating in vacuum in the positive \(z\) direction, specifying both
Show that there is exactly one radius at which a light beam can move in a circular orbit around a spherical black hole, and find this radius. Then show that the orbit is unstable, by showing that a
Write the Lagrangian of a charged particle in terms of potentials in the case where we use the static gauge condition, and show that it appears to be different than the Lagrangian in the absence of
Show that for a Gaussian probability distribution\[p(x)=\frac{e^{-\frac{\left(x-x_{0}\right)^{2}}{2 a^{2}}}}{\sqrt{2 \pi a^{2}}}\]all the moments are given by
(a) Show that for any probability distribution, if we compute the generating function \(Z(\beta) \equiv\left\langle e^{\beta X}\rightangle\) for arbitrary \(\beta\) and \(X\) being the stochastic
For the stochastic equation studied in the text, show that\[\overline{X(t)^{2}}=\frac{\sigma^{2}}{\alpha^{2}}\left(t+\frac{1}{2 \alpha} e^{-2 \alpha t}\right) \rightarrow
Using the generating function \(Z\) introduced in an earlier problem, show that:(a) If \(X\) is a stochastic variable with a gaussian distribution with mean \(x_{0}\) and variance \(\sigma^{2}\),
Show that if the initial condition \(C\) of the linear stochastic differential equation introduced in the text has a Gaussian distribution, so does the solution of the stochastic differential
Show that the case of a particle executing a random walk as described by the statistical moments of its position computed in the text, the probability function \(p(x, t)\) satisfies the so-called
From statistical mechanics, for each degree of freedom \(q\) of a free system in thermal equilibrium at temperature \(T\), the corresponding thermal fluctuations of \(\dot{q}\) is given byHere \(m\)
A team of researchers has long tracked the path of a star named S2 that orbits the supermassive black hole Sagittarius A* at the center of our Milky Way galaxy. (The orbit is one of those shown on
Suppose that the orbit of Star S2, as described in the preceding problem, lies in a plane that is perpendicular to our line of sight. Then at periastron, when S2 is a distance 120 au from the central
Find the Legendre transform \(B(x, z)\) of the function \(A(x, y)=x^{4}-(y+a)^{4}\), and verify that \(-\partial A / \partial x=\partial B / \partial x\).
In thermodynamics the enthalpy \(H\) (no relation to the Hamiltonian \(H\) ) is a function of the entropy \(S\) and pressure \(P\) such that \(\partial H / \partial S=T\) and \(\partial H / \partial
In thermodynamics, for a system such as an enclosed gas, the internal energy \(U(S, V)\) can be expressed in terms of the independent variables of entropy \(S\) and volume \(V\), such that \(d U=T d
The energy of a relativistic free particle is the Hamiltonian H = \(\sqrt{p^{2} c^{2}+m^{2} c^{4}}\) in terms of the particle's momentum and mass.(a) Using one of Hamilton's equations in one
The Lagrangian for a particular system is\[L=\dot{x}^{2}+a \dot{y}+b \dot{x} \dot{z}\]where \(a\) and \(b\) are constants. Find the Hamiltonian, identify any conserved quantities, and write out
A system with two degrees of freedom has the Lagrangian\[L=\dot{q}_{1}^{2}+\alpha \dot{q}_{1} \dot{q}_{2}+\beta q_{2}^{2} / 2,\]where \(\alpha\), and \(\beta\) are constants. Find the Hamiltonian,
Write the Hamiltonian and find Hamilton's equations of motion for a simple pendulum of length \(\ell\) and mass \(m\). Sketch the constant \(H\) contours in the \(\theta, p_{\theta}\) phase plane.
(a) Write the Hamiltonian for a spherical pendulum of length \(\ell\) and mass \(m\), using the polar angle \(\theta\) and azimuthal angle \(\varphi\) as generalized coordinates. (b) Then write out
A Hamiltonian with one degree of freedom has the form\[H=\frac{p^{2}}{2 m}+\frac{k q^{2}}{2}-2 a q^{3} \sin \alpha t\]where \(m, k, a\), and \(\alpha\) are constants. Find the Lagrangian
A particle of mass \(m\) slides on the inside of a frictionless vertically-oriented cone of semi-vertical angle \(\alpha\). (a) Find the Hamiltonian \(H\) of the particle, using generalized
A particle of mass \(m\) is attracted to the origin by a force of magnitude \(k / r^{2}\). Using plane polar coordinates, find the Hamiltonian and Hamilton's equations of motion. Sketch constant-
A double pendulum consists of two strings of equal length \(\ell\) and two bobs of equal mass \(m\). The upper string is attached to the ceiling, while the lower end is attached to the first bob. One
A double Atwood's machine consists of two massless pulleys, each of radius \(R\), some massless string, and three weights, with masses \(m_{1}, m_{2}\), and \(m_{3}\). The axis of pulley 1 is
A massless unstretchable string is slung over a massless pulley. A weight of mass \(2 m\) is attached to one end of the string and a weight of mass \(m\) is attached to the other end. One end of a
(a) A particle is free to move only in the \(x\) direction, subject to the potential energy \(U=U_{0} e^{-\alpha x^{2}}\), where \(\alpha\) and \(U_{0}\) are positive constants. Sketch
A cyclic coordinate \(q_{k}\) is a coordinate absent from the Lagrangian (even though \(\dot{q}_{k}\) is present in L.) (a) Show that a cyclic coordinate is likewise absent from the Hamiltonian. (b)
Show that the Poisson bracket of two constants of the motion is itself a constant of the motion, even when the constants depend explicitly on time.
Prove the anticommutativity and distributivity of Poisson brackets by showing that (a) \(\{A, B\}_{\mathrm{q}, \mathrm{p}}=-\{B, A\}_{\mathrm{q}, \mathrm{p}}\) (b) \(\{A, B+C\}_{\mathrm{q},
Show that Hamilton's equations of motion can be written in terms of Poisson brackets as\[\dot{q}=\{q, H\}_{\mathrm{q}, \mathrm{p}}, \quad \dot{p}=\{p, H\}_{\mathrm{q}, \mathrm{p}}\]
A Hamiltonian has the form\[H=q_{1} p_{1}-q_{2} p_{2}+a q_{1}^{2}-b q_{2}^{2}\]where \(a\) and \(b\) are constants. (a) Using the method of Poisson brackets, show that\[f_{1} \equiv q_{1} q_{2} \quad
Show, using the Poisson bracket formalism, that the Laplace-Runge-Lenz vector\[\mathbf{A} \equiv \mathbf{p} \times \mathbf{L}-\frac{m k \mathbf{r}}{r}\]is a constant of the motion for the Kepler
A beam of protons with a circular cross-section of radius \(r_{0}\) moves within a linear accelerator oriented in the \(x\) direction. Suppose that the transverse momentum components \(\left(p_{y},
A large number of particles, each of mass \(m\), move in response to a uniform gravitational field \(g\) in the negative \(z\) direction. At time \(t=0\), they are all located within the corners of a
In an electron microscope, electrons scattered from an object of height \(z_{0}\) are focused by a lens at distance \(D_{0}\) from the object and form an image of height \(z_{1}\) at a distance
Show directly that the transformation\[Q=\ln \left(\frac{1}{q} \sin p\right) \quad P=q \cot p\]is canonical.
Show that if the Hamiltonian and some quantity \(Q\) are both constants of the motion, then the \(n^{\text {th }}\) partial derivative of \(Q\) with respect to time must also be a constant of the
Prove the Jacobi identity for Poisson brackets,\[\left\{A,\{B, C\}_{\mathrm{q}, \mathrm{p}}\right\}_{\mathrm{q}, \mathrm{p}}+\left\{B,\{C, A\}_{\mathrm{q}, \mathrm{p}}\right\}_{\mathrm{q},
(a) Find the Hamiltonian for a projectile of mass \(m\) moving in a uniform gravitational field \(g\), using coordinates \(x, y\). (b) Then find Hamilton's equations of motion and solve them.
(a) Find the Hamiltonian for a projectile of mass \(m\) moving in a force field with potential energy \(U(ho, \varphi, z)\), where \(ho, \varphi, z\) are cylindrical coordinates. (b) Find Hamilton's
Consider a particle of mass \(m\) with relativistic Hamiltonian \(H=\) \(\sqrt{p^{2} c^{2}+m^{2} c^{4}}+U(x, y, z)\) where \(U\) is its relativistic potential energy. Find the particle's equations of
We found Hamilton's equations by starting with the Lagrangian \(L\left(q_{i}, \dot{q}_{i}, t\right)\) and using a Legendre transformation to define the Hamiltonian \(H\left(q_{i}, p_{i}, t\right)\).
Suppose that for some situations the coordinates \(p, q\) are canonical. Show that the transformed coordinates \(P=\frac{1}{2}\left(p^{2}+q^{2}\right), Q=\tan ^{-1}(q / p)\) are also canonical.
Prove that if one makes two successive canonical transformations, the result is also canonical.
Prove that the Poisson bracket is invariant under a canonical transformation.
A plane pendulum consists of a rod of length \(R\) and negligible mass supporting a plumb bob of mass \(m\) that swings back and forth in a uniform gravitational field \(g\). The point of support at
A plane pendulum consists of a string supporting a plumb bob of mass \(m\) free to swing in a vertical plane and free to swing subject to uniform gravity \(g\). The upper end of the string is
At time \(t=0\) a large number of particles, each of mass \(m\), is strung out along the \(x\) axis from \(x=0\) to \(x=\Delta x\), with momenta \(p_{x}\) varying from \(p=p_{0}\) to \(p=p_{0}+\Delta
Any spherically symmetric function of the canonical coordinate and momentum of a particle can depend only on \(r^{2}, p^{2}\), and \(\mathbf{r} \cdot \mathbf{p}\). Show that the Poisson bracket of
Write the Hamiltonian of a free particle of mass \(m\) in a reference frame that is rotating uniformly with angular velocity \(\boldsymbol{\omega}\) with respect to an inertial frame.
Three objects, starting from rest at the same altitude, roll without slipping down an inclined plane. One is a ring of mass \(M\) and radius \(R\); another is a uniform-density disk of mass \(2 M\)
A cylindrical pole is inserted into a frozen lake so the pole stands vertically. One end of a rope is attached to a point on the surface of the pole near where it enters the ice, and the rope is then
Humanity collectively uses energy at the average rate of about 18 Terawatts. (a) At that rate, after one year how long would the length of the day have increased if during that year we were able to
In a supernova explosion, the core of a heavy star collapses and the outer layers are blown away. Before collapse, suppose the core of a given star has twice the mass of the sun and the same radius
In some theoretical models of pulsars, which are rotating neutron stars, the braking torque slowing the pulsar's spin rate is proportional to the \(n^{\text {th }}\) power of the pulsar's angular
Tidal effects of the moon on the earth have caused the earth's rotation rate to slow, thus reducing the spin angular momentum of the earth leading to an increase in earth's day by \(0.1 \mathrm{~s}\)
Compute the moment of inertia matrix of a solid circular cylinder of height \(H\) and base radius \(R\), and of uniform mass density \(ho=ho_{0}\). In this expression, the cylinder is arranged so
A rod of length \(\ell\) and mass \(m\) is attached to a pivot on one end. The rim of a disc of radius \(R\) and mass \(M\) is attached to its other end in such a way that the disc can pivot in the
(a) Using Euler angles, write the constraint of rolling without slipping for a sphere of radius \(R\) moving on a flat surface. (b) Write the Lagrangian and equations of motion using Lagrange
(a) Find the moment of inertia \(I_{z z}\) of a thin disk of mass \(m\) and radius \(R\) about an axis through its center and perpendicular to the plane of the disk. (b) What are \(I_{x x}\) and
A private plane has a single propeller in front, which rotates in the clockwise sense as seen by the pilot. Flying horizontally, the pilot causes the tail rudder to extend out to the left from the
A uniform-density cone has mass M, base radius R, and height H. Find its inertia matrix if the origin is at the center of the circular base in the x, y plane, the axis of symmetry is along the z
The Crab Nebula is a bright, reddish nebula consisting of the debris from a supernova explosion observed on earth in \(1054 \mathrm{AD}\). The estimated total power it emits, mostly in X-rays, UV,
A cylindrical space station is a hollow cylinder of mass \(M\), radius \(R\), and length \(D\), and endcaps of negligible mass. It spins about its symmetry axis ( \(z\) axis) with angular velocity
(a) Find all elements of the principal moment of inertia matrix for a thin uniform rod of mass \(\Delta m\) and length \(D\) if the rod is oriented along the \(x\) axis and the origin of coordinates
(a) Find the principal moments of inertia for a thin disk of mass \(\Delta m\) and radius \(R\), if its mass density is uniform, the origin of coordinates is at the center of the disk, the \(x\) and
(a) Find all elements of the moment of inertia matrix for a cube of mass \(M\) and edge length \(\ell\) using its principal axes. (b) Then find all elements of the moment of inertia matrix for the
If the entire human race were to leave their current habitats, estimate how much the length of the day would be changed if (a) they gathered at the equator; (b) they gathered at the poles.
Consider a square plane lamina with coordinate axes \(x, y\) in the plane with origin at the center of the square and which are perpendicular to edges of the square. If the moment of inertia about
An equilateral triangle of mass \(M\) and side-length \(L\) is cut from uniformdensity sheet metal. (a) Draw the triangle along with the three perpendicular bisectors, each of which extends from the
Using the rotation matrices appropriate for each of the three Euler angles, find the overall \(3 \times 3\) rotation matrix for arbitrary rotations in terms of the angles \(\varphi, \theta\), and
A rigid body has principal moments of inertia \(I_{x x}=I_{0}, I_{y y}=I_{z z}=2 I_{0} / 3\). (a) Find all elements of the moment of inertia matrix in a reference frame that has been rotated by
Show that any antisymmetric part of the moment of inertia matrix of a rigid body does not contribute to the body's equations of motion. Therefore we may safely assume that the moment of inertia
(a) Write the Lagrangian for the Euler problem of a rigid body undergoing torque-free precession. (b) Write the equations of motion and show that they agree with those in the text.
Write the six equations of motion for the Lagrangian of a hoop attached to a spring.
Show that the magnitude of the angular momentum vector for the torque-free rigid body dynamics case is given by \(L=I_{3} p_{\psi} / \cos \theta\).
Show that \(\dot{u}=0\) if \(g=0\) from equation 12.200.Data from equation 12.200 1 (p-up) + 2 R 1 (2H13 - (p) 2Mgl, 2 113
Show that if\[I_{1} \geq I_{2} \geq I_{3}\]for a torque-free rigid body, we then have\[\sqrt{2 T I_{3}} \leq L \leq \sqrt{2 T I_{1}}\]
Show that \(u \rightarrow 1\) is a stable point for the gyroscope, and find the corresponding nutation. Show that there is a criticial angular momnetum \(p_{\psi}=2 \sqrt{M g l I}\).
A rigid body has an axis of symmetry, which we designate as axis 1. The principal moment of inertia about this axis is \(I_{1}\), while the principal moments of inertia about the remaining two
Using Euler’s equations, show that a rigid body rotating without applied torque has a total angular momentum whose magnitude is constant.
Two blocks, of masses \(m\) and \(M\), are connected by a single spring of forceconstant \(k\). The blocks are free to slide on a frictionless table. Beginning with the Lagrangian, find the
Two blocks, of masses \(m\) and \(2 m\), are connected together linearly by three springs of equal force-constants \(k\). The outer springs are also attached to stationary walls, while the middle
Reconsider the problem of two equal-mass blocks and three springs, in a straight line with the outer springs attached to stationary walls. Now suppose the outer springs have the same force-constant
A hypothetical linear molecule of four atoms is free to move in three dimensions. How many degrees of freedom are there? How many translational modes? How many rotational modes? How many vibrational
Find the normal modes of oscillation for small-amplitude motions of a double pendulum (a lower mass \(m\) hanging from an upper mass \(M\) ) where the pendulum lengths are equal. Find the normal mode
A uniform horizontal rod of mass \(m\) and length \(\ell\) is supported against gravity by two identical springs, one at each end of the rod. Assuming the motion is confined to the vertical plane,
The voltage across a capacitor is \(V_{C}=q / C\), where \(C\) is the capacitance and \(q\) is the charge on the capacitor. The voltage across an inductor is \(V_{L}=L d I / d t\), where \(L\) is the

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