Classical Mechanics by Konstantin K. Likharev - HTML preview
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Konstantin K. Likharev
Essential Graduate Physics
Lecture Notes and Problems
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Part CM:
Classical Mechanics
Last edit: July 2, 2024
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© K. Likharev
CM: Classical Mechanics
Table of Contents
Chapter 1. Review of Fundamentals (14 pp.)
1.0. Terminology: Mechanics and dynamics
1.1. Kinematics: Basic notions
1.2. Dynamics: Newton laws
1.3. Conservation laws
1.4. Potential energy and equilibrium
1.5. OK, can we go home now?
1.6. Self-test problems (14)
Chapter 2. Lagrangian Analytical Mechanics (14 pp.)
2.1. Lagrange equation
2.2. Three simple examples
2.3. Hamiltonian function and energy
2.4. Other conservation laws
2.5. Exercise problems (11)
Chapter 3. A Few Simple Problems (22 pp.)
3.1. One-dimensional and 1D-reducible systems
3.2. Equilibrium and stability
3.3. Hamiltonian 1D systems
3.4. Planetary problems
3.5. Elastic scattering
3.6. Exercise problems (27)
Chapter 4. Rigid Body Motion (32 pp.)
4.1. Translation and rotation
4.2. Inertia tensor
4.3. Fixed-axis rotation
4.4. Free rotation
4.5. Torque-induced precession
4.6. Non-inertial reference frames
4.7. Exercise problems (37)
Chapter 5. Oscillations (38 pp.)
5.1. Free and forced oscillations
5.2. Weakly nonlinear oscillations
5.3. Reduced equations
5.4. Self-oscillations and phase locking
5.5. Parametric excitation
5.6. Fixed point classification
5.7. Numerical approaches
5.8. Higher-harmonic and subharmonic oscillations
5.9. Relaxation oscillations
5.10. Exercise problems (22)
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CM: Classical Mechanics
Chapter 6. From Oscillations to Waves (30 pp.)
6.1. Two coupled oscillators
6.2. N coupled oscillators
6.3. 1D waves
6.4. Acoustic waves
6.5. Standing waves
6.6. Wave decay and attenuation
6.7. Nonlinear and parametric effects
6.8. Exercise problems (26)
Chapter 7. Deformations and Elasticity (38 pp.)
7.1. Strain
7.2. Stress
7.3. Hooke’s law
7.4. Equilibrium
7.5. Rod bending
7.6. Rod torsion
7.7. 3D acoustic waves
7.8. Elastic waves in thin rods
7.9. Exercise problems (23)
Chapter 8. Fluid Mechanics (30 pp.)
8.1. Hydrostatics
8.2. Surface tension effects
8.3. Kinematics
8.4. Dynamics: Ideal fluids
8.5. Dynamics: Viscous fluids
8.6. Turbulence
8.7. Exercise problems (27)
Chapter 9. Deterministic Chaos (14 pp.)
9.1. Chaos in maps
9.2. Chaos in dynamic systems
9.3. Chaos in Hamiltonian systems
9.4. Chaos and turbulence
9.5. Exercise problems (5)
Chapter 10. A Bit More of Analytical Mechanics (16 pp.)
10.1. Hamilton equations
10.2. Adiabatic invariance
10.3. The Hamilton principle
10.4. The Hamilton-Jacobi equation
10.5. Exercise problems (10)
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CM: Classical Mechanics
* * *
Supplemental file Exercise Problems with Model Solutions (202 problems, 304 pp.) is available online:
https://essentialgraduatephysics.org/Files/CM%20exercises.pdf .
B/W paperback copies of these materials are available on Amazon.com:
https://www.amazon.com/gp/product/B0D7ZC7CVX .
Additional
file
Test Problems with Model Solutions (45 problems, 42 pp.)
is available for course instructors from the author upon request – see Front Matter.
* * *
Introductory Remarks
This course mostly follows the well-established traditions of teaching classical mechanics to physics graduate students. Its most distinguishing feature is substantial attention to the mechanics of physical continua, including the discussions of 1D waves in Chapter 6, deformations and elasticity (including 3D waves) in Chapter 7, and fluid dynamics in Chapter 8. A natural extension of the discussion of turbulence in the last of these chapters becomes possible after a brief introduction to deterministic chaos in Chapter 9.
Another not-quite-standard feature of this course is that the introduction to analytical mechanics, starting with the Lagrangian formalism in Chapter 2, is based on the experiment-based Newton’s laws rather than general concepts such as the Hamilton principle, which is discussed only at the end of the course (Sec. 10.3). I feel that this route emphasizes better the experimental roots of physics, and the secondary nature of any general principles – regardless of their aesthetic and heuristic value.
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Chapter 1. Review of Fundamentals
After a brief discussion of the title and contents of the course, this introductory chapter reviews the basic notions and facts of the non-relativistic classical mechanics, that are supposed to be known to the reader from their undergraduate studies. 1 Due to this reason, the discussion is very short.
1.0. Terminology: Mechanics and dynamics
A more fair title for this course would be Classical Mechanics and Dynamics, because the notions of mechanics and dynamics, though much intertwined, are still somewhat different. The term mechanics, in its narrow sense, means the derivation of equations of motion of point-like particles and their systems (including solids and fluids), the solution of these equations, and an interpretation of the results. Dynamics is a more ambiguous term; it may mean, in particular:
(i) the part of physics that deals with motion (in contrast to statics); (ii) the part of physics that deals with reasons for motion (in contrast to kinematics); (iii) the part of mechanics that focuses on its two last tasks, i.e. the solution of the equations of motion and discussion of the results.2
Because of this ambiguity, after some hesitation, I have opted to use the traditional name Classical Mechanics, with the word Mechanics in its broad sense that includes (similarly to Quantum Mechanics and Statistical Mechanics) studies of dynamics of some non-mechanical systems as well.
1.1. Kinematics: Basic notions
The basic notions of kinematics may be defined in various ways, and some mathematicians pay much attention to alternative systems of axioms and the relations between them. In physics, we typically stick to less rigorous ways (in order to proceed faster to solving particular problems) and end debating any definition as soon as “everybody in the room” agrees that we are all speaking about the same thing –
at least in the context in which they are being discussed. Let me hope that the following notions used in classical mechanics do satisfy this criterion in our “room”:
1 The reader is advised to perform (perhaps after reading this chapter as a reminder) a self-check by solving a few problems of those listed in Sec. 1.6. If the results are not satisfactory, it may make sense to start with some remedial reading. For that, I could recommend, e.g., J. Marion and S. Thornton, Classical Dynamics of Particles and Systems, 5th ed., Saunders, 2003; and D. Morin, Introduction to Classical Mechanics, Cambridge U., 2008.
2 The reader may have noticed that the last definition of dynamics is suspiciously close to the part of mathematics devoted to differential equation analysis; what is the difference? An important bit of philosophy: physics may be defined as an art (and a bit of science :-) of describing Mother Nature by mathematical means; hence in many cases the approaches of a mathematician and a physicist to a problem are very similar. The main difference between them is that physicists try to express the results of their analyses in terms of the properties of the systems under study , rather than the functions describing them, and as a result develop a sort of intuition (“gut feeling”) about how other similar systems may behave, even if their exact equations of motion are somewhat different – or not known at all. The intuition so developed has enormous heuristic power, and most discoveries in physics have been made through gut-feeling-based insights rather than by plugging one formula into another one.
© K. Likharev
CM: Classical Mechanics
(i) All the Euclidean geometry notions, including the point, the straight line, the plane, etc.3
(ii) Reference frames: platforms for observation and mathematical description of physical phenomena. A reference frame includes a coordinate system used for measuring the point’s position (namely, its radius vector r that connects the coordinate origin to the point – see Fig. 1) and a clock that measures time t. A coordinate system may be understood as a certain method of expressing the radius vector r of a point as a set of its scalar coordinates. The most important of such systems (but by no means the only one) are the Cartesian (orthogonal, linear) coordinates 4 rj of a point, in which its radius vector may be represented as the following sum:
3
r n r ,
(1.1)
Cartesian
j j
coordinates
j1
where n1, n2, and n3 are unit vectors directed along the coordinate axis – see Fig. 1.5
r z
3
n
point
3
r
0
n
Fig. 1.1. Cartesian coordinates of a point.
r y
n
2
2
1
r x
1
(iii) The absolute (“Newtonian”) space/time,6 which does not depend on the matter distribution.
The space is assumed to have the Euclidean metric, which may be expressed as the following relation between the length r of any radius vector r and its Cartesian coordinates: 3
r
2
2
r
2
r ,
(1.2)
Euclidean
j
metric
j1
while time t is assumed to run similarly in all reference frames. These assumptions are critically revised in the relativity theory (which, in this series, is discussed only starting from EM Chapter 9.) 3 All these notions are of course abstractions: simplified models of the real objects existing in Nature. But please always remember that any quantitative statement made in physics (e.g., a formula) may be strictly valid only for an approximate model of a physical system. (The reader should not be disheartened too much by this fact: experiments show that many models make extremely precise predictions of the behavior of the real systems.) 4 In this series, the Cartesian coordinates (introduced in 1637 by René Descartes, a.k.a. Cartesius) are denoted either as either { r 1, r 2, r 3} or { x, y, z}, depending on convenience in each particular case. Note that axis numbering is important for operations like the vector (“cross”) product; the “correct” (meaning generally accepted) numbering order is such that the rotation n1 n2 n3 n1… looks counterclockwise if watched from a point with all rj > 0 – like the one shown in Fig. 1.
5 Note that representation (1) is also possible for locally orthogonal but curvilinear (for example, polar/cylindrical and spherical) coordinates, which will be extensively used in this series. However, such coordinates are not Cartesian, and for them some of the relations given below are invalid – see, e.g., MA Sec. 10.
6 These notions were formally introduced by Sir Isaac Newton in his main work, the three-volume Philosophiae Naturalis Principia Mathematica published in 1686-1687, but are rooted in earlier ideas by Galileo Galilei, published in 1632.
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(iv) The (instant) velocity of the point,
Velocity
dr
v( t)
r ,
(1.3)
dt
and its acceleration:
dv
Acceleration
a( t)
v r .
(1.4)
dt
(v) Transfer between reference frames. The above definitions of vectors r, v, and a depend on the chosen reference frame (are “reference-frame-specific”), and we frequently need to relate those vectors as observed in different frames. Within Euclidean geometry, the relation between the radius vectors in two frames with the corresponding axes parallel at the moment of interest (Fig. 2), is very simple:
Radius
vector’s
r
'
0
in
r
0
in
r
.
(1.5)
0
'
0
in
trans-
formation
point
r
in 0'
r in 0
Fig. 1.2. Transfer between two reference frames.
'
0
r
0
0 in 0'
If the frames move versus each other by translation only (no mutual rotation!), similar relations are valid for the velocities and accelerations as well:
v
'
0
in
v
0
in
v
,
(1.6)
0
'
0
in
a
'
0
in
a 0
in
a
.
(1.7)
0
'
0
in
Note that in the case of mutual rotation of the reference frames, the transfer laws for velocities and accelerations are more complex than those given by Eqs. (6) and (7). Indeed, in this case, notions like v0in 0 ’ are not well defined: different points of an imaginary rigid body connected to frame 0 may have different velocities when observed in frame 0 ’. It will be more natural for me to discuss these more general relations at the end of Chapter 4 devoted to rigid body motion.
(vi) A particle (or “point particle”): a localized physical object whose size is negligible, and whose shape is irrelevant to the given problem. Note that the last qualification is extremely important.
For example, the size and shape of a spaceship are not too important for the discussion of its orbital motion but are paramount when its landing procedures are being developed. Since classical mechanics neglects the quantum mechanical uncertainties,7 in it, the position of a particle at any particular instant t may be identified with a single geometrical point, i.e. with a single radius vector r( t). The formal final goal of classical mechanics is finding the laws of motion r( t) of all particles in the given problem.
7 This approximation is legitimate when the product of the coordinate and momentum scales of the particle motion is much larger than Planck’s constant ~ 10-34 Js. More detailed conditions of the classical mechanics’
applicability depend on a particular system – see, e.g., the QM part of this series.
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1.2. Dynamics: Newton’s laws
Generally, the classical dynamics is fully described (in addition to the kinematic relations discussed above) by three Newton’s laws. In contrast to the impression some textbooks on theoretical physics try to create, these laws are experimental in nature, and cannot be derived from purely theoretical arguments.
I am confident that the reader of these notes is already familiar with Newton’s laws,8 in some formulation. Let me note only that in some formulations, the 1st Newton’s law looks just like a particular case of the 2nd law – when the net force acting on a particle equals zero. To avoid this duplication, the 1st law may be formulated as the following postulate:
There exists at least one reference frame, called inertial, in which any free particle (i.e. a 1st Newton’s
particle fully isolated from the rest of the Universe) moves with v = const, i.e. with a = 0.
law
Note that according to Eq. (7), this postulate immediately means that there is also an infinite number of inertial reference frames – because all frames 0 ’ moving without rotation or acceleration relative to the postulated inertial frame 0 (i.e. having a0in 0 ’ = 0) are also inertial.
On the other hand, the 2nd and 3rd Newton’s laws may be postulated together in the following elegant way. Each particle, say number k, may be characterized by a scalar constant (called mass mk), such that at any interaction of N particles (isolated from the rest of the Universe), in any inertial system, N
N
Total
P p m v
(1.8)
momentum
k
const.
k
k
and its
k 1
k 1
conservation
(Each component of this sum,
p m v ,
(1.9)
Particle’s
k
k
k
momentum
is called the mechanical momentum 9 of the corresponding particle, while the sum P, the total momentum of the system.)
Let us apply this postulate to just two interacting particles. Differentiating Eq. (8) written for this case, over time, we get
p p
.
(1.10)
1
2
Let us give the derivative p (which is a vector) the name of the force F exerted on particle 1. In our 1
current case, when the only possible source of the force is particle 2, it may be denoted as F12: p F .
1
12
Similarly, F p , so Eq. (10) becomes the 3rd Newton’s law
21
2
F F .
(1.11) 3rd Newton’s
12
21
law
Plugging Eq. (1.9) into these force definitions, and differentiating the products mkv k, taking into account that particle masses are constants,10 we get that for the k and k’ taking any of values 1, 2, 8 Due to the genius of Sir Isaac, these laws were formulated in the same Principia (1687), well ahead of the physics of his time.
9 The more extended term linear momentum is typically used only in cases when there is a chance of confusion with the angular momentum of the same particle/system – see below. The present-day definition of linear momentum and the term itself belong to John Wallis (1670), but the concept may be traced back to more vague notions of several previous scientists – all the way back to at least a 570 AD work by John Philoponus.
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m v m a F ,
where k' k..
(1.12)
k
k
k
k
kk '
Now, returning to the general case of several interacting particles, and making an additional (but very natural) assumption that all partial forces F kk’ acting on particle k add up as vectors, we may generalize Eq. (12) into the 2nd Newton’s law
2nd
Newton’s
m a p F F ,
(1.13)
k
k
k
kk '
k
law
k ' k
that allows a clear interpretation of the mass as a measure of a particle’s inertia.
As a matter of principle, if the dependence of all pair forces F kk’ of particle positions (and generally of time as well) is known, Eq. (13) augmented with the kinematic relations (2) and (3) allows calculation of the laws of motion r k( t) of all particles of the system. For example, for one particle the 2nd law (13) gives an ordinary differential equation of the second order:
mr F(r, t) ,
(1.14)
which may be integrated – either analytically or numerically.
In certain cases, this is very simple. As an elementary example, for local motions with r << r, Newton’s gravity force11
mm'
Newton’s
F G
R
(1.15)
3
gravity law
R
(where R r – r ’ is the distance between particles of masses m and m’)12 may be approximated as Uniform
F g
m ,
(1.16)
gravity field
with the vector g –( Gm’/ R 3)R being constant.13 As a result, m in Eq. (13) cancels, it is reduced to just r g = const, and may be easily integrated twice:
t
t
2
t
r( t) v( t) g dt' v(0) g t v(0), r( t) v( t' ) dt' r(0) g
v(0) t r(0)
,
(1.17)
2
0
0
thus giving the generic solution to all those undergraduate problems on the projectile motion, which should be so familiar to the reader.
10 Note that this may not be true for composite bodies of varying total mass M (e.g., rockets emitting jets, see Problem 11), in these cases the momentum’s derivative may differ from Ma.
11 Introduced in the same famous Principia!
12 The fact that the masses participating in Eqs. (14) and (16) are equal, the so-called weak equivalence principle, is actually highly nontrivial, but has been repeatedly verified with gradually improved relative accuracy, starting from ~10-3 in Isaac Newton’s own experimentation and all the way down to 1.510-15 from recent satellite experiments – see P. Touboul et al., Phys. Rev. Lett. 129, 121102 (2022).
13 Of course, the most important particular case of Eq. (16) is the gravity field near the Earth’s surface. In this case, using the fact that Eq. (15) remains valid for the gravity field created by a spherically uniform sphere, we get g = GM
2
E/ R E , where M E and R E are the Earth’s mass and radius. Plugging in their values, M E 5.971024 kg and R E 6.37106 m, we get g 9.82 m/s2. The experimental value of g varies from 9.78 to 9.83 m/s2 at various locations on the surface (due to the deviations of Earth’s shape from a sphere, and the location-dependent effect of the centrifugal “inertial force” – see Sec. 4.5 below), with an average value of approximately 9.807 m/s2.
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CM: Classical Mechanics
All this looks (and indeed is) very simple, but in most other cases, Eq. (13) leads to more complex calculations. As an example, let us think about how would we use it to solve another simple problem: a bead of mass m sliding, without friction, along a round ring of radius R in a gravity field obeying Eq. (16) – see Fig. 3. (This system is equivalent to the usual point pendulum, i.e. a point mass suspended from point 0 on a light rod or string, and constrained to move in one vertical plane.) R
initial
0
position,
v = 0
N
intermediate
position
v
final
Fig. 1.3. A bead sliding along a vertical ring.
position, mg
v = ?
Suppose we are only interested in the bead’s velocity v at the lowest point after it has been dropped from the rest at the rightmost position. If we want to solve this problem using only the Newton laws, we have to take the following steps:
(i) consider the bead in an arbitrary intermediate position on a ring, described, for example by the angle θ shown in Fig. 3;
(ii) draw all the forces acting on the particle – in our current case, the gravity force mg and the reaction force N exerted by the ring – see Fig. 3 above
(iii) write the Cartesian components of the 2nd Newton’s law (14) for the bead acceleration: max
= Nx, may = Ny – mg,
(iv) recognize that in the absence of friction, the force N should be normal to the ring, so that we can use two additional equations, Nx = – N sin and Ny = N cos ; (v) eliminate unknown variables N, Nx, and Ny from the resulting system of four equations, thus getting a single second-order differential equation for one variable, for example, :
mR mg sin ;
(1.18)
(vi) use the mathematical identity d 2
/ 2/
d to integrate this equation over once to get
an expression relating the velocity and the angle ; and, finally,
(vii) using our specific initial condition ( 0 at / 2 ), find the final velocity as v
R at
0 .
All this is very much doable, but please agree that the procedure it too cumbersome for such a simple problem. Moreover, in many other cases even writing equations of motion along relevant coordinates is very complex, and any help the general theory may provide is highly valuable. In many cases, such help is given by conservation laws; let us review the most general of them.
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1.3. Conservation laws
(i) Energy conservation is arguably the most general law of physics, but in mechanics, it takes a more humble form of mechanical energy conservation, which has limited applicability. To derive it, we first have to define the kinetic energy of a particle as14
Kinetic
m
energy
2
T
v ,
(1.19)
2
and then recast its differential as15
m
m
dr dv
dp
2
dT d v d v v mv dv m
dr
.
(1.20)
2
2
dt
dt
Now plugging in the momentum’s derivative from the 2nd Newton’s law, dp/ dt = F, where F is the full force acting on the particle, we get dT = F dr. The integration of this equality along the particle’s trajectory connecting some points A and B gives the formula that is sometimes called the work-energy principle:
B
Work-
energy
Δ T T (r ) T (r ) F r
d ,
(1.21)
B
A
principle
A
where the integral on the right-hand side is called the work of the force F on the path from A to B.
The next step may be made only for a potential (also called “conservative”) force that may be represented as the (minus) gradient of some scalar function U(r), called the potential energy. 16 The vector operator (called either del or nabla) of spatial differentiation17 allows a very compact expression of this fact:
Force vs
F
potential
U .
(1.22)
energy
For example, for the uniform gravity field (16),
U mgh const,
(1.23)
where h is the vertical coordinate directed “up” – opposite to the direction of the vector g.
Integrating the tangential component F of the vector F given by Eq. (22), along an arbitrary path connecting the points A and B, we get
B
B
F dr F r
d U (r ) U (r )
,
(1.24)
A
B
A
A
14 In such quantitative form, the kinetic energy was introduced (under the name “living force”) by Gottfried Leibniz and Johann Bernoulli (circa 1700), though its main properties (21) and (27) had not been clearly revealed until an 1829 work by Gaspard-Gustave de Coriolis. The modern term “kinetic energy” was coined only in 1849-1851 by Lord Kelvin (born William Thomson).
15 In these notes, ab denotes the scalar (or “dot-”) product of vectors a and b – see, e.g., MA Eq. (7.1).
16 Note that because of its definition via the gradient, the potential energy is only defined up to an arbitrary additive constant. This notion had been used already by G. Leibniz, though the term we are using for it nowadays was introduced much later (in the mid-19th century) by William Rankine.
17 Its basic properties are listed in MA Sec. 8.
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i.e. work of potential forces may be represented as the difference of values of the function U(r) in the initial and final points of the path. (Note that according to Eq. (24), the work of a potential force on any closed path, with rA = rB, is zero.)
Now returning to Eq. (21) and comparing it with Eq. (24), we see that
T (r ) T (r ) U (r ) U (r ), i.e. T (r ) U (r ) T (r ) U (r ) , (1.25)
B
A
A
B
A
A
A
A
so the total mechanical energy E, defined as
Total
E T U ,
(1.26) mechanical
energy
is indeed conserved:
Mechanical
E(r ) E(r ) ,
(1.27) energy:
A
B
conservation
but for conservative forces only. (Non-conservative forces may change E by either transferring energy from its mechanical form to another form, e.g., to heat in the case of friction, or by pumping the energy into the system under consideration from another, “external” system.)
Mechanical energy conservation allows us to return for just a second to the problem shown in Fig. 3 and solve it in one shot by writing Eq. (27) for the initial and final points:18
m
0
2
mgR
v .
0
(1.28)
2
The (elementary) solution of Eq. (28) for v immediately gives us the desired answer. Let me hope that the reader agrees that this way of problem’s solution is much simpler, and I have earned their attention to discuss other conservation laws – which may be equally effective.
(ii) Linear momentum. The conservation of the full linear momentum of any system of particles isolated from the rest of the world was already discussed in the previous section, and may serve as the basic postulate of classical dynamics – see Eq. (8). In the case of one free particle, the law is reduced to the trivial result p = const, i.e. v = const. If a system of N particles is affected by external forces F(ext), we may write
N
F (ext)
F
F .
(1.29)
k
k
kk'
k 1
If we sum up the resulting Eqs. (13) for all particles of the system then, due to the 3rd Newton’s law (11) valid for any indices k k’, the contributions of all internal forces F kk’ to the resulting double sum on the right-hand side cancel, and we get the following equation:
N
System’s
(ext)
P F
,
where
(ext)
(ext)
F
F .
(1.30) momentum
k
evolution
k 1
It tells us that the translational motion of the system as a whole is similar to that of a single particle, under the effect of the net external force F(ext). As a simple sanity check, if the external forces have a zero sum, we return to the postulate (8). Just one reminder: Eq. (30), as its precursor Eq. (13), is only valid in an inertial reference frame.
18 Here the arbitrary constant in Eq. (23) is chosen so that the potential energy is zero at the final point.
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I hope that the reader knows numerous examples of the application of the linear momentum’s conservation law, including all these undergraduate problems on car collisions, where the large collision forces are typically not known so the direct application of Eq. (13) to each car is impracticable.
(iii) The angular momentum of a particle19 is defined as the following vector:20
Angular
momentum:
definition
L r p,
(1.31)
where ab means the vector (or “cross-“) product of the vector operands.21 Differentiating Eq. (31) over time, we get
L r p r .
p
(1.32)
In the first product, r is just the velocity vector v, parallel to the particle momentum p = mv, so this term vanishes since the vector product of any two parallel vectors equals zero. In the second product, p is equal to the full force F acting on the particle, so Eq. (32) is reduced to Angular
momentum:
L τ,
(1.33)
evolution
where the vector
Torque
τ r F,
(1.34)
is called the torque exerted by force F.22 (Note that the torque is reference-frame specific – and again, the frame has to be inertial for Eq. (33) to be valid, because we have used Eq. (13) for its derivation.) For an important particular case of a central force F that is directed along the radius vector r of a particle, the torque vanishes, so (in that particular reference frame only!) the angular momentum is Angular conserved:
momentum:
L const.
(1.35)
conservation
For a system of N particles, the total angular momentum is naturally defined as System’s
angular
N
momentum:
L L .
(1.36)
k
definition
k 1
Differentiating this equation over time, using Eq. (33) for each L , and again partitioning each force per k
Eq. (29), we get
N
N
(ext)
L r F τ
where
,
(ext)
τ
(ext)
r
F
(1.37)
k
kk
.
'
k
k
k , k ' 1
k 1
k ' k
The first (double) sum may be always divided into pairs of the type (r k F kk’ + r k’ F k’k). With a natural assumption of the central forces, F kk’ (r k – r k’), each of these pairs equals zero. Indeed, in this case, 19 Here we imply that the internal motions of the particle, including its rotation about its axis, are negligible.
(Otherwise, it could not be represented by a point, as was postulated in Sec. 1.) 20 This explicit definition of angular momentum (in different mathematical forms, and under the name of
“moment of rotational motion”) appeared in scientific publications only in the 1740s, though the fact of its conservation (35) in the field of central forces, in the form of the 2nd Kepler law (see Fig. 3.4 below), had been proved already by I. Newton in his Principia.
21 See, e.g., MA Eq. (7.3).
22 Alternatively, especially in mechanical engineering, torque is called the force moment. This notion may be traced all the way back to Archimedes’ theory of levers developed in the 3rd century BC.
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each component of the pair is a vector perpendicular to the plane containing the positions of both particles and the reference frame origin, i.e. to the plane of the drawing in Fig. 4.
)
(ext
F
F k
kk'
r k
/2
)
(ext
F
F k'
h h
k'k
kk'
k'k
r k'
Fig. 1.4. Internal and external forces, and
the internal torque cancellation in a system
of two particles.
0
Also, due to the 3rd Newton’s law (11), these two forces are equal and opposite, and the magnitude of each term in the sum may be represented as Fkk’ hkk’, with equal “lever arms” hkk’ = hk’k.
As a result, each sum (r kF kk’ + r k’F k’k), and hence the whole double sum in Eq. (37) vanish, and it is reduced to a very simple result,
System’s
(ext)
L τ
,
(1.38) angular
momentum:
evolution
which is similar to Eq. (33) for a single particle, and is the angular analog of Eq. (30).
In particular, Eq. (38) shows that if the full external torque (ext) vanishes for some reason (e.g. if the system of particles is isolated from the rest of the Universe), the conservation law (35) is valid for the full angular momentum L even if its individual components L k are not conserved due to inter-particle interactions.
Please note again that since the conservation laws may be derived from Newton’s laws (as was done above), they do not introduce anything new to the dynamics of any system. Indeed, from the mathematical point of view, the conservation laws discussed above are just the first integrals of the second-order differential equations of motion following from Newton’s laws. However, for a physicist, thinking about particular systems in terms of the conserved (or potentially conserved) quantities frequently provides decisive clues on their dynamics.
1.4. Potential energy and equilibrium
Another important role of the potential energy U, especially for dissipative systems whose total mechanical energy E is not conserved because it may be drained to the environment, is finding the positions of equilibrium (sometimes called the fixed points) of the system and analyzing their stability with respect to small perturbations. For a single particle, this is very simple: the force (22) vanishes at each extremum (either minimum or maximum) of the potential energy.23 (Of those fixed points, only the minimums of U(r) are stable – see Sec. 3.2 below for a discussion of this point.) A slightly more subtle case is a particle with an internal potential energy U(r), subjected to an additional external force F(ext)(r). In this case, the stable equilibrium is reached at the minimum of not the function U(r), but of what is sometimes called the Gibbs potential energy 23 Assuming that the additional, non-conservative forces (such as viscosity) responsible for the mechanical energy drain, vanish at equilibrium – as they typically do. (The static friction is one counter-example.) Chapter 1
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r
Gibbs’
potential
U r U r F
r '
r
d '
(1.39)
G
ext
,
energy
which is defined, just as U(r) is, to an arbitrary additive constant.24 The proof of Eq. (39) is very simple: in an extremum of this function, the total force acting on the particle, r
tot
ext
ext
F
F F
U F
' r d ' r U
(1.40)
G
vanishes, as it is necessary for equilibrium.
Physically, the difference U G – U specified by Eq. (39) is the r-dependent part of the potential energy U(ext) of the external system responsible for the force F(ext), so U G is just the total potential energy U + U(ext), excluding its part that does not depend on r and hence is irrelevant for the analysis.
According to the 3rd Newton’s law, the force exerted by the particle on the external system equals (–
F(ext)), so its work (and hence the change of U(ext) due to the change of r) is given by the second term on the right-hand side of Eq. (39). Thus the condition of equilibrium, U G = 0, is just the condition of an extremum of the total potential energy, U + U(ext) + const, of the two interacting systems.
For the simplest (and very frequent) case when the applied force is independent of the particle’s position, the Gibbs potential energy (39) is just25
U r U r F
r
.
(1.41)
G
ext
const
As the simplest example, consider a 1D deformation of the usual elastic spring providing the returning force (– x), where x is the deviation from its equilibrium. As follows from Eq. (22), its potential energy is U = x 2/2 + const, so its minimum corresponds to x = 0. Now let us apply an additional external force F, say independent of x. Then the equilibrium deformation of the spring, x 0 = F/, corresponds to the minimum of not U, but rather of the Gibbs potential energy (41), in our particular case taking the form x 2
U U Fx
Fx .
(1.42)
G
2
1.5. OK, we’ve got it – can we go home now?
Sorry, not yet. In many cases, the conservation laws discussed above provide little help, even in systems without dissipation. As a simple example, consider a generalization of the bead-on-the-ring problem shown in Fig. 3, in which the ring is rotated by external forces, with a constant angular velocity
, about its vertical diameter.26 In this problem (to which I will repeatedly return below, using it as an 24 Unfortunately, in most textbooks, the association of the (unavoidably used) notion of U G with the glorious name of Josiah Willard Gibbs is postponed until a course of statistical mechanics and/or thermodynamics, where U G is a part of the Gibbs free energy, in contrast to U, which is a part of the Helmholtz free energy – see, e.g., SM
Sec. 1.4. I use this notion throughout my series, because the difference between U G and U, and hence that between the Gibbs and Helmholtz free energies, has nothing to do with statistics or thermal motion, and belongs to the whole physics, including not only mechanics but also electrodynamics and quantum mechanics.
25 Eq. (41) is a particular case of what mathematicians call the Legendre transformations.
26 This is essentially a simplified model of the mechanical control device called the centrifugal (or “flyball”, or
“centrifugal flyball”) governor – see, e.g., http://en.wikipedia.org/wiki/Centrifugal_governor. (Sometimes the device is called the “Watt’s governor”, after the famous James Watts who used it in 1788 in one of his first steam Chapter 1
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analytical mechanics “testbed”), none of the three conservation laws listed in the last section, holds. In particular, the bead’s energy,
m
E
v 2 mgh ,
(1.43)
2
is not constant, because the external forces rotating the ring may change it. Of course, we still can solve the problem using Newton’s laws, but this is even more complex than for the above case of the ring at rest, in particular because the force N exerted on the bead by the ring now may have three rather than two Cartesian components, which are not simply related. On the other hand, it is clear that the bead still has just one degree of freedom (say, the angle ), so its dynamics should not be too complicated.
This case gives us a clue on how situations like this one can be simplified: if we only could exclude the so-called reaction forces such as N, that take into account external constraints imposed on the particle motion, in advance, that should help a lot. Such a constraint exclusion may be provided by analytical mechanics, in particular its Lagrangian formulation, to which we will now proceed.
Of course, the value of the Lagrangian approach goes far beyond simple systems such as the bead on a rotating ring. Indeed, this system has just two externally imposed constrains: the fixed distance of the bead from the center of the ring, and the instant angle of rotation of the ring about its vertical diameter. Now let us consider the motion of a rigid body. It is essentially a system of a very large number, N >> 1, of particles (~1023 of them if we think about atoms in a 1-cm-scale body). If the only way to analyze its motion would be to write Newton’s laws for each of the particles, the situation would be completely hopeless. Fortunately, the number of constraints imposed on its motion is almost similarly huge. (At negligible deformations of the body, the distances between each pair of its particles should be constant.) As a result, the number of actual degrees of freedom of such a body is small (at negligible deformations, just six – see Sec. 4.1), so with the kind help from analytical mechanics, the motion of the body may be, in many important cases, analyzed even without numerical calculations.
One more important motivation for analytical mechanics is given by the dynamics of “non-mechanical” systems, for example, of the electromagnetic field – possibly interacting with charged particles, conducting bodies, etc. In many such systems, the easiest (and sometimes the only practicable) way to find the equations of motion is to derive them from either the Lagrangian or Hamiltonian function of the system. Moreover, the Hamiltonian formulation of the analytical mechanics (to be reviewed in Chapter 10 below) offers a direct pathway to deriving quantum-mechanical Hamiltonian operators of various systems, which are necessary for the analysis of their quantum properties.
1.6. Self-test problems
1.1. A bicycle, ridden with velocity v on wet pavement, has no mudguards on its wheels. How far behind should the following biker ride to avoid being splashed over? Neglect the air resistance effects.
engines, though it had been used in European windmills at least since the early 1600s.) Just as a curiosity: the now-ubiquitous term cybernetics was coined by Norbert Wiener in 1948 from the word “governor” (or rather from its Ancient-Greek original ή) exactly in this meaning because the centrifugal governor had been the first well-studied control device.
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1.2. Two round disks of radius R are firmly connected with a coaxial
R
T
cylinder of a smaller radius r, and a thread is wound on the resulting spool.
r
The spool is placed on a horizontal surface, and the thread’s end is being O
pooled out at angle – see the figure on the right. Assuming that the spool
does not slip on the surface, what direction would it roll?
1.3.* Calculate the equilibrium shape of a flexible heavy rope of
d
length l, with a constant mass per unit length, if it is hung in a
, l
uniform gravity field between two points separated by a horizontal
distance d – see the figure on the right.
g
1.4. A uniform, long, thin bar is placed horizontally on two
similar round cylinders rotating toward each other with the same
angular velocity and displaced by distance d – see the figure on
the right. Calculate the laws of relatively slow horizontal motion of
the bar within the plane of the drawing, for both possible directions
d
of cylinder rotation, assuming that the kinetic friction force
g
R
R
between the slipping surfaces of the bar and each cylinder obeys
the simple Coulomb approximation 27 F = N, where N is the normal pressure force between them, and
is a constant (velocity-independent) coefficient. Formulate the condition of validity of your result.
1.5. A small block slides, without friction, down a smooth slide
that ends with a round loop of radius R – see the figure on the right.
R
What smallest initial height h allows the block to make its way around h the loop without dropping from the slide if it is launched with negligible
initial velocity?
g
1.6. A satellite of mass m is being launched from height H over
v0
the surface of a spherical planet with radius R and mass M >> m – see H
the figure on the right. Find the range of initial velocities v
m
0 (normal to
the radius) providing closed orbits above the planet’s surface.
M, R
1.7. Prove that the thin-uniform-disk model of a galaxy allows for small sinusoidal (“harmonic”) oscillations of stars inside it, along the direction normal to the disk, and calculate the frequency of these oscillations in terms of Newton’s gravitational constant G and the average density of the disk’s matter.
27 It was suggested in 1785 by the same Charles-Augustin de Coulomb who discovered the famous Coulomb law of electrostatics, and hence pioneered the whole quantitative science of electricity – see EM Ch. 1.
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1.8. Derive differential equations of motion for small oscillations of two
similar pendula coupled with a spring (see the figure on the right), within their l
l
common vertical plane. Assume that at the vertical position of both pendula, the F d
spring is not stretched ( d = 0).
g
m
m
1.9. One of the popular futuristic concepts of travel is digging a straight railway tunnel through the Earth and letting a train go through it, without initial velocity – driven only by gravity. Calculate the train’s travel time through such a tunnel, assuming that the Earth’s density is constant, and neglecting the friction and planet-rotation effects.
1.10. A small bead of mass m may slide, without friction,
2 d
along a light string stretched with force T >> mg, between two points separated by a horizontal distance 2 d – see the figure on the T
T
right. Calculate the frequency of oscillations of the bead about its
m
equilibrium position, within the vertical plane.
g
1.11. For a rocket accelerating (in free space) due to its working jet motor (and hence spending the jet fuel), calculate the relation between its velocity and the remaining mass.
Hint: For the sake of simplicity, consider the 1D motion.
1.12. Prove the following virial theorem:28 for a set of N particles performing a periodic motion, N
1
T F r ,
k
k
2 k1
where the top bar means averaging over time – in this case over the motion period. What does the virial theorem say about:
(i) a 1D motion of a particle in the confining potential29 U( x) = ax 2 s, with a > 0 and s > 0, and (ii) an orbital motion of a particle in the central potential U( r) = – C/ r?
N
Hint: Explore the time derivative of the following scalar function of time: G t p r .
k
k
k 1
1.13. As will be discussed in Chapter 8, if a solid body moves through a fluid with a sufficiently high velocity v, the fluid’s drag force is approximately proportional to v 2. Use this approximation (introduced by Sir Isaac Newton himself) to find the velocity as a function of time during the body’s vertical fall in the air near the Earth’s surface.
1.14. A particle of mass m, moving with velocity u, collides head-on with a particle of mass M, initially at rest, increasing its internal energy by E. Calculate the velocities of both particles after the collision, if u is barely sufficient for such an internal energy increase.
28 It was first stated by Rudolf Clausius in 1870.
29 Here and below I am following the (regretful) custom of using the single word “potential” for the potential energy of the particle – just for brevity. This custom is also common in quantum mechanics, but in electrodynamics, these two notions should be clearly distinguished – as they are in the EM part of this series.
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Chapter 2. Lagrangian Analytical Mechanics
The goal of this chapter is to describe the Lagrangian formalism of analytical mechanics, which is extremely useful for obtaining the differential equations of motion ( and sometimes their first integrals)
not only for mechanical systems with holonomic constraints but also for some other dynamic systems.
2.1. Lagrange equations
In many cases, the constraints imposed on the 3D motion of a system of N particles may be described by N vector (i.e. 3 N scalar) algebraic equations
r r ( q , q ,..., q ,..., q , t), 1
with k N,
(2.1)
k
k
1
2
j
J
where qj are certain generalized coordinates that (together with constraints) completely define the system position. Their number J ≤ 3 N is called the number of the actual degrees of freedom of the system. The constraints that allow such a description are called holonomic.1
For example, for the problem already mentioned in Section 1.5, namely, the bead sliding along a rotating ring (Fig. 1), J = 1, because with the constraints imposed by the ring, the bead’s position is uniquely determined by just one generalized coordinate – for example, its polar angle .
0
Fig. 2.1. A bead on a rotating ring as an
R
y
x
example of a system with just one
degree of freedom ( J = 1).
mg
z
Indeed, selecting the reference frame as shown in Fig. 1 and using the well-known formulas for the spherical coordinates,2 we see that in this case, Eq. (1) has the form
r x, y, z R sin cos, R sin sin, R cos, where t const ,
(2.2)
with the last constant depending on the exact selection of the axes x and y and the time origin. Since the angle , in this case, is a fixed function of time, and R is a fixed constant, the particle’s position in space 1 Possibly, the simplest counter-example of a non-holonomic constraint is a set of inequalities describing the hard walls confining the motion of particles in a closed volume. Non-holonomic constraints are better dealt with by other methods, e.g., by imposing proper boundary conditions on the (otherwise unconstrained) motion.
2 See, e.g., MA Eq. (10.7).
© K. Likharev
CM: Classical Mechanics
at any instant t is completely determined by the value of its only generalized coordinate . (Note that its dimensionality is different from that of Cartesian coordinates!)
Now returning to the general case of J degrees of freedom, let us consider a set of small variations (alternatively called “virtual displacements”) qj allowed by the constraints. Virtual displacements differ from the actual small displacements (described by differentials dqj proportional to time variation dt) in that qj describes not the system’s motion as such, but rather its possible variation –
see Fig. 1.
q
possible
j
motion
actual
motion
q
dq
j
j
Fig. 2.2. Actual displacement dq
j vs. the
dt
virtual one (i.e. variation) qj.
t
Generally, operations with variations are the subject of a special field of mathematics, the calculus of variations.3 However, the only math background necessary for our current purposes is the understanding that operations with variations are similar to those with the usual differentials, though we need to watch carefully what each variable is a function of. For example, if we consider the variation of the radius vectors (1), at a fixed time t, as functions of independent variations qj, we may use the usual formula for the differentiation of a function of several arguments:4
r
k
r
q
.
(2.3)
k
j
j
q j
Now let us break the force acting upon the k th particle into two parts: the frictionless, constraining part N k of the reaction force and the remaining part F k – including the forces from other sources and possibly the frictional part of the reaction force. Then the 2nd Newton’s law for the k th particle of the system may be rewritten as
m v F N .
(2.4)
k
k
k
k
Since any variation of the motion has to be allowed by the constraints, its 3 N-dimensional vector with N
3D-vector components r k has to be perpendicular to the 3 N- dimensional vector of the constraining forces, also having N 3D-vector components N k. (For example, for the problem shown in Fig. 1, the virtual displacement vector r k may be directed only along the ring, while the constraining force N
exerted by the ring, has to be perpendicular to that direction.) This condition may be expressed as 3 For a concise introduction to the field see, e.g., either I. Gelfand and S. Fomin, Calculus of Variations, Dover, 2000, or L. Elsgolc, Calculus of Variations, Dover, 2007. An even shorter review may be found in Chapter 17 of Arfken and Weber – see MA Sec. 16. For a more detailed discussion, using many examples from physics, see R.
Weinstock, Calculus of Variations, Dover, 2007.
4 See, e.g., MA Eq. (4.2). Also, in all formulas of this section, summations over j are from 1 to J, while those over the particle number k are from 1 to N, so for the sake of brevity, these limits are not explicitly specified.
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N r 0,
(2.5)
k
k
k
where the scalar product of 3 N-dimensional vectors is defined exactly like that of 3D vectors, i.e. as the sum of the products of the corresponding components of the operands. The substitution of Eq. (4) into Eq. (5) results in the so-called D’Alembert principle:5
D’Alembert
principle
( m v F )r 0.
(2.6)
k
k
k
k
k
Plugging Eq. (3) into Eq. (6), we get
r k
m v
F
q
,
(2.7)
k
k
j
0
k
q j
j
j
where the scalars F j, called the generalized forces, are defined as follows:6
r
Generalized
F
F
(2.8)
j
k .
k
force
k
q j
Now we may use the standard argument of the calculus of variations: for the left-hand side of Eq. (7) to be zero for an arbitrary selection of independent variations qj, the expression in the curly brackets, for every j, should equal zero. This gives us the desired set of J 3 N equations
r
m v k F 0 ;
(2.9)
k
k
j
k
q j
what remains is just to recast them in a more convenient form.
First, using the differentiation by parts to calculate the following time derivative: d
r
r
d r
v k v k v k ,
(2.10)
k
k
k
dt
q
q
dt
q
j
j
j
we may notice that the first term on the right-hand side is exactly the scalar product in the first term of Eq. (9).
Second, let us use another key fact of the calculus of variations (which is, essentially, evident from Fig. 3): the differentiation of a variable over time and over the generalized coordinate variation (at a fixed time) are interchangeable operations. As a result, in the second term on the right-hand side of Eq.
(10), we may write
d r
dr v
k
k
k
.
(2.11)
dt q
q
dt
q
j
j
j
5 It was spelled out in a 1743 work by Jean le Rond d’Alembert, though the core of this result has been traced to an earlier work by Jacob (Jean) Bernoulli (1667 – 1748) – not to be confused with his son Daniel Bernoulli (1700-1782) who is credited, in particular, for the Bernoulli equation for ideal fluids, to be discussed in Sec. 8.4 below.
6 Note that since the dimensionality of generalized coordinates may be arbitrary, that of generalized forces may also differ from the newton.
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f f
( df ) d ( f )
f
f
df
Fig. 2.3. The variation of the differential (of
any smooth function f) is equal to the
dt
differential of its variation.
t
Finally, let us differentiate Eq. (1) over time:
dr
r
k
k
v
r
q
k
.
(2.12)
k
dt
q
j
j
t
j
This equation shows that particle velocities v k may be considered to be linear functions of the generalized velocities q considered as independent variables, with proportionality coefficients j
v
r
k
k
.
(2.13)
q
q
j
j
With the account of Eqs. (10), (11), and (13), Eq. (9) turns into
d
v
v
m v k m v
.
(2.14)
k
k
k F 0
dt k
k
k
q
j
q
j
k
j
This result may be further simplified by making, for the total kinetic energy of the system, m
2
1
k
T
v m v v ,
(2.15)
k
k
k
k
k
2
2 k
the same commitment as for v k, i.e. considering T a function of not only the generalized coordinates qj and time t but also of the generalized velocities q – as variables independent of q i
j and t. Then we may
calculate the partial derivatives of T as
T
v
T
v
k
m v
,
m v
(2.16)
k
k
k ,
q
q
q
q
j
k
k
k
j
j
k
j
and notice that they are exactly the two sums participating in Eq. (14). As a result, we get a system of J
Lagrange equations,7
d
T
T
General
F ,
0
for j ,
1 ,...,
2
J .
(2.17) Lagrange
dt q
q
j
equations
j
j
Their big advantage over the initial Newton’s-law equations (4) is that the Lagrange equations do not include the constraining forces N k, and thus there are only J of them – typically much fewer than 3 N.
7 They were derived in 1788 by Joseph-Louis Lagrange, who pioneered the whole field of analytical mechanics –
not to mention his key contributions to number theory and celestial mechanics.
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This is as far as we can go for arbitrary forces. However, if all the forces may be expressed in a form similar to, but somewhat more general than Eq. (1.22): F k = – kU(r1, r2,…, r N, t), where U is the effective potential energy of the system,8 and k denotes the spatial differentiation over coordinates of the k th particle, we may recast Eq. (8) into a simpler form:
r
U
x
U
y
U
z
U
k
k
k
i
F F
.
(2.18)
j
k
k
q
x
q
y
q
z
q
q
j
k
k
j
k
j
i
j
j
Since we assume that U depends only on particle coordinates (and possibly time), but not velocities:
U / q ,
0 with the substitution of Eq. (18), the Lagrange equation (17) may be represented in the so-j
called canonical form:
Canonical
d L
L
,
0
(2.19a)
Lagrange
dt q
equations
q
j
j
where L is the Lagrangian function (sometimes called just the “Lagrangian”), defined as Lagrangian
function
L T U .
(2.19b)
(It is crucial to distinguish this function from the mechanical energy (1.26), E = T + U.) Note also that according to Eq. (2.18), for a system under the effect of an additional generalized external force F j( t) we have to use, in all these relations, not the internal potential energy U(int) of the system, but its Gibbs potential energy U U(int) – F jqj – see the discussion in Sec. 1.4.
Using the Lagrangian approach in practice, the reader should always remember, first, that each system has only one Lagrange function (19b), but is described by J 1 Lagrange equations (19a), with j taking values 1, 2,…, J, and second, that differentiating the function L, we have to consider the generalized velocities as its independent arguments, ignoring the fact they are actually the time derivatives of the generalized coordinates.
2.2. Three simple examples
As the first, simplest example, consider a particle constrained to move along one axis (say, x): m
2
T
x ,
U U ( x, t).
(2.20)
2
In this case, it is natural to consider x as the (only) generalized coordinate, and x as the generalized velocity, so
m
2
L T U
x U ( x, t).
(2.21)
2
Considering x and x as independent variables, we get L
/ x
mx , and L
/ x
U
/ x
, so Eq. (19)
(the only Lagrange equation in this case of the single degree of freedom!) yields 8 Note that due to the possible time dependence of U, Eq. (17) does not mean that the forces F k have to be conservative – see the next section for more discussion. With this understanding, I will still use for function U the convenient name of “potential energy”.
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d
U
x
m
,
0
(2.22)
dt
x
evidently the same result as the x-component of the 2nd Newton’s law with Fx = – U/ x. This example is a good sanity check, but it also shows that the Lagrange formalism does not provide too much advantage in this particular case.
Such an advantage is, however, evident in our testbed problem – see Fig. 1. Indeed, taking the polar angle for the (only) generalized coordinate, we see that in this case, the kinetic energy depends not only on the generalized velocity but also on the generalized coordinate:9
m 2
T
R 2
2
sin2 ,
U mgz const mgR cos const,
2
(2.23)
m 2
L T U
R 2
2
sin2 mgR cos
.
const
2
Here it is especially important to remember that at substantiating the Lagrange equation, and have to be treated as independent arguments of L, so
L
L
2
mR ,
2
2
mR sin cos mgR sin,
(2.24)
giving us the following equation of motion:
d
2
mR
2
2
mR sin cos mgR sin .
0
(2.25)
dt
As a sanity check, at = 0, Eq. (25) is reduced to the equation (1.18) of the usual pendulum: 1/ 2
g
2
Ω sin ,
0
Ω
where
.
(2.26)
R
We will explore Eq. (25) in more detail later, but please note how simple its derivation was – in comparison with writing the 3D Newton’s law and then excluding the reaction force.
Next, though the Lagrangian formalism was derived from Newton’s law for mechanical systems, the resulting equations (19) are applicable to other dynamic systems, especially those for which the kinetic and potential energies may be readily expressed via some generalized coordinates. As the simplest example, consider the well-known connection of a capacitor with capacitance C to an inductive coil with self-inductance L 10 (Electrical engineers frequently call it the LC tank circuit.) I
Q
V
L
C
Fig. 2.4. LC tank circuit.
9 The above expression for T ( m / )(
2
2
2
2
x y z ) may be readily obtained either by the formal differentiation of Eq. (2) over time, or just by noticing that the velocity vector has two perpendicular components: one (of magnitude
R ) along the ring, and another one (of magnitude R sin ) normal to the ring’s plane.
10 A fancy font is used here to avoid any chance of confusion between the inductance and the Lagrange function.
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As the reader (hopefully :-) knows from their undergraduate studies, at relatively low frequencies we may use the so-called lumped-circuit approximation, in which the total energy of this system is the sum of two components, the electric energy E e localized inside the capacitor, and the magnetic energy E m localized inside the inductance coil:
2
2
Q
L I
E
,
E
.
(2.27)
e
2
m
C
2
Since the electric current I through the coil and the electric charge Q on the capacitor are related by the charge continuity equation dQ/ dt = I (evident from Fig. 4), it is natural to declare Q the generalized coordinate of the system, and the current, its generalized velocity. With this choice, the electrostatic energy E e ( Q) may be treated as the potential energy U of the system, and the magnetic energy E m( I), as its kinetic energy T. With this attribution, we get
T
E
T
E
U
E
Q
m
L I L Q,
m
,
0
e
,
(2.28)
q
I
q
Q
q
Q
C
so the Lagrange equation (19) becomes
d
Q
1
L Q ,0
i.e. Q
Q 0 .
(2.29)
dt
C
L C
Note, however, that the above choice of the generalized coordinate and velocity is not unique.
Instead, one can use, as the generalized coordinate, the magnetic flux through the inductive coil, related to the common voltage V across the circuit (Fig. 4) by Faraday’s induction law V = – d/ dt. With this choice, (- V) becomes the generalized velocity, E m = 2/2L should be understood as the potential energy, and E e = CV 2/2 treated as the kinetic energy. For this choice, the resulting Lagrange equation of motion is equivalent to Eq. (29). If both parameters of the circuit, L and C, are constant in time, Eq.
(29) describes sinusoidal oscillations with the frequency
1
.
(2.30)
0
L C1/2
This is of course a well-known result, which may be derived in a more standard way – by equating the voltage drops across the capacitor ( V = Q/ C) and the inductor ( V = –L dI/ dt –L d 2 Q/ dt 2).
However, the Lagrangian approach is much more convenient for more complex systems – for example, for the general description of the electromagnetic field and its interaction with charged particles.11
2.3. Hamiltonian function and energy
The canonical form (19) of the Lagrange equation has been derived using Eq. (18), which is formally similar to Eq. (1.22) for a potential force. Does this mean that the system described by Eq. (19) always conserves energy? Not necessarily, because the “potential energy” U that participates in Eq.
(18), may depend not only on the generalized coordinates but on time as well. Let us start the analysis of this issue with the introduction of two new (and very important!) notions: the generalized momentum corresponding to each generalized coordinate qj,
11 See, e.g., EM Secs. 9.7 and 9.8.
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L
p
,
(2.31) Generalized
j
q
momentum
j
and the Hamiltonian function 12
L
Hamiltonian
H
q L
p q
L .
(2.32)
j
function:
j
j
j
q
definition
j
j
To see whether the Hamiltonian function is conserved during the motion, let us differentiate both sides of its definition (32) over time:
dH
d L
L
dL
q
q
.
(2.33)
dt
j
j
j dt
q
q
dt
j
j
If we want to make use of the Lagrange equation (19), the last derivative has to be calculated considering L as a function of independent arguments q , q
j
, and t, so
j
dL
L
L
L
q
q
,
(2.34)
dt
j
j
j
q
q
t
j
j
where the last term is the derivative of L as an explicit function of time. We see that the last term in the square brackets of Eq. (33) immediately cancels with the last term in the parentheses of Eq. (34).
Moreover, using the Lagrange equation (19a) for the first term in the square brackets of Eq. (33), we see that it cancels with the first term in the parentheses of Eq. (34). As a result, we arrive at a very simple and important result:
dH
L
Hamiltonian
.
(2.35) function:
dt
t
time
evolution
The most important corollary of this formula is that if the Lagrangian function does not depend on time explicitly ( L / t ),
0 the Hamiltonian function is an integral of motion:
H const.
(2.36)
Let us see how this works, using the first two examples discussed in the previous section. For a 1D particle, the definition (31) of the generalized momentum yields
L
p
mv ,
(2.37)
x
v
so it coincides with the usual linear momentum – or rather with its x-component. According to Eq. (32), the Hamiltonian function for this case (with just one degree of freedom) is
p
2
2
x
m
p
H p v L p
x U
x
U ,
(2.38)
x
x m 2
2 m
12 It is named after Sir William Rowan Hamilton, who developed his approach to analytical mechanics in 1833, on the basis of the Lagrangian mechanics. This function is sometimes called just the “Hamiltonian”, but it is advisable to use the full term “Hamiltonian function” in classical mechanics, to distinguish it from the Hamiltonian operator used in quantum mechanics, whose abbreviation to Hamiltonian is extremely common.
(The relation of these two notions will be discussed in Sec. 10.1 below.)
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i.e. coincides with the particle’s mechanical energy E = T + U. Since the Lagrangian does not depend on time explicitly, both H and E are conserved.
However, it is not always that simple! Indeed, let us return again to our testbed problem (Fig. 1).
In this case, the generalized momentum corresponding to the generalized coordinate is L
2
p
mR ,
(2.39)
and Eq. (32) yields:
m
2
2
2
H p L mR
R 2
2
sin2 mgR cos const
2
(2.40)
m 2
R 2
2
sin2 mgR cos
.
const
2
This means that (as soon as 0 ), the Hamiltonian function differs from the mechanical energy m
2
E T U
R 2
2
sin2 mgR cos const .
(2.41)
2
The difference, E – H = mR 22sin2 (besides an inconsequential constant), may change at the bead’s motion along the ring, so although H is an integral of motion (since L/ t = 0), the energy is generally not conserved.
In this context, let us find out when these two functions, E and H, do coincide. In mathematics, there is a notion of a homogeneous function f( x 1, x 2,…) of degree , defined in the following way: for an arbitrary constant a,
f ( ax , ax ,...) a
f ( x , x ,...).
(2.42)
1
2
1
2
Such functions obey the following Euler theorem:13
f
x f ,
(2.43)
j
j x j
which may be simply proved by differentiating both parts of Eq. (42) over a and then setting this parameter to the particular value a = 1. Now, consider the case when the kinetic energy is a quadratic form of all generalized velocities q :
j
T t ( q , q ,..., t) q q , (2.44)
jj'
1
2
j
j'
j, j'
with no other terms. It is evident that such T satisfies the definition (42) of a homogeneous function of the velocities with = 2,14 so the Euler theorem (43) gives
T
q 2 T.
(2.45)
q
j
j
j
13 This is just one of many theorems bearing the name of their author – the genius mathematician Leonhard Euler (1707-1783).
14 Such functions are called quadratic-homogeneous.
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But since U is independent of the generalized velocities, L
/ q
T
/ q
, and the left-hand side of
j
j
Eq. (45) is exactly the first term in the definition (32) of the Hamiltonian function, so in this case H 2 T L 2 T ( T U ) T U E.
(2.46)
So, for a system with a kinetic energy of the type (44), for example, a free particle with T
considered as a function of its Cartesian velocities,
m
T
2 2 2
v v v ,
(2.47)
x
y
2
z
the notions of the Hamiltonian function and mechanical energy are identical. Indeed, some textbooks, very regrettably, do not distinguish these notions at all! However, as we have seen from our bead-on-the-rotating-ring example, these variables do not always coincide. For that problem, the kinetic energy, in addition to the term proportional to 2
, has another, velocity-independent term – see the first of Eqs.
(23) – and hence is not a quadratic-homogeneous function of the angular velocity, giving E H.
Thus, Eq. (36) expresses a new conservation law, generally different from that of mechanical energy conservation.
2.4. Other conservation laws
Looking at the Lagrange equation (19), we immediately see that if L T – U is independent of some generalized coordinate qj, L/ qj = 0,15 then the corresponding generalized momentum is an integral of motion:16
L
p
.
const
(2.48)
j
q j
For example, for a 1D particle with the Lagrangian (21), the momentum px is conserved if the potential energy is constant (and hence the x-component of force is zero) – of course. As a less obvious example, let us consider a 2D motion of a particle in the field of central forces. If we use polar coordinates r and
in the role of generalized coordinates, then the Lagrangian function17
m
L T U
2 2 2
r r U ( r)
(2.49)
2
is independent of , and hence the corresponding generalized momentum,
L
2
p
mr
,
(2.50)
15 Such coordinates are frequently called cyclic, because in some cases (like in Eq. (49) below) they represent periodic coordinates such as angles. However, this terminology is somewhat misleading, because some “cyclic”
coordinates (e.g., x in our first example) have nothing to do with rotation.
16 This fact may be considered a particular case of a more general mathematical statement called the Noether theorem – named after its author, Emmy Nöther, sometimes called the “greatest woman mathematician ever lived”. Unfortunately, because of time/space restrictions, for its discussion I have to refer the interested reader elsewhere – for example to Sec. 13.7 in H. Goldstein et al., Classical Mechanics, 3rd ed. Addison Wesley, 2002.
17 Note that here 2
r is the square of the scalar derivative r, rather than the square of the vector r = v.
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is conserved. This is just a particular (2D) case of the angular momentum conservation – see Eq. (1.24).
Indeed, for the 2D motion within the [ x, y] plane, the angular momentum vector, n
n
n
x
y
z
L r p x
y
z ,
(2.51)
x
m
y
m
z
m
has only one component different from zero, namely the component normal to the motion plane: L x ( y
m) y ( x
m ).
(2.52)
z
Differentiating the well-known relations between the polar and Cartesian coordinates, x r cos,
y r sin,
(2.53)
over time, and plugging the result into Eq. (52), we see that
2
L mr p .
(2.54)
z
Thus the Lagrangian formalism provides a powerful way of searching for non-evident integrals of motion. On the other hand, if such a conserved quantity is obvious or known a priori, it is helpful for the selection of the most appropriate generalized coordinates, giving the simplest Lagrange equations.
For example, in the last problem, if we knew in advance that p had to be conserved, this could provide sufficient motivation for using the angle as one of the generalized coordinates.
2.5. Exercise problems
In each of Problems 1-11, for the given system:
(i) introduce a convenient set of generalized coordinates qj,
(ii) write down the Lagrangian L as a function of q , q
j , and (if appropriate) time,
j
(iii) write down the Lagrange equation(s) of motion,
(iv) calculate the Hamiltonian function H; find out whether it is conserved, (v) calculate the mechanical energy E; is E = H?; is the energy conserved?
(vi) any other evident integrals of motion?
2.1. A double pendulum – see the figure on the right. Consider only the motion
within the vertical plane containing the suspension point.
l
m
l
g
m
2.2. A stretchable pendulum (i.e. a massive particle hung on an elastic cord that exerts force F = –( l – l
l
0), where and l 0 are positive constants), also confined to the vertical plane:
g
m
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x
0( t)
2.3. A fixed-length pendulum hanging from a point whose motion law x 0( t) in the horizontal direction is fixed. (No vertical plane constraint here.)
l
g
m
2.4. A pendulum of mass m, hung on another point mass m’ that may slide, without m'
friction, along a straight horizontal rail – see the figure on the right. The motion is confined l
to the vertical plane that contains the rail.
g
m
2.5. A point-mass pendulum of length l, attached to the rim of a disk of
radius R, which is rotated in a vertical plane with a constant angular velocity
– see the figure on the right. (Consider only the motion within the disk’s plane.) R
l m
g
2.6. A bead of mass m, sliding without friction along a light
2 d
string with a fixed tension T, hung between two horizontally
displaced supports – see the figure on the right. Here, in contrast to T
T
the similar Problem 1.10, the tension T may be comparable with the
bead’s weight mg, and the motion is not restricted to the vertical g
m
plane.
2.7. A bead of mass m, sliding without friction along a light string of a 2 d
fixed length 2 l, that is hung between two support points displaced horizontally by distance 2 d < 2 l – see the figure on the right. As in the previous problem, l
2
the motion is not restricted to the vertical plane.
g
m
2.8. A block of mass m that can slide, without friction, along the m
inclined plane surface of a heavy wedge with mass m’. The wedge is free to m'
move, also without friction, along a horizontal surface – see the figure on the
g
right. (Both motions are within the vertical plane containing the steepest slope line.)
2.9. The two-pendula system that was the subject of Problem 1.8 – see
l
l
the figure on the right.
g
m
m
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M
2.10. A system of two similar, inductively coupled LC circuits –
C
C
see the figure on the right.
L
L
2.11.*A small Josephson junction – the system consisting of two S
superconductors (S) weakly coupled by Cooper-pair tunneling through a E , C
I
J
thin insulating layer (I) that separates them – see the figure on the right.
S
Hints:
(i) At not very high frequencies (whose quantum is lower than the binding energy 2 of the Cooper pairs), the Josephson effect in a sufficiently small junction may be described by the following coupling energy:
U E cos const ,
J
where the constant E J describes the coupling strength, while the variable (called the Josephson phase difference) is connected to the voltage V across the junction by the famous frequency-to-voltage relation d
2 e
V ,
dt
where e 1.60210-19 C is the fundamental electric charge and 1.05410-34 Js is the Planck constant.18
(ii) The junction (as any system of two close conductors) has a substantial electric capacitance C.
18 More discussion of the Josephson effect and the physical sense of the variable may be found, for example, in EM Sec. 6.5 and QM Secs. 1.6 and 2.8, but the given problem may be solved without that additional information.
Chapter 2
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Chapter 2
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Chapter 3. A Few Simple Problems
The objective of this chapter is to solve a few simple but very important particle dynamics problems that may be reduced to 1D motion. They notably include the famous “planetary” problem of two particles interacting via a spherically symmetric potential, and the classical particle scattering problem. In the process of solution, several methods that will be very essential for the analysis of more complex systems are also discussed.
3.1. One-dimensional and 1D-reducible systems
If a particle is confined to motion along a straight line (say, axis x), its position is completely determined by this coordinate. In this case, as we already know, the particle’s Lagrangian function is given by Eq. (2.21):
m
L T x U ( x, t), T x
2
x ,
(3.1)
2
so the Lagrange equation of motion given by Eq. (2.22),
U
( x, t)
x
m
(3.2)
x
is just the x-component of the 2nd Newton’s law.
It is convenient to discuss the dynamics of such really-1D systems as a part of a more general class of effectively-1D systems. This is a system whose position, due to either holonomic constraints and/or conservation laws, is also fully determined by one generalized coordinate q, and whose Lagrangian may be represented in a form similar to Eq. (1):
Effectively-
m
ef
2
L T ( q
1D system
) U ( q, t),
T
q ,
(3.3)
ef
ef
ef
2
where m ef is some constant which may be considered as the effective mass of the system, and the function U ef, its effective potential energy. In this case, the Lagrange equation (2.19), describing the system’s dynamics, has a form similar to Eq. (2):
U
( q, t)
ef
m q
.
(3.4)
ef
q
As an example, let us return to our testbed system shown in Fig. 2.1. We have already seen that for this system, having one degree of freedom, the genuine kinetic energy T, expressed by the first of Eqs. (2.23), is not a quadratically-homogeneous function of the generalized velocity. However, the system’s Lagrangian function (2.23) still may be represented in the form (3),
m
m
2
2
2
2
L
R
R sin 2 mgR cos const
T U ,
(3.5)
2
2
ef
ef
provided that we take
© K. Likharev
CM: Classical Mechanics
m
m
2
2
T
R ,
2
2
U
R sin 2 mgR cos
.
const
(3.6)
ef