1.14. A particle of mass m, moving with velocity u, collides head-on with a particle of mass M, initially at rest. Calculate the velocities of both particles after the collision, if the initial energy of the system is barely sufficient for an increase of its internal energy by E.
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.
Chapter 1
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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 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 that for the sake of brevity, these limits are not explicitly specified.
Chapter 2
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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.
Chapter 2
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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 the 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 the 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 that
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 that 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”.
Chapter 2
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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 that
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 that 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 j
j
definition
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 that
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 that 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 bead’s motion along the ring, so that 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 that 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 that 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 2.1-2.11, 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 horizontal support whose motion law
x 0( t) 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, that 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 stretched by a fixed force T between two horizontally
displaced points – see the figure on the right. Here, in contrast to the T
T
similar Problem 1.10, the string’s tension T may be comparable
with the bead’s weight mg, and the motion is not restricted to the g
m
vertical 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 points displaced horizontally by distance 2 d < 2 l – see the figure on the right. As in the previous problem, the l
2
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
Chapter 2
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Essential Graduate Physics
CM: Classical Mechanics
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 of this series, 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 that 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
2
ef
2
In this new partitioning of the function L, which is legitimate because U ef depends only on the generalized coordinate , but not on the corresponding generalized velocity, T ef includes only a part of the genuine kinetic energy T of the bead, while U ef includes not only its real potential energy U in the gravity field but also an additional term related to ring rotation. (As we will see in Sec. 4.6, this term may be interpreted as the effective potential energy due to the inertial centrifugal “force” arising at the problem’s solution in the non-inertial reference frame rotating with the ring.)
Returning to the general case of effectively-1D systems with Lagrangians of the type (3), let us calculate their Hamiltonian function, using its definition (2.32):
L
2
H
q L m q ( T U ) T U .
(3.7)
ef
ef
ef
ef
ef
q
So, H is expressed via T ef and U ef exactly as the energy E is expressed via genuine T and U.
3.2. Equilibrium and stability
Autonomous systems are defined as dynamic systems whose equations of motion do not depend on time explicitly. For the effectively-1D (and in particular the really-1D) systems obeying Eq. (4), this means that their function U ef, and hence the Lagrangian function (3) should not depend on time explicitly. According to Eqs. (2.35), in such systems, the Hamiltonian function (7), i.e. the sum T ef + U ef, is an integral of motion. However, be careful! Generally, this conclusion is not valid for the genuine mechanical energy E of such a system; for example, as we already know from Sec. 2.2, for our testbed problem, with the generalized coordinate q = (Fig. 2.1), E is not conserved.
According to Eq. (4), an autonomous system, at appropriate initial conditions, may stay in equilibrium at one or several stationary (alternatively called fixed) points qn, corresponding to either the minimum or a maximum of the effective potential energy (see Fig. 1):
dU
ef q
(3.8)
Fixed-point
n
.
0
dq
condition
1
~2
q
ef
2
U ( q)
ef
Fig. 3.1. An example of the effective
potential energy profile near stable ( q 0, q 2)
and unstable ( q 1) fixed points, and its
quadratic approximation (10) near point q 0.
q
q
q
q
0
1
2
In order to explore the stability of such fixed points, let us analyze the dynamics of small deviations
q~ t
( ) q t
( ) q
(3.9)
n
from one of such points. For that, let us expand the function U ef(q) in the Taylor series at qn: Chapter 3
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CM: Classical Mechanics
dU
d U
ef
1 2
~
ef
~
U ( q) U ( q )
( q ) q
( q ) 2
q ....
(3.10)
ef
ef
n
dq
n
2
2
dq
n
The first term on the right-hand side, U ef( qn), is an arbitrary constant and does not affect motion. The next term, linear in the deviation q~ , equals zero – see the fixed point’s definition (8). Hence the fixed point’s stability is determined by the next term, quadratic in q~ , more exactly by its coefficient, 2
d U
ef
( q ) ,
(3.11)
ef
2
n
dq
which is frequently called the effective spring constant. Indeed, neglecting the higher terms of the Taylor expansion (10),1 we see that Eq. (4) takes the familiar form:
~
~
m q q 0.
(3.12)
ef
ef
I am confident that the reader of these notes knows everything about this equation, but since we will soon run into similar but more complex equations, let us review the formal procedure of its solution. From the mathematical standpoint, Eq. (12) is an ordinary linear differential equation of the second order, with constant coefficients. The general theory of such equations tells us that its general solution (for any initial conditions) may be represented as
~
t
t
q t
( )
c e c e
,
(3.13)
where the constants c are determined by initial conditions, while the so-called characteristic exponents
are completely defined by the equation itself. To calculate these exponents, it is sufficient to plug just one partial solution, e t, into the equation. In our simple case (12), this yields the following characteristic equation:
2
m 0 .
(3.14)
ef
ef
If the ratio k ef /m ef is positive, i.e. the fixed point corresponds to the minimum of potential energy (e.g., see points q 0 and q 2 in Fig. 1), the characteristic equation yields 1/ 2
ef
i ,
with
,
(3.15)
0
0
m ef
(where i is the imaginary unit, i 2 = –1), so that Eq. (13) describes harmonic (sinusoidal) oscillations of the system,2
q~ t
( )
c e i t
0
c e i t 0 c cos t c sin t ,
(3.16)
c
0
s
0
1 Those terms may be important only in very special cases then ef is exactly zero, i.e. when a fixed point is also an inflection point of the function U ef( q).
2 The reader should not be scared of the first form of (16), i.e. of the representation of a real variable (the deviation from equilibrium) via a sum of two complex functions. Indeed, any real initial conditions give c–* = c+, so that the sum is real for any t. An even simpler way to deal with such complex representations of real functions will be discussed in the beginning of Chapter 5, and then used throughout this series.
Chapter 3
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CM: Classical Mechanics
with the frequency 0, about the fixed point – which is thereby stable.3 On the other hand, at the potential energy maximum ( k ef < 0, e.g., at point q 1 in Fig. 1), we get 1/ 2
ef
~
,
where
,
that
so
q ( t)
t
t
c e
c e
.
(3.17)
m
ef
Since the solution has an exponentially growing part,4 the fixed point is unstable.
Note that the quadratic expansion of function U ef( q), given by the truncation of Eq. (10) to the three displayed terms, is equivalent to a linear Taylor expansion of the effective force: dU
ef
~
F
q,
(3.18)
ef
ef
dq
immediately resulting in the linear equation (12). Hence, to analyze the stability of a fixed point qn, it is sufficient to linearize the equation of motion with respect to small deviations from the point, and study possible solutions of the resulting linear equation. This linearization procedure is typically simpler to carry out than the quadratic expansion (10).
As an example, let us return to our testbed problem (Fig. 2.1) whose function U ef we already know – see the second of Eqs. (6). With it, the equation of motion (4) becomes
dU
2
ef
2
mR
mR 2
cos Ω2 sin,
i.e. 2
cos Ω2 sin,
(3.19)
d
where ( g/ R)1/2 is the frequency of small oscillations of the system at = 0 – see Eq. (2.26).5 From Eq. (8), we see that on any 2-long segment of the angle , 6 the system may have four fixed points; for example, on the half-open segment (-, +] these points are
2
Ω
1
,
0
,
cos
.
(3.20)
0
1
2,3
2
The last two fixed points, corresponding to the bead shifted to either side of the rotating ring, exist only if the angular velocity of the rotation exceeds . (In the limit of very fast rotation, >> , Eq. (20) yields 2,3 /2, i.e. the stationary positions approach the horizontal diameter of the ring – in accordance with our physical intuition.)
~
To analyze the fixed point stability, we may again use Eq. (9), in the form , plug it n
into Eq. (19), and Taylor-expand both trigonometric functions of up to the term linear in ~ :
~
2
~
2
~
cos sin
.
(3.21)
n
n
sin cos
n
n
3 This particular type of stability, when the deviation from the equilibrium oscillates with a constant amplitude, neither growing nor decreasing in time, is called either the orbital, or “neutral”, or “indifferent” stability.
4 Mathematically, the growing part vanishes at some special (exact) initial conditions which give c+ = 0. However, the futility of this argument for real physical systems should be obvious to anybody who has ever tried to balance a pencil on its sharp point.
5 Note that Eq. (19) coincides with Eq. (2.25). This is a good sanity check illustrating that the procedure (5)-(6) of moving a term from the potential to the kinetic energy within the Lagrangian function is indeed legitimate.
6 For this particular problem, the values of that differ by a multiple of 2, are physically equivalent.
Chapter 3
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Generally, this equation may be linearized further by purging its right-hand side of the term proportional
~
to 2
; however in this simple case, Eq. (21) is already convenient for analysis. In particular, for the fixed point 0 = 0 (corresponding to the bead’s position at the bottom of the ring), we have cos 0 = 1
and sin0 = 0, so that Eq. (21) is reduced to a linear differential equation
~
2
2
~
– Ω ,
(3.22)
whose characteristic equation is similar to Eq. (14) and yields
2
2
2
– Ω , for .
(3.23a)
0
This result shows that if 2 < 2, both roots are imaginary, so that this fixed point is orbitally stable.
However, if the rotation speed is increased so that 2 < 2, the roots become real: = (2 – 2)1/2, with one of them positive, so that the fixed point becomes unstable beyond this threshold, i.e. as soon as fixed points 2,3 exist. Absolutely similar calculations for other fixed points yield
Ω2 2
,
0
for ,
2
1
(3.23b)
Ω2 2
,
for .
2,3
These results show that the fixed point 1 (the bead on the top of the ring) is always unstable – just as we could foresee, while the side fixed points 2,3 are orbitally stable as soon as they exist – at 2 < 2.
Thus, our fixed-point analysis may be summarized very simply: an increase of the ring rotation speed beyond a certain threshold value, equal to given by Eq. (2.26), causes the bead to move to one of the ring sides, oscillating about one of the fixed points 2,3. Together with the rotation about the vertical axis, this motion yields quite a complex (generally, open) spatial trajectory as observed from a lab frame, so it is fascinating that we could analyze it quantitatively in such a simple way.
Later in this course, we will repeatedly use the linearization of the equations of motion for the analysis of the stability of more complex dynamic systems, including those with energy dissipation.
3.3. Hamiltonian 1D systems
Autonomous systems that are described by time-independent Lagrangians are frequently called Hamiltonian ones because their Hamiltonian function H (again, not necessarily equal to the genuine mechanical energy E!) is conserved. In our current 1D case, described by Eq. (3), m
ef
2
H
q U ( q) const .
(3.24)
2
ef
From the mathematical standpoint, this conservation law is just the first integral of motion. Solving Eq.
(24) for q , we get the first-order differential equation,
1/ 2
dq
2
m 1/2
ef
dq
H U ( q)
,
i.e.
,
(3.25)
ef
dt
m
2
ef
H U ( )
ef
dt
q 1/ 2
which may be readily integrated:
Chapter 3
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CM: Classical Mechanics
1/ 2 q( )
m
t
dq'
ef
t t
.
(3.26)
2 q( t ) H U ( q' )
ef
1/2
0
0
Since the constant H (as well as the proper sign before the integral – see below) is fixed by initial conditions, Eq. (26) gives the reciprocal form, t = t( q), of the desired law of system motion, q( t). Of course, for any particular problem the integral in Eq. (26) still has to be worked out, either analytically or numerically, but even the latter procedure is typically much easier than the numerical integration of the initial, second-order differential equation of motion, because at the addition of many values (to which any numerical integration is reduced7) the rounding errors are effectively averaged out.
Moreover, Eq. (25) also allows a general classification of 1D system motion. Indeed:
(i) If H > U ef( q) in the whole range of our interest, the effective kinetic energy T ef (3) is always positive. Hence the derivative dq/ dt cannot change its sign, so that this effective velocity retains the sign it had initially. This is an unbound motion in one direction (Fig. 2a).
(a)
(b)
(c)
H
H
H
A
A
B
U ( q)
ef
U ( q)
U ( q)
ef
ef
U min
q 0
(d)
U ( )
ef
H 2
2 mgR
0
1.5
Ω
B
H 1
B'
A
1
2 1
0
1
2
3
/
Fig. 3.2. Graphical representations of Eq. (25) for three different cases: (a) an unbound motion, with the velocity sign conserved, (b) a reflection from a “classical turning point”, accompanied by the velocity sign change, and (c) bound, periodic motion between two turning points – schematically. (d) The effective potential energy (6) of the bead on the rotating ring (Fig. 2.1) for a particular case with 2 < 2.
(ii) Now let the particle approach a classical turning point A where H = U ef( q) – see Fig. 2b.8
According to Eq. (25), at that point the particle velocity vanishes, while its acceleration, according to Eq. (4), is still finite. This means that the particle’s velocity sign changes its sign at this point, i.e. it is reflected from it.
7 See, e.g., MA Eqs. (5.2) and (5.3).
8 This terminology comes from quantum mechanics, which shows that a particle (or rather its wavefunction) actually can, to a certain extent, penetrate “classically forbidden” regions where H < U ef( q).
Chapter 3
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(iii) If, after the reflection from some point A, the particle runs into another classical turning point B (Fig. 2c), the reflection process is repeated again and again, so that the particle is bound to a periodic motion between two turning points.
The last case of periodic oscillations presents a large conceptual and practical interest, and the whole Chapter 5 will be devoted to a detailed analysis of this phenomenon and numerous associated effects. Here I will only note that for an autonomous Hamiltonian system described by Eq. (4), Eq. (26) immediately enables the calculation of the oscillation period:
1/ 2
Oscillation
m A
dq
period
T 2
ef
,
(3.27)
2
[ H U ( q)]1/ 2
B
ef
where the additional front factor 2 accounts for two time intervals: of the motion from B to A and back –
see Fig. 2c. Indeed, according to Eq. (25), at each classically allowed point q, the velocity’s magnitude is the same, so these time intervals are equal to each other.
(Note that the dependence of points A and B on H is not necessarily continuous. For example, for our testbed problem, whose effective potential energy is plotted in Fig. 2d for a particular value of >
, a gradual increase of H leads to a sudden jump, at H = H 1, of the point B to a new position B’, corresponding to a sudden switch from oscillations about one fixed point 2,3 to oscillations about two adjacent fixed points – before the beginning of a persistent rotation around the ring at H > H 2.) Now let us consider a particular, but a very important limit of Eq. (27). As Fig. 2c shows, if H is reduced to approach U min, the periodic oscillations take place at the very bottom of this potential well, about a stable fixed point q 0. Hence, if the potential energy profile is smooth enough, we may limit the Taylor expansion (10) to the displayed quadratic term. Plugging it into Eq. (27), and using the mirror symmetry of this particular problem about the fixed point q 0, we get
1/ 2 A
~
m
dq
d
ef
4
1
T 4
I
I
(3.28)
2
H
~2
U
q / 2
min
ef
,
with
1/ 2
1/ 2
2
0
0
0
,
1
where
q~
/ A , with A (2/ef)1/2( H – U min)1/2 being the classical turning point, i.e. the oscillation amplitude, and 0 the frequency given by Eq. (15). Taking into account that the elementary integral I in that equation equals /2,9 we finally get
2
T
,
(3.29)
0
as it should be for the harmonic oscillations (16). Note that the oscillation period does not depend on the oscillation amplitude A, i.e. on the difference ( H – U min) – while it is sufficiently small.
3.4. Planetary problems
Leaving a more detailed study of oscillations for Chapter 5, let us now discuss the so-called planetary systems 10 whose description, somewhat surprisingly, may be also reduced to an effectively 1D
9 Indeed, introducing a new variable as sin , we get d = cos d = (1–2)1/2 d, so that the function under the integral is just d, and its limits are = 0 and = /2.
Chapter 3
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