Classical Mechanics: a graduate course

131 Pages·1999·0.524 MB·English

by  Rosu H.

Los Alamos Electronic Archives: physics/9909035
CLASSICAL MECHANICS
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9 HARET C. ROSU
9 [email protected]
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Copyright c 1999 H.C. Rosu
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Leo(cid:19)n, Guanajuato, Mexico
v1: September 1999.
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CONTENTS
1. THE \MINIMUM" PRINCIPLES ... 3.
2. MOTION IN CENTRAL FORCES ... 19.
3. RIGID BODY ... 32.
4. SMALL OSCILLATIONS ... 52.
5. CANONICAL TRANSFORMATIONS ... 70.
6. POISSON PARENTHESES... 79.
7. HAMILTON-JACOBI EQUATIONS... 82.
8. ACTION-ANGLE VARIABLES ... 90.
9. PERTURBATION THEORY ... 96.
10. ADIABATIC INVARIANTS ... 111.
11. MECHANICS OF CONTINUOUS SYSTEMS ... 116.
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1. THE \MINIMUM" PRINCIPLES
Forward: The historyof\minimum"principlesin physicsislongand inter-
esting. The studyofsuch principlesisbased ontheideathatthenatureacts
always in such a way that the important physical quantities are minimized
whenever arealphysicalprocess takesplace. The mathematicalbackground
for these principles is the variational calculus.
CONTENTS
1. Introduction
2. The principle of minimum action
3. The principle of D’Alembert
4. Phase space
5. The space of con(cid:12)gurations
6. Constraints
7. Hamilton’s equations of motion
8. Conservation laws
9. Applications of the action principle
3
1. Introduction
Theempiricalevidence hasshownthatthe motionofaparticleinaninertial
system is correctly described by Newton’s second law F~ = dp~=dt, whenever
possible to neglect the relativistice(cid:11)ects. When the particle happens not to
be forced to a complicated motion, the Cartesian coordinates are su(cid:14)cient
to describe the movement. If none of these conditions are ful(cid:12)lled, rather
complicated equations of motion are to be expected.
In addition,when the particlemovesona givensurface,certain forcescalled
constraint forces must exist to maintain the particle in contact with the
surface. Such forces are not so obvious from the phenomenological point
of view; they require a separate postulate in Newtonian mechanics, the one
of action and reaction. Moreover, other formalisms that may look more
general have been developed. These formalisms are equivalent to Newton’s
laws when applied to simple practical problems, but they provide a general
approach for more complicated problems. The Hamilton principle is one
of these methods and its corresponding equations of motion are called the
Euler-Lagrange equations.
If the Euler-Lagrange equations are to be a consistent and correct descrip-
tion of the dynamics of particles, they should be equivalent to Newton’s
equations. However, Hamilton’s principle can also be applied to phenom-
ena generally not related to Newton’s equations. Thus, although HP does
not give a new theory, it uni(cid:12)es many di(cid:11)erent theories which appear as
consequences of a simple fundamental postulate.
The (cid:12)rst \minimum"principle was developed in the (cid:12)eld of opticsby Heron
of Alexandria about 2,000 years ago. He determined that the law of the
re(cid:13)ection of light on a plane mirror was such that the path taken by a light
raytogofromagiveninitialpointtoagiven(cid:12)nalpointisalwaystheshortest
one. However, Heron’s minimum path principle does not give the right law
of re(cid:13)ection. In 1657, Fermat gave another formulation of the principle by
stating that the light ray travels on paths that require the shortest time.
Fermat’s principle of minimal time led to the right laws of re(cid:13)ection and
refraction. The investigations of the minimum principles went on, and in
the last half of the XVII century, Newton, Leibniz and Bernoulli brothers
initiatedthe development of the variationalcalculus. In the followingyears,
Lagrange(1760)wasabletogiveasolidmathematicalbasetothisprinciple.
4
In 1828, Gauss developed a method of studying Mechanics by means of his
principle of minimum constraint. Finally, in a sequence of works published
during 1834-1835,Hamilton presented the dynamical principle of minimum
action.This principle has always been the base of all Mechanics and also of
a big part of Physics.
Action is a quantity of dimensions of length multiplied by the momentum
or energy multiplied by time.
2. The action principle
The most general formulation of the law of motion of mechanical systems
is the action or Hamilton principle. According to this principle every me-
chanical system is characterized by a function de(cid:12)ned as:
L q1;q2;:::;qs;q(cid:1)1;q(cid:1)2;q(cid:1)s;t ;
(cid:16) (cid:17)
or shortly L q;q(cid:1);t , and the motion of the system satis(cid:12)es the following
(cid:18) (cid:19)
condition: assume that at the moments t and t the system is in the posi-
1 2
tions given by the set of coordinates q(1) y q(2); the system moves between
these positions in such a way that the integral
t2
S = L q;q(cid:1);t dt (1)
Zt1 (cid:18) (cid:19)
takes the minimum possible value. The function L is called the Lagrangian
of the system, and the integral (1) is known as the action of the system.
The Lagrange function contains only q and q(cid:1), and no other higher-order
derivatives. This is because the mechanical state is completely de(cid:12)ned by
its coordinates and velocities.
Letusestablishnowthedifererentialequationsthatdeterminetheminimum
of the integral(1). For simplicitywe begin by assumingthat the system has
only one degree of freedom, therefore we are looking for only one function
q(t). Let q = q(t) be the function for which S is a minimum. This means
that S grows when one q(t) is replaced by an arbitrary function
q(t)+(cid:14)q(t); (2)
where (cid:14)q(t) is a smallfunction through the intervalfromt to t [itis called
1 2
the variation of the function q(t)]. Since at t and t all the functions (2)
1 2
should take the same values q(1) and q(2), one gets:
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(cid:14)q(t ) = (cid:14)q(t )= 0: (3)
1 2
What makes S change when q is replaced by q+(cid:14)q is given by:
t2 t2
L q+(cid:14)q;q(cid:1) +(cid:14) q(cid:1);t dt L q;q(cid:1);t dt:
(cid:0)
Zt1 (cid:18) (cid:19) Zt1 (cid:18) (cid:19)
An expansion in series of this di(cid:11)erence in powers of (cid:14)q and (cid:14) q(cid:1) begins by
terms of (cid:12)rst order. The necessary condition of minimum (or, in general,
extremum) for S is that the sum of allterms turns to zero; Thus, the action
principle can be written down as follows:
t2
(cid:14)S = (cid:14) L q;q(cid:1);t dt = 0; (4)
Zt1 (cid:18) (cid:19)
or by doing the variation:
t1 @L @L
(cid:14)q+ (cid:14) q(cid:1) dt = 0 :
Zt2 @q @ q(cid:1) !
Taking into account that (cid:14) q(cid:1)= d=dt((cid:14)q), we make an integration by parts
to get:
@L t2 t1 @L d @L
(cid:14)S = (cid:14)q + (cid:14)qdt = 0 : (5)
"@ q(cid:1) #t1 Zt2 @q (cid:0) dt@ q(cid:1)!
Considering the conditions (3), the (cid:12)rst term of this expresion disappears.
Only the integral remains that should be zero for all values of (cid:14)q. This is
possible only if the integrand is zero, which leads to the equation:
@L d @L
= 0 :
@q (cid:0) dt@ q(cid:1)
For more degrees of freedom, the s di(cid:11)erent functions q (t) should vary
i
independently. Thus, it is obvious that one gets s equations of the form:
d @L @L
= 0 (i= 1;2;:::;s) (6)
dt @ q(cid:1)i!(cid:0) @qi
These are the equations we were looking for; in Mechanics they are called
Euler-Lagrange equations. If the Lagrangian of a given mechanical system
6
is known, then the equations (6) form the relationship between the accel-
erations, the velocities and the coordinates; in other words, they are the
equations of motion of the system. From the mathematical point of view,
the equations (6) form a system of s di(cid:11)erential equations of second order
for s unknown functions q (t). The general solution of the system contains
i
2s arbitrary constants. To determine them means to completely de(cid:12)ne the
movement of the mechanical system. In order to achieve this, it is neces-
sary to know the initial conditions that characterize the state of the system
at a given moment (for example, the initial values of the coordinates and
velocities.
3. D’Alembert principle
The virtual displacement of a system is the change in its con(cid:12)gurational
space under an arbitraryin(cid:12)nitesimalvariationof the coordinates (cid:14)r ;which
i
is compatible with the forces and constraints imposed on the system at the
given instant t. It is called virtual in order to distinguish it from the real
one, which takes place in a time interval dt, during which the forces and the
constraints can vary.
Theconstraintsintroducetwotypesofdi(cid:14)cultiesinsolvingmechanicsprob-
lems:
(1) Not all the coordinates are independent.
(2)Ingeneral,theconstraintforcesarenotknowna priori;theyaresome
unknowns of the problem and they should be obtained from the solution
looked for.
In thecase ofholonomicconstraintsthedi(cid:14)culty(1)isavoidedby introduc-
ing a set of independent coordinates (q q q , where m is the number of
1; 2;:::; m
degreesoffreedominvolved). Thismeansthatiftherearemconstraintequa-
tions and 3N coordenates (x ;:::;x ), we can eliminate these n equations
1 3N
by introducing the independent variables (q ;q ;::;;q ). A transformation
1 2 n
of the following form is used
x = f (q ;:::;q ;t)
1 1 1 m
.
.
.
x = f (q ;:::;q ;t) ;
3N 3N 1 n
where n = 3N m.
(cid:0)
To avoid the di(cid:14)culty (2) Mechanics needs to be formulated in such a way
that the forces of constraint do not occur in the solution of the problem.
This is the essence of the \principle of virtual work".
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Virtual work: We assume that a system of N particles is described by
3N coordenates (x ;x ;:::;x ) and let F F F be the components of
1 2 3N 1; 2;:::; 3N
the forces acting on each particle. If the particles of the system display
in(cid:12)nitesimal and instantaneous displacements (cid:14)x ;(cid:14)x ;:::;(cid:14)x under the
1 2 3N
action of the 3N forces, then the performed work is:
3N
(cid:14)W = F (cid:14)x : (7)
j j
j=1
X
Such displacements are known as virtual displacements and (cid:14)W is called
virtual work; (7) can be also written as:
N
(cid:14)W = F (cid:14)r : (8)
(cid:11)
(cid:1)
(cid:11)=1
X
(e)
Forces of constraint: besides the applied forces F , the particles can be
(cid:11)
acted on by forces of constraint F .
(cid:11)
The principleof virtual work: LetF be the forceactingontheparticle
(cid:11)
(e)
(cid:11) of the system. If we separate F in a contribution from the outside F
(cid:11) (cid:11)
and the constraint R
(cid:11)
F = F(e)+R : (9)
(cid:11) (cid:11) (cid:11)
and if the system is in equilibrium, then
F = F(e) +R = 0 : (10)
(cid:11) (cid:11) (cid:11)
Thus, the virtual work due to all possible forces F is:
(cid:11)
N N
W = F (cid:14)r = F(e)+R (cid:14)r = 0 : (11)
(cid:11)(cid:1) (cid:11) (cid:11) (cid:11) (cid:1) (cid:11)
(cid:11)X=1 (cid:11)X=1(cid:16) (cid:17)
If the system is such that the constraint forces do not make virtual work,
then from (11) we obtain:
N
F(e) (cid:14)r = 0 : (12)
(cid:11) (cid:1) (cid:11)
(cid:11)=1
X
Taking into account the previous de(cid:12)nition, we are now ready to introduce
the D’Alembert principle. According to Newton, the equation of motion is:
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F =p(cid:1)
(cid:11) (cid:11)
and can be written in the form
F p(cid:1) = 0 ;
(cid:11) (cid:11)
(cid:0)
whichtellsthattheparticlesofthesystemwouldbeinequilibriumunderthe
action of a force equal to the real one plus an inverted force p(cid:1) . Instead
i
(cid:0)
of (12) we can write
N
F p(cid:1) (cid:14)r = 0 (13)
(cid:11) (cid:11) (cid:11)
(cid:0) (cid:1)
(cid:11)=1(cid:18) (cid:19)
X
and by doing the same decomposition in applied and constraint forces (f ),
(cid:11)
we obtain:
N N
F(e) p(cid:1) (cid:14)r + f (cid:14)r = 0 :
(cid:11) (cid:0) (cid:11) (cid:1) (cid:11) (cid:11)(cid:1) (cid:11)
(cid:11)=1(cid:18) (cid:19) (cid:11)=1
X X
Again, let us limit ourselves to systems for which the virtual work due to
the forces of constraint is zero leading to
N
F(e) p(cid:1) (cid:14)r = 0 ; (14)
(cid:11) (cid:0) (cid:11) (cid:1) (cid:11)
(cid:11)=1(cid:18) (cid:19)
X
which is the D’Alembert’s principle. However, this equation does not have
a useful form yet for getting the equations of motion of the system. There-
fore, we should change the principle to an expression entailing the virtual
displacements ofthe generalized coordinates, which being independent from
each other, imply zero coe(cid:14)cients for (cid:14)q . Thus, the velocity in terms of
(cid:11)
the generalized coordinates reads:
dr @r @r
(cid:11) (cid:11) (cid:11)
v(cid:11) = = q(cid:1)k + where r(cid:11) = r(cid:11)(q1;q2;:::;qn;t) :
dt @q @t
k
k
X
Similarly,thearbitraryvirtualdisplacement(cid:14)r canberelatedtothevirtual
(cid:11)
displacements (cid:14)q through
j
@r
(cid:11)
(cid:14)r = (cid:14)q :
(cid:11) j
@q
j
j
X
9
Then, the virtualworkF expressed in termsof the generalized coordinates
(cid:11)
will be:
N
@r
(cid:11)
F (cid:14)r = F (cid:14)q = Q (cid:14)q ; (15)
(cid:11) (cid:11) (cid:11) j j j
(cid:1) (cid:1) @q
(cid:11)=1 j;(cid:11) j j
X X X
where the Q are the so-called components of the generalized force, de(cid:12)ned
j
in the form
@r
(cid:11)
Q = F :
j (cid:11)
(cid:1) @q
(cid:11) j
X
Now if we see eq. (14) as:
p(cid:1) (cid:14)r(cid:11) = m(cid:11) r(cid:1)(cid:11)(cid:1) (cid:14)r(cid:11) (16)
(cid:1) (cid:1)
(cid:11) (cid:11)
X X
and by substituting in the previous results we can see that (16) can be
written:
d @v @v d @T @T
(cid:11) (cid:11)
m v m v = Q (cid:14)q = 0:
(cid:11) (cid:11) (cid:11) (cid:11) j j
(cid:11) (dt (cid:1) @ q(cid:1)j!(cid:0) (cid:1) @qj ) j "(dt @ q(cid:1)j!(cid:0) @qj)(cid:0) #
X X
(17)
The variables q can be an arbitrary system of coordinates describing the
j
motion of the system. However, if the constraintsare holonomic,it is possi-
ble to (cid:12)nd systems of independent coordinates q containing implicitly the
j
constraint conditions already in the equations of transformation x = f if
i i
one nulli(cid:12)es the coe(cid:14)cients by separate:
d @T @T
= Q : (18)
j
dt @ q(cid:1)!(cid:0) @q(cid:11)
There are m equations. The equations (18) are sometimes called the La-
grange equations, although this terminology is usually applied to the form
they take when the forces are conservative (derived from a scalar potential
V)
F = V:
(cid:11) i
(cid:0)r
Then Q can be written as:
j
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