Classical Mechanics

Abstract

In addition to the standard examples treated with Newtonian and Lagrangian mechanics, the highlights of this chapter are (Newtonian and Stokesian) friction, velocity-dependent forces, (gauge) transformations and generating functions, dissipation, phase space considerations, the mathematical pendulum (with elliptic integrals), oscillations (with Laplace transform and Green functions), the time-shift matrix (Floquet operator), stability of solutions (including Lyapunov exponents), Matthieu’s and Hill’s differential equations, Matthieu functions, and parametric resonance. Hamilton mechanics, canonical transformations, the Liouville equation, and the Hamilton–Jacobi theory are introduced along with the notion of arbitrary “coordinates” in phase space. Reference is often made to the limited phase space elements required by quantum theory. There is a list of 48 problems.

Appendices

Problems

Problem 2.1

Determinethe $3\times 3$ matrix of the rotation operator D for a body as a function of the Euler angles $\alpha$, $\beta$, $\gamma$, which are introduced in Fig. 1.10 on p. 30. (2 P)

Problem 2.2

Verify the result for the following 7 special cases: no rotation, $180^\circ$ rotation about the x-, y-, z-axis, and $90^\circ$ rotation about the x-, y-, z-axis. Which Euler angles belong to these 7 cases?

Hint: Here, occasionally only $\alpha +\gamma$ or $\alpha -\gamma$ are determined. (7 P)

Problem 2.3

Which Euler angles $\{\overline{\alpha }$, $\overline{\beta }$, $\overline{\gamma }\}$ belong to the inverse rotations? (Note that $0\le \overline{\alpha }<2\pi$, $0\le \overline{\beta }\le \pi$, and $0\le \overline{\gamma }<2\pi$.) (3 P)

Problem 2.4

The rotation operator and Euler angles are needed to describe a top. For this application, the original coordinate system is the laboratory system, the new system is the body-fixed system. Let the unit vectors be $\mathbf {l}_x$, $\mathbf {l}_y$, $\mathbf {l}_z$ or $\mathbf {k}_x$, $\mathbf {k}_y$, $\mathbf {k}_z$. Let $\mathbf {\omega }_\gamma =\dot{\gamma }\mathbf {k}_z$ and determine the corresponding decomposition into $\mathbf {\omega }_\beta =\dot{\beta }\mathbf {e}_{\text {line of nodes}}$ and $\mathbf {\omega }_\alpha =\dot{\alpha }\mathbf {l}_z$ in the body-fixed and lab-fixed systems.

Let a rotation be $D=D_\alpha D_\beta$. How do the vector A and the matrix M given by

$$\begin{aligned} A=\begin{pmatrix}a_x\\ a_y\\ a_z\end{pmatrix} \qquad \text{ and }\qquad M=\begin{pmatrix}0&{}a_z&{}-a_y\\ -a_z&{}0&{}a_x\\ a_y&{}-a_x&{}\end{pmatrix} \end{aligned}$$

transform under the rotation D? Note that the fact that $M'=DMD^{-1}=DM\widetilde{D}$ shows the same behavior under a rotation as $A'=DA$ is connected to the notion of axial vector. (8 P)

Problem 2.5

Is the tensor force $\mathbf {F}=\frac{3\mu _0}{4\pi }\;\mathbf {T}$ with (see p. 56)

$$\begin{aligned} \mathbf {T}(\mathbf {r})=\frac{(\mathbf {m}\cdot \mathbf {r})\, \mathbf {m}'+(\mathbf {m}'\cdot \mathbf {r})\,\mathbf {m}+(\mathbf {m}\cdot \mathbf {m}')\,\mathbf {r}}{r^5} \;-\,5\;\frac{(\mathbf {m}\cdot \mathbf {r})(\mathbf {m}'\cdot \mathbf {r})\,\mathbf {r}}{r^7} \end{aligned}$$

curl-free?

Hint: To investigate the singularity for ${\mathbf {r}=\mathbf {0}}$, we may encircle the origin and apply Stokes’s theorem $\mathbf {\nabla }\times \mathbf {F}=\lim _{A\rightarrow 0} \frac{1}{A}\int _{(A)}\mathrm {d}\mathbf {r}\cdot \mathbf {F}$.) (8 P)

Problem 2.6

Determine the potential energy V for Problem 2.5, and check that $\mathbf {F}=-\mathbf {\nabla }V$. (4 P)

Problem 2.7

A circular disk of radius R rolls, without sliding, on the x, y-plane. In addition to the two coordinates (x, y) of the point of contact, the three Euler angles $\alpha$, $\beta$, $\gamma$ arise, because the normal to the circular disc has spherical coordinates ($\beta ,\alpha$), and $\gamma$ describes the rotation of the disc. The problem requires five coordinates with finite ranges, having five degrees of freedom (see Fig. 2.35). However, the static friction also delivers two differential conditions between the coordinates on the infinitely small scale:

• How do the constraints for the virtual displacements read?

• How many degrees of freedom does the disc have on the infinitely small scale?

• Why do the equations $\Phi (\alpha ,\beta ,\gamma ,x,y) =0$ here lead to inner contradictions? (Why are the constraints non-holonomous)?

(8 P)

Fig. 2.35

Rolling circular disc with normal $\mathbf {n}$. The Euler angles $\alpha$ and $\beta$ are shown, but not the rolling angle $\gamma$. The orbit arises from the rotation about the normal in the positive sense

Problem 2.8

How do the Lagrange equations of the first kind read in statics if the constraints are given only in differential form (as in the last problem), namely through ? Here n counts the $3N\!-\!f$ constraints.

Use this for Problem 2.7 to determine the Lagrangian parameters, and interpret the connection found between the generalized forces. Show in particular that, in the contact plane, a tangential force acts on the disc, that $F_\gamma$ cancels its torque, and that both $F_\alpha$ and $F_\beta$ are equal to zero. (8 P)

Problem 2.9

How strong does the force $F_2(F_1,\varphi )$ at the crank in Fig. 2.36 have to be for equilibrium? Determine this using the principle of virtual work. (4 P)

Fig. 2.36

Crank motion. The pinion runs on a circle of radius R and moves a connecting rod of length L

Problem 2.10

What does one obtain for this crank from the Lagrange equations (Cartesian coordinates with origin at the center of rotation)? Do the results agree? (8 P)

Problem 2.11

How much does the eccentricity $\varepsilon$ differ from 1 for a given axis ratio $b:a\le 1$ of an ellipse? Relate the difference between the distances at aphelion and perihelion for this ellipse to the mean value of these distances, and compare this with the axis ratio b / a. What follows then for small $\varepsilon$ if we account only for linear, but no squared terms in $\varepsilon$? (For the orbits of planets around the Sun, $\varepsilon <0.1$.) For Comet Halley, $\varepsilon =0.967\,276\,0$. How are the two axes related to each other, and what is the ratio of the lowest to the highest velocity? (2 P)

Problem 2.12

Let the polar angle $\varphi =0$ be associated with the aphelion of the orbit of the Earth (astronomers associate $\varphi =0$ with the perihelion) and the polar angle $\varphi _\text {F}$ with the beginning of spring. Then $\varphi$ increases by $\pi /2$ at the beginning of the summer, autumn, and winter, respectively. In Hamburg, Germany, the lengths of the seasons are $T_\text {sp}=92\,\mathrm {d}\;20.5\,{\text {h}}$, $T_\text {su}=93\,\mathrm {d}\;14.5\,{\text {h}}$, $T_\text {fa}=89\,\mathrm {d}\;18.5\,{\text {h}}$, $T_\text {wi}=89\,\mathrm {d}\; 0.5\,{\text {h}}$. Determine $\varphi _\text {sp}$ and $\varepsilon$, neglecting squared terms in $\varepsilon$ compared to the linear ones. (6 P)

Problem 2.13

By how much is the sidereal day shorter than the solar day (the time between two highest altitudes of the Sun)? By how much does the length of the solar day change in a year? (The result should be determined at least to a linear approximation as a function of $\varepsilon$. Then for $\varepsilon =1/60$, the difference between the longest and shortest solar day follows absolutely.) (4 P)

Problem 2.14

Why are the following three theorems valid for the acceleration $\ddot{\mathbf {r}}=-k\mathbf {r}$ (and constant $k>0)$?

• The orbit is an ellipse with the center $\mathbf {r}=\mathbf {0}$.

• The ray $\mathbf {r}$ moves over equal areas in equal time spans.

• The period T does not depend on the form of the orbital ellipse, but only on k.

Hint: Show that at certain times $\mathbf {r}$ and $\dot{\mathbf {r}}$ are perpendicular to each other. With such a time as the zero time, the problem simplifies enormously. (8 P)

Problem 2.15

What is the kinetic energy $T'_{2\text {L}}$ of a mass $m_2$ in the laboratory system after the collision with another mass $m_1$ initially at rest, taken relative to its kinetic energy $T_{2\text {L}}$ before the collision as a function of the scattering angle $\theta _\text {S}$ (in the center-of-mass system) and of the heat tone Q or the parameter $\xi =\sqrt{1+(m_1\!+\!m_2)/m_1\cdot Q/T_{2\text {L}}}$? How does this ratio read for equal masses and elastic scattering as a function of the scattering angle in the laboratory system? (4 P)

Problem 2.16

What is the angle between the directions of motion of two particles in the laboratory system after the collision? Consider the special case of elastic scattering and in particular of equal masses. (4 P)

Problem 2.17

Two smooth spheres with radii $R_1$ and $R_2$ collide with each other with the collision parameter s (see Fig. 2.37). How large is the scattering angle $\theta _\text {S}$? (2 P)

Problem 2.18

How high is the mass $m_1$ of a body initially at rest, which has collided elastically with another body of mass $m_2$ and momentum $\mathbf {p}_2$, if it is scattered by $\theta _{2\text {L}}=90^\circ$ and keeps only the fraction q of its kinetic energy in the laboratory system? (3 P)

Problem 2.19

A spherical rain drop falls in a homogeneous gravitational field without friction through a saturated cloud. Its mass increases in proportion to its surface area with time. Which inhomogeneous linear differential equation follows for the velocity v, if instead of the time, we take the radius as independent variable? What is the solution of this differential equation? (In practice, we consider only the momentum $\propto r^3v$ as an unknown function.) Compare with the free fall of a constant mass. (7 P)

Problem 2.20

How can we show using Legendre polynomials that the gravitational potential is constant within an inhomogeneous, but spherically symmetric hollow sphere, and therefore that it does not exert there a gravitational force on a test body. How does the potential read if a sphere with radius $r_1$ and homogeneous density $\rho _1$ is covered by a hollow sphere of homogeneous density $\rho _2$ and external radius $r_2$? (The Earth has a core, mainly of iron, and a mantle of SiO$_2$, MgO, FeO, and others, approximately 2900 km thick.) (6 P)

Fig. 2.37

For the collision of two smooth spheres in the center-of-mass system, the component of $\mathbf {p}$ in the direction of the line connecting the sphere centers becomes reversed, the component perpendicular to it is conserved (see Problem 2.17)

Problem 2.21

What height is reached by a ball thrown vertically upwards with velocity $v_0$? Consider the friction with the air (Newtonian friction) and determine the frictional work done, by integration as well as by comparing heights with and without friction. (8 P)

Problem 2.22

A horizontal plate oscillates harmonically up and down with amplitude A and oscillation period T. What inequality is obeyed by A and T if a loosely attached body on the plate does not lift off? (2 P)

Problem 2.23

A car at a speed of 20 km/h runs into a wall and is then evenly decelerated, until it stops, at which point it has been deformed by 30 cm. What is the deceleration during the collision? Can a weightlifter who can lift twice the weight of his body protect himself from hitting the steering wheel? If two such cars with relative velocity 40 km/h hit each other head-on, are the same processes valid for the single drivers as above, or do double or fourfold forces arise? (4 P)

Problem 2.24

Prove the following theorem: For each plane mass distribution, the moment of inertia with respect to the normal of the plane is equal to the sum of the moments of inertia with respect to two mutually perpendicular axes in the plane. (1 P)

Problem 2.25

Derive from that the main moments of inertia of a homogeneous cuboid with edge lengths a, b, and c. (1 P)

Problem 2.26

Determine the moment of inertia of the cuboid with respect to the edge c using three methods:

• As in Problem 2.25.

• Using Problem 2.25 but dividing up a correspondingly larger cuboid.

• Using Steiner’s theorem.

(2 P)

Problem 2.27

Decide whether the following claim is correct: The moment of inertia of a rod of mass M and length l perpendicular to the axis does not depend on the cross-section A, and with respect to an axis of rotation on the face is four times as large as with respect to an axis of rotation through the center of mass. (2 P)

Problem 2.28

Prove the following: Rotations about the axes of the highest and lowest moments of inertia are stable motions, while rotations about the axis of the middle moment of inertia are unstable.

Hint: Use the Euler equations for the rigid body, and make an ansatz for the angular velocity with constant $\mathbf {\omega }_1$ along a principal axis of the moment of inertia under small perturbations perpendicular to it. This implies a constraint for $\lambda (I_1,I_2,I_3,\omega _1$). (4 P)

Problem 2.29

How high is the Coriolis acceleration of a sphere shot horizontally with velocity $v_0$ at the north pole? Through which angle $\varphi$ is it deflected during the time t? Through which angle does the Earth rotate during the same time? (2 P)

Problem 2.30

A uniform heavy rope of length l and mass $\mu \,l$ hangs on a pulley of radius R and moment of inertia I, with the two rope ends initially at the same height. Then the pulley gets pushed with $\dot{\theta }(0)=\omega _0$. Neglect the friction of the pulley about its horizontal axis. As long as the rope presses on the pulley with the total force $F\ge F_0$, the static friction leads to the same (angular) velocity of rope and pulley—after that the rope slides down faster. How does the (angular) velocity depend on the time, up until the rope starts sliding? What is the difference in height of the ends of the rope at this time? (8 P)

Problem 2.31

Show that the homogeneous magnetic field $\mathbf {B}=B{\mathbf {e}}_z$ may be associated with the two vector potentials $\mathbf {A}_1=\frac{1}{2}(\mathbf {B}\times \mathbf {r})$ and $\mathbf {A}_2=Bx\mathbf {e}_y$ (gauge invariance). What scalar field $\psi$ leads to $\mathbf {\nabla }\psi =\mathbf {A}_1-\mathbf {A}_2$? What is the difference between the associated Lagrange functions $L_1$ and $L_2?$ Why is it that this difference does not affect the motion of a particle of charge q and mass m in the magnetic field $\mathbf {B}$? (6 P)

Fig. 2.38

Double pendulum made from rods of mass $m_1$ and $m_2$ and with moments of inertia $I_1$ and $I_2$ with respect to the hinges ($\bullet$), which are separated by a distance l. The distances of the centers of mass of the rods from the hinges are $s_1$ and $s_2$, respectively

Problem 2.32

Two point masses interact with $V(|\mathbf {r}_1-\mathbf {r}_2|)$ and are not subject to any external forces. How do the Lagrange equations (of the second kind) read in the center-of-mass and relative coordinates? (4 P)

Problem 2.33

For the double pendulum in Fig. 2.38, determine T as well as V as a function of $\theta _1$, $\theta _2$, $\dot{\theta }_1$, and $\dot{\theta }_2$. How are these expressions simplified for small amplitudes? (6 P)

Problem 2.34

In the last problem, let $\theta _1=\theta _2=0$ for $t<0$. At time $t=0$, the upper pendulum obtains an impulse, in fact with angular momentum $\mathbf {L}$ with respect to its hinge. What initial values follow for $\dot{\theta }_1$ and $\dot{\theta }_2$, in particular for the mathematical pendulum? (4 P)

Problem 2.35

A homogeneous sphere of mass M and radius r rolls on the inclined plane shown in Fig. 2.39 (with $\mathbf {g}\cdot \mathbf {e}_x=0$). Its moment of inertia is $I=\frac{2}{5}\,Mr^2$. Determine its Lagrange function and the equations of motion for the coordinates (x, z) at the point of contact. (Here we use z instead of y, in anticipation of the next problem.) (3 P)

Fig. 2.39

Oblique plane with inclination angle $\alpha$ (the angle between the downwards oriented normal and the vertical), whence $\mathbf {g}\cdot \mathbf {e}_z=g\sin \alpha$

Problem 2.36

Treat the corresponding problem if the plane is deformed into a cylindrical groove with radius R and axis parallel to ${\mathbf {e}}_z$ (see Fig. 2.40). (Instead of x, it is better to adopt the cylindrical coordinate $\varphi$ with $\varphi =0$ at the lowest position.) How large may $\dot{\varphi }(0)$ be at most, if we always have $\vert \varphi \vert \le \frac{1}{2}\,\pi$ and $\varphi (0)=0$? (4 P)

Fig. 2.40

Sphere in a groove. A sphere with radius r rolls on a circle of radius R, then with the angles $\psi$ and $\varphi$ shown, the relation $r\,(\psi +\varphi )=R\,\varphi$ holds

Problem 2.37

Determine the resonance angular frequency $\omega _\text {R}$ of a forced damped oscillation and show that the frequency $\omega _0$ of the undamped oscillation is higher than $\omega _\text {R}$. What is the ratio of the oscillation amplitude for $\omega _\text {R}$ to that for $\omega _0$? What is the approximate result for $\gamma \ll \omega _0\,$? (4 P)

Problem 2.38

What differential equation and initial values are valid for the Green function $G(\tau )$ for the differential equation $\ddot{x}+2\gamma \dot{x}+\omega _0^2x=f(t)$ of the forced damped oscillation with solution written as $x(t)=\int _{-\infty }^tf(t')\,G(t-t')\,\mathrm {d}t'$? Which $G(\tau )$ is the most general solution, independent of f(t)? With this, the solution of the differential equation may be traced back to a simple integration—check this for the example $f(t)=c\,\cos (\omega t)$ in the special case $\gamma =\omega _0\;(>0)$. (9 P)

Problem 2.39

What equation of motion is supplied by the Lagrange formalism for the double pendulum investigated in Problem 2.33 in the angle coordinates $\theta _1$ and $\theta _2$, exactly on the one hand, and for restriction to small oscillations on the other (i.e., taking $\theta _1$ and $\theta _2$ and their derivatives to be small quantities)? (4 P)

Problem 2.40

Which normal frequencies $\omega _\pm$ result for this double pendulum? Determine the matrices A and B. Investigate also the special case of the mathematical pendulum with $s_1=l$, and use the abbreviation $\sigma =s_2/s_1$ and $\mu =m_2/m_1$, where the normal frequencies are given here at best as multiples of the eigenfrequency $\omega _1$ of the upper pendulum. (6 P)

Problem 2.41

Determine the normal frequencies and the matrices A, B, and C for the mathematical double pendulum with $s_2=s_1=l$ and $m_2\ll m_1$. (Here one should use the fact that $\mu \ll 1$ holds in C—why does one have to calculate $\omega _\pm$ “more precisely by one order”?) (6 P)

Problem 2.42

What functions $\theta _1(t)$ and $\theta _2(t)$ belong to the just investigated mathematical double pendulum (with $\mu \ll 1$) for the following initial values: $\theta _1(0)=\theta _2(0)=0$, $\dot{\theta }_1(0)=-\dot{\theta }_2(0) ={\Omega }$, which according to Problem 2.34 correspond to a collision against the upper pendulum for the double pendulum initially at rest? (Why do we only have to consider here the behavior of the normal coordinates?) Which angular frequencies do the beats have, and how does the amplitude of $\theta _1$ behave in comparison to that of $\theta _2$? (4 P)

Problem 2.43

Prove the Jacobi identity $[u,[v,w]]+[v,[w,u]]+[w,[u,v]]=0$ for the Poisson brackets, with $[u,v]=u_xv_p-u_pv_x$ and $u_x=\partial u/\partial x$, etc. (3 P)

Problem 2.44

Determine the Poisson brackets of the angular momentum component $L_x$ with x, y, z, $p_x,p_y,p_z$, and $L_y$. Note that, by cyclic commutation, $\bigl [\mathbf {L},\mathbf {r}\,\bigr ]$, $\bigl [\mathbf {L},\mathbf {p}\,\bigr ]$, and $\bigl [L_i,L_k\bigr ]$ are then also proven. (5 P)

Problem 2.45

Under which constraints is the transformation $x'=\arctan (\alpha x/p)$, $p\,{}'=\beta x^2\!+\!\gamma p^2$ a canonical one? (3 P)

Problem 2.46

Is the transformation $x'=x^\alpha \cos \beta p$, $p\,{}'=x^\alpha \sin \beta p$ canonical?

(2 P)

Problem 2.47

Using the generating function

$$ G(x,p_x{}',p_y,p_y{}')=x\,\frac{1}{2}\,\{p_x{}'+p_y{}'\}\;-\;p_y\,\{p_x{}'-p_y{}'\}/(qB)\;, $$

show that the Hamilton function

$$ H=\frac{1}{2m}\;\{(p_x+\frac{1}{2}\,qBy)^2+(p_y-\frac{1}{2}\,qBx)^2\} $$

for a charged point mass in the plane perpendicular to a homogeneous magnetic field $B\mathbf {e}_z$ can be written as the Hamilton function of a linear harmonic oscillation. (3 P)

Problem 2.48

From this derive the transformation on p. 128. Show also, without using the generating function, that this transformation is canonical. Why does it not suffice here to compare the four derivatives $\partial x'/\partial x$, $\partial x'/\partial y$, $\partial y'/\partial x$, and $\partial y'/\partial y$ with $\partial p_x/\partial p_x{}'$, $\partial p_x/\partial p_y{}'$, $\partial p_y/\partial p_x{}'$, and $\partial p_y/\partial p_y{}'$, as seems to suffice according to p. 126? (Whence an additional comment is missing here.) (4 P)

List of Symbols

We stick closely to the recommendations of the International Union of Pure and Applied Physics (IUPAP) and the Deutsches Institut für Normung (DIN). These are listed in Symbole, Einheiten und Nomenklatur in der Physik (Physik-Verlag, Weinheim 1980) and are marked here with an asterisk. However, one and the same symbol may represent different quantities in different branches of physics. Therefore, we have to divide the list of symbols into different parts (Table 2.3).

Table 2.3 Symbols used in mechanics