D’Alembert’s principle applies to the constraining forces: a For all virtual displacements a force of constraint does no work n α=1 The following two statements are equivalent: b A forc
Trang 112 Remarks on Non-integrable Systems: Chaos
Systems for which the number of independent conserved quantities agrees
with the number f of degrees of freedom are termed integrable They are quasi especially simple and set the standards in many respects.
If one fixes the values of the independent conserved quantities, then in 2f -dimensional phase-space (with the canonical phase-space variables p1 , , p f
and q1 , , q f ) one generates an f -dimensional hypersurface, which for f = 2 has the topology of a torus.
However, it is obvious that most systems are non-integrable, since gener-ically the number of degrees of freedom is larger than the number of con-servation theorems This applies, e.g., to so-called three-body problems, and
it would also apply to the asymmetric heavy top, as mentioned above, or to the double pendulum It is no coincidence that in the usual textbooks1 little
attention is paid to non-integrable systems because they involve complicated
relations, which require more mathematics than can be assumed on a more
or less elementary level.2
Linear systems are, as we have seen above, always simple, at least in
principle In contrast, for non-linear non-integrable systems chaotic behavior occurs In most cases this behavior is qualitatively typical and can often be understood from simple examples or so-called scenarios One of these scenar-ios concerns the so-called sensitive dependence on the initial conditions, e.g.
as follows
Consider a non-linear system of differential equations
dX(t)
dt =f(t; X0) , whereX0are the initial values ofX(t) for t = t0 We then ask whether the orbits X(t) of this non-linear “dynamical system” depend continuously on
the initial values in the limit t → ∞; or we ask how long the orbits remain
in anε-neighborhood of the initial values.
1
This text makes no exception
2 It is mostly unknown and symptomatic for the complexity of nonintegrable sys-tems that Sommerfeld, who was one of the greatest mathematical physicists of the time, wrote in the early years of the twentieth century a voluminous book containing three volumes on the “Theory of the Spinning Top”
Trang 286 12 Remarks on Non-integrable Systems: Chaos
In particular one might ask for the properties of the derivative
dX(t; X0+ε)
dε
as a function of t For “malicious” behavior of the system e.g., for turbulent
flow, weather or stock-exchange or traffic forecasting etc., the limits ε → 0
and t → ∞ cannot be interchanged This has the practical consequence that
very small errors in the data accumulation or in the weighting can have irreparable consequences beyond a characteristic time for the forecasting or
turbulent flow (the so-called butterfly effect ).
However, these and other topics lead far beyond the scope of this text and shall therefore not be discussed in detail, especially since a very readable book on chaotic behavior already exists [9]
Some of these aspects can be explained by the above example of the
spherical pendulum, see section 8.2 As long as the length of the thread of the pendulum is constant, the system is conservative in the mechanical sense; or
in a mathematical sense it is describable by an autonomous system of two coupled ordinary differential equations for the two variables ϕ(t) and ϑ(t) Thus the number of degrees of freedom is f = 2, which corresponds perfectly
to the number of independent conserved quantities, these being the vertical component of the angular momentum plus the sum of kinetic and potential
energies: the system is integrable.
However, the integrability is lost if the length of the pendulum depends
explicitly on t and/or the rotational invariance w.r.t the azimuthal angle ϕ
is destroyed3 In the first case the system of coupled differential equations
for the variables ϑ and ϕ becomes non-autonomous; i.e., the time variable t
must be explicitly considered as a third relevant variable
In the following we shall consider autonomous systems In a 2f -dimen-sional phase space Φ of (generalized) coordinates and momenta we consider
a two-dimensional sub-manifold, e.g., a plane, and mark on the plane the points, one after the other, where for given initial values the orbit intersects
the plane In this way one obtains a so-called Poincar´ e section, which gives
a condensed impression of the trajectory, which may be very “chaotic” and may repeatedly intersect the plane For example, for periodic motion one obtains a deterministic sequence of a finite number of discrete intersections
with the plane; for nonperiodic motion one typically has a more or less chaotic
or random sequence, from which, however, on detailed inspection one can sometimes still derive certain nontrivial quantitative laws for large classes of similar systems
3 This applies, e.g., to the so-called Henon-Heiles potential V (x1, x2) := x2+
x2+ ε · (x2
x2− x33), which serves as a typical example of non-integrability and
“chaos” in a simple two-dimensional system; for ε = 0 the potential is no longer
rotationally invariant, but has only discrete triangular symmetry.
Trang 312 Remarks on Non-integrable Systems: Chaos 87
A typical nonintegrable system, which has already been mentioned, is
the double pendulum, i.e., with L(ϑ1, ϑ2, ˙ ϑ1, ˙ ϑ2), which was discussed in Sect. 11.2 If here the second pendulum sometimes “flips over”, the sequence of these times is of course deterministic, but practically “random”, i.e.,
non-predictable, such that one speaks of deterministic chaos; this can easily be
demonstrated experimentally
Trang 413 Lagrange Formalism II: Constraints
13.1 D’Alembert’s Principle
Consider a system moving in n-dimensional space with (generalized) coor-dinates q1, ,q n , but now under the influence of constraints The constraints can be either (i) holonomous or (ii) anholonomous In the first case a single constraint can be formulated, as follows: f (q1 , q2, , q n , t)= 0, i.e., the con-!
straint defines a time-dependent (n −1)-dimensional hypersurface in R n(q) If
one has two holonomous constraints, another condition of this kind is added, and the hypersurface becomes (n − 2)-dimensional, etc
In the second case we may have:
n
α=1
a α (q1, , q n , t) dq α + a0(q1, , q n , t) dt = 0 ,! (13.1)
where (in contrast to holonomous constraints, for which necessarily a α ≡ ∂f
∂q α,
and generally but not necessarily, a0 ≡ ∂f
∂t) the conditions of integrability,
∂a α
∂q β − ∂a β
∂q α ≡ 0, ∀α, β = 1, , n , are not all satisfied; thus in this case one has only local hypersurface elements,
which do not fit together
If the constraints depend (explicitly) on the time t, they are called rheonomous, otherwise skleronomous.
In the following we shall define the term virtual displacement : in contrast
to real displacements, for which the full equation (13.1) applies and which we describe by exact differentials dq α , the virtual dispacements δq α are written
with the variational sign δ, and instead of using the full equation (13.1), for the δq α the following shortened condition is used:
n
a α (q1 , , q2, t)δq α
!
Trang 590 13 Lagrange Formalism II: Constraints
In the transition from equation (13.1) to (13.2) we thus always put δt ≡ 0 (although dt may be = 0), which corresponds to the special role of time in
a Galilean transformation1, i.e., to the formal limit c → ∞.
To satisfy a constraint, the system must exert an n-dimensional force
of constraint Z.2 (For more than one constraint, μ = 1, , λ, of course,
a corresponding set of forces Z (μ) would be necessary, but for simplicity in
the following we shall only consider the case n = 1 explicitly, where the index
μ can be omitted.)
D’Alembert’s principle applies to the constraining forces:
a) For all virtual displacements a force of constraint does no work
n
α=1
The following two statements are equivalent:
b) A force of constraint is always perpendicular to the instantaneous
hyper-surface or to the local virtual hyperhyper-surface element, which corresponds to
the constraint; e.g., we have δf (q1 , , q n , t) ≡ 0 or equation (13.2) c) A so-called Lagrange multiplier λ exists, such that for all
α = 1, , n : Z α = λ · a α (q1 , , q n , t)
These three equivalent statements have been originally formulated in cartesian coordinates; but they also apply to generalized coordinates, if the
term force of constraint is replaced by a generalized force of constraint.
Now, since without constraints
L = T − V
while for cartesian coordinates the forces are
F α=− ∂ V
∂x α
it is natural to modify the Hamilton principle of least action in the presence
of a single holonomous constraint, as follows:
dS[q1+ εδq1, , q n + εδq n]
=
t2
t=t1
dt {L(q1, , q n , ˙ q1, , ˙ q n , t) + λ · f(q1, , q n , t) } !
= 0 (13.4)
1 This is also the reason for using a special term in front of dt in the definition
(13.1) of anholonomous constraints
2 As stated below, here one should add a slight generalization: force → general-ized force; i.e.,Pn
β=1 Z˜β δx β ≡Pn
α=1 Z α δq α, where the ˜Z β are the (cartesian)
components of the constraining force and the Z α the components of the related generalized force
Trang 613.2 Exercise: Forces of Constraint for Heavy Rollerson an Inclined Plane 91
Analogous relations apply for more than one holonomous or anholonomous
constraint Instead of the Lagrange equations of the second kind we now have:
− d dt
∂ L
∂ ˙ q α
+ ∂ L
∂q α
+ λ · ∂f
∂q α
For anholonomous constraints the term
λ · ∂f
∂q α
is replaced by λ · a α, and for additional constraints one has a sum of similar
μ
λ (μ) · a (μ)
α
These are the Lagrange equations of the 2nd kind with constraints
Ori-ginally they were formulated only in cartesian coordinates as so-called
La-grange equations of the 1st kind, which were based on the principle of d’ Alembert and from which the Lagrange equations of the 2nd kind were
de-rived Many authors prefer this historical sequence
Textbook examples of anholonomous constraints are not very common.
We briefly mention here the example of a skater In this example the con-straint is such that the gliding direction is given by the angular position of the skates The constraints would be similar for skiing
13.2 Exercise: Forces of Constraint for Heavy Rollers
on an Inclined Plane
For the above-mentioned problem of constraints additional insight can again
be gained from the seemingly simple problem of a roller on an inclined plane Firstly we shall consider the Euler angles:
– ϕ is the azimuthal rotation angle of the symmetry axis ±e3 of the roller about the fixed vertical axis (±ˆz-axis); this would be a “dummy” value,
if the plane were not inclined We choose the value ϕ ≡ 0 to correspond
to the condition that the roller is just moving down the plane, always in
the direction of steepest descent.
– ϑ is the tilt angle between the vector ˆ e3and the vector ˆz; usually ϑ = π/2.
– Finally, ψ is the azimuthal rotation angle corresponding to the distance
Δs = R · Δψ moved by the perimeter of the roller; R is the radius of the roller.
Trang 792 13 Lagrange Formalism II: Constraints
In the following we consider the standard assumptions3
ϕ ≡ 0 and ϑ ≡ π
2 , i.e., we assume that the axis of the circular cylinder lies horizontally on the plane, which may be not always true
In each case we assume that the Lagrangian may be written
L = T − V = M
2 v
2
s +Θ ||
2
˙
ψ2+ M gs · sin α , where α corresponds to the slope of the plane, and s is the distance
corre-sponding to the motion of the center of mass, i.e., of the axis of the roller
We now consider three cases, with vs= ˙s:
a) Let the plane be perfectly frictionless, i.e., the roller slides down the inclined plane The number of degrees of freedom is, therefore, f = 2; they correspond to the generalized coordinates s and ψ As a consequence, the
equation
d
dt
∂ L
∂ ˙s − ∂ L
∂s = 0
results in
M ¨ s = M g sin α
In contrast the angle ψ is cyclic, because
∂ L
∂ψ = 0 ;
therefore one has
vs= v s |0 + geff · (t − t0) and ψ = constant ,˙
i.e., the heavy roller slides with constant angular velocity and with effec-tive gravitational acceleration
geff := g sin α , which is given by the slope tan α of the inclined plane.
b) In contrast, let the plane be perfectly rough, i.e., the cylinder rolls down the plane Now we have f ≡ 1; and since
˙
ψ = vs
R :L ≡ M
2 v
2
s +Θ ||
2
v2 s
R2 + M geffs
3 One guesses that the problem can be made much more complex, if the roller does
not simply move down the plane in the direction of steepest descent, but if ϕ and/or ϑ were allowed to vary; however, even to formulate these more complex
problems would take some effort
Trang 813.2 Exercise: Forces of Constraint for Heavy Rollerson an Inclined Plane 93
Therefore
vs≡ R ˙ψ = v s |0+M · geff
Meff · (t − t0) , with Meff := M ·
1 + Θ ||
M R2
.
The holonomous (and skleronomous) constraint s − Rψ = 0 has been
explicitly eliminated, such that for the remaining degree of freedom the simple Lagrange equation of the second kind without any constraint could
be used directly
But how do the constraining forces originate? (These are frictional forces responsible for the transition from sliding to rolling.) The answer is ob-tained by detailed consideration, as follows:
c) The plane is rough, but initially 0≤ Rsψ < v˙
s(e.g., the rolling is slow or
zero) Now consider the transition from f = 2 to f ≡ 1.
We have,
(1), M ˙vs= F g − F f r and , (2), Θ || ψ = R¨ · F g
(Here F g = M g sin α is the constant gravitational force applied to the axis,
directed downwards, while (−F f r ) is the frictional force, ∝ ˙ψ, applied to
the tangential point of rolling, and with upward direction.)
According to (b), the angular velocity ˙ψ increases (e.g., from zero) ∝ t as
the cylinder rolls downwards; at the same time the gravitational force is
constant; thus, after a certain time τ c the frictional force counteracts the
gravitational one and vs = R ˙ ψ As a consequence, a weighted sum of the
two above equations yields after this time:
M + Θ ||
R2
˙vs = F g
Here one sees explicity how d’Alembert’s principle (that the forces of
con-straint do no work, one considers virtual displacements) becomes satisfied after τ c: Then we have
δA = F f r · δr + D ψ δψ = −F g ds + RF f r dψ = 0 for ds = R dψ Another consequence of the above facts is that a soccer player should avoid letting the football roll on the grass, because the ball is slowed down due to rolling:
M + Θ ||
R2
.
Trang 914 Accelerated Reference Frames
14.1 Newton’s Equation
in an Accelerated Reference Frame
Thus far we have considered reference frames moving at a constant velocity.
In the following, however, we shall consider the transition to accelerated coor-dinate systems, where the accelerated coorcoor-dinates and basis vectors are again
denoted by a prime, whiler0(t) is the radius vector of the accelerated origin.
Keeping to the limit v2 c2, i.e., to the Newtonian theory, the basic equation is
mt
d2
dt2
*
r0(t) +
3
i=1
x
i (t) e
i (t)
+
≡ F
The forceF on the r.h.s will be called the true force1, in contrast to ficti-tious forces, which are also called inertial forces, appearing below in equation (14.1) on the r.h.s., all multiplied by the inertial mass mt.
By systematic application of the product rule and the relation
de i
dt =ω × e
i , with v (t) =
3
i=1
˙x
i (t) e
i (t) ,
the following result for the velocity is obtained:
v(t) = v0(t) +v (t) + ω × r .
In the same way one obtains for the acceleration:
a(t) = a0(t) +a (t) + 2 ω × v +ω × [ω × r ] + ˙ω × r .
Newton’s equation of motion mt a = F , transformed to the primed
(ac-celerated) system, is therefore given by:
mta ≡ F
−mt¨r0(t)− 2mtω × r − mtω × [ω × r ]− mtω × r˙ (14.1)
1
Here we remind ourselves that in General Relativity, [7] and [8], all inertial forces and the gravitational part of the “true” forces are transformed into geometrical properties of a curved Minkowski spacetime
Trang 1096 14 Accelerated Reference Frames
This is the equation of motion in the accelerated (i.e., linearly
acceler-ated and/or rotating) reference frame On the r.h.s of (14.1) the first term
represents the true force (involving, e.g., the gravitational mass), from which the following terms are subtracted: these terms are the fictitous forces, re-cognizable by the factor mt, the inertial mass of the corresponding point,
i.e
– the so-called elevator force, mt · ¨r0(t);
– the Coriolis force mt · 2ω × v;
– the centrifugal force mt· ω × [ω × r ],
– and finally, a force mt· ˙ω × r , which has no specific name.
Since a major part of the true force, the gravitational force, is propor-tional to the gravitapropor-tional mass ms, this can be compensated by the inertial forces, because of the equality (according to the pre-Einstein viewpoint: not
identity!2) ms = mt.
In particular, for a linearly accelerated system, which corresponds to an elevator falling with acceleration−gˆz downwards in “free fall”, the difference
between the gravitational “pull” −ms· gˆz and the inertial elevator “push”
is exactly zero From this thought experiment, Einstein, some years after he had formulated his special theory of relativity, was led to the postulate that
a) no principal difference exists between gravitational and inertial forces
(Einstein’s equivalence principle); moreover,
b) no global inertial frames as demanded by Mach exist, but only “free
falling” local inertial frames, or more accurately: relative to the given
gravitating bodies freely moving local inertial frames exist, where for small trial masses the gravitational forces are exactly compensated by the in-ertial forces corresponding to the “free motion” in the gravitational field;
in particular
c) the general motion of a small trial point of infinitesimal mass mt = ms takes place along extremal paths in a curved Minkowski space, where the proper time does not obey, as in a “flat” Minkowski space, the formula
−ds2
≡ c2dτ2 = c2dt2− dx2− dy2− dz2, but a more general formula corresponding to a nontrivial differential geometry in a curved Minkowski manifold, i.e.,
−ds2
≡ c2dτ2 =
4
i=1
g i,k(˜x) dx i dx k ,
with a so-called metric fundamental tensor3 g i,k(˜x), which depends in
a nontrivial manner on the distribution of the gravitating masses, and
2 It was again Einstein, who postulated in 1910 that the equality should actually
be replaced by an identity
3 As to the sign and formulation of ds2 there are, unfortunately, different equiva-lent conventions