Classical Mechanics Joel phần 8 ppt
Classical Mechanics Joel phần 8 ppt
... is unchanged, it must be
independent of q
I
,andq
I
is an ignorable coordinate.
6 .8. ACTION-ANGLE VARIABLES 187
Exercises
6.1 In Exercise 2.6, we discussed the connection between two Lagrangians,
L
1
and ... in the 2n + 1 dimensional extended phase space which includes time
as one of its coordinates.
188 CHAPTER 6. HAMILTON’S EQUATIONS
A 1-form acts on such a vector by
dx
j
(D
f
)=f
j
.
Sh...
Classical Mechanics Joel phần 4 ppt
... 88 CHAPTER 4. RIGID BODY MOTION
and we may conclude from the fact that the ˆe
j
are linearly independent
that ... rather than Euclidean space arise often
in applications in quantum mechanics and quantum field theory, in ad-
dition to the classical problems we will consider such as gyroscopes and
tops.
4.1 ... are very well
approximated by the assumption that the distances between the atom...
Classical Mechanics Joel phần 10 pptx
... 2nd
edition, 1969. QA805.L 283 /1976.
[7] Jerry B. Marion and Stephen T. Thornton. Classical Dynam-
ics. Harcourt Brace Jovanovich, San Diego, 3rd ed edition, 1 988 .
QA845.M 38/ 1 988 .
[8] R. A. Matzner ... Math. Methods of Classical Mechanics. Springer-
Verlag, New York, 1 984 . QA805.A 681 3.
[3] R. Creighton Buck. Advanced Calculus. McGraw-Hill, 1956.
[4] Herbert Goldstein. Classi...
Classical Mechanics Joel phần 1 pdf
... will need to go beyond classical mechanics, we
begin with a few words of justification for investing effort in under-
standing classical mechanics.
First of all, classical mechanics is still very ... 181
6 .8 Action-AngleVariables 185
7 Perturbation Theory 189
7.1 Integrablesystems 189
7.2 CanonicalPerturbationTheory 194
7.2.1 TimeDependentPerturbationTheory 196
7.3 AdiabaticInv...
Classical Mechanics Joel phần 2 ppsx
... arose in 183 5 in his unification of the language
of optics and mechanics. It too had a usefulness far beyond its origin,
and the Hamiltonian is now most familiar as the operator in quantum
mechanics ... begin to feel the asym-
metry of the potential, and
the orbits become egg-shaped.
1
-1
x
p
1.210 .8
0.6
0.40.2-0.2-0.4
0
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
x
U
U(x)
Figure 1.3. Motion in a cubic...
Classical Mechanics Joel phần 3 potx
... fields
E(r,t)and
B(r,t),
F = q
E +
v
c
×
B
. (2. 18)
If the motion of a charged particle is described by Lagrangian mechanics
with a potential U(r,v, t), Lagrange’s equation says
0=
d
dt
∂L
∂v
i
−
∂L
∂r
i
= ... discussed in Chapter 8, but until then we will assume that
the phase space (q, p), or cotangent bundle, is equivalent to the tangent bundle,
i.e. the space of (q,...
Classical Mechanics Joel phần 5 potx
... generally used in quantum mechanics. Many of the standard
classical mechanics texts
10
take the second rotation to be about the x
1
-
axis instead of y
1
, but quantum mechanics texts
11
avoid ... add or subtract, and many choices for the Euler angles (φ, ψ) will give the
same full rotation.
1 18 CHAPTER 4. RIGID BODY MOTION
Then E becomes
E =
1
2
I
1
˙
θ
2
+ U
(θ)+
1
2
I
3
ω
2
3
,...
Classical Mechanics Joel phần 6 docx
... expression in (4.29). [For
this and exercise 4 .8, you might want to use a computer algebra program
such as mathematica or maple, if one is available.]
4 .8 Find the expression for ω in terms of φ, ... 1 28 CHAPTER 5. SMALL OSCILLATIONS
Note that A is a constant symmetric real matrix.
The kinetic energy ... messy.]
4.9 A diamond shaped object is shown in top, front, and side views. It is
a...
Classical Mechanics Joel phần 7 doc
... canonical.
The Poisson bracket plays such an important role in classical me-
chanics, and an even more important role in quantum mechanics, that
it is worthwhile to discuss some of its abstract ... R
n
and totally antisymmetric
in its action on the k copies, and is a smooth function of x ∈M.Ata
1 58 CHAPTER 6. HAMILTON’S EQUATIONS
where f
j
are an arbitrary set of functions on phase space...
Classical Mechanics Joel phần 9 doc
... 2 08 CHAPTER 7. PERTURBATION THEORY
Consider the evolution in time, un-
der the full hamiltonian, of ... magnetic field. It is interesting to note that in
Bohr-Sommerfeld quantization in the old quantum mechanics, used be-
fore the Schr¨odinger equation clarified such issues, the quantization of
bound ... +
1
ω
2
ξ
η(t)=
1
ω
∂U
∂q
cos ωt +
1
ω
2
η
and assume ¯q and ¯p obey Hamiltons equat...