PoissonBrackets
Docsity.com
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Poisson Brackets - Classical Mechanics - Lecture Slides, Slides of Classical Mechanics
These main points are discussed in these Lecture Slides : Poisson Brackets, Matrix Form, Single Set, Hamilton’s Equations, Symplectic, Return the Lagrangian, Dynamical Variable, Angular Momentum, Two Dimensional Harmonic, Oscillator
Typology: Slides
2012/2013
1 / 9
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Poisson Brackets - Classical Mechanics - Lecture Slides and more Slides Classical Mechanics in PDF only on Docsity!
Poisson
Brackets
Matrix
Form
•^
The
dynamic
variables
can
be
assigned
to
a
single
set.
q^1
,^ q
, …, 2
qn
,^ p
, 1 p
, …, 2
pn
z^1
,^ z
, …, 2
z
2 n
•^
Hamilton’s
equations
can
be
written
in
terms
of
z
Symplectic
2
n^
x^
2 n
matrix
Return
the
Lagrangian
0^ I
I
J
^
z
t z H
J
z^
j j
j j^
q p
d dt
t z H
q p
t z H z J z t z z L
Angular
Momentum
Example •^
The
two
dimensional
harmonic
oscillator
can
be
put
in
normalized
coordinates.
m
=
k
=
1
•^
Find
the
change
in
angular
momentum
l
It’s
conserved
^
^
^
(^21) 1 2
(^22)
(^21)
1 2
q
q
p
p
H
1 2
2 1
p q
p q l^
q q q q p p p p
dl dt
H q l p
H p l q
dl dt
H z
J
l z
dl dt
i
i
i
i
Docsity.com
Poisson
Bracket
•^
The
time
‐independent
part
of
the
expansion
is
the
Poisson
bracket
of
F
with
H
•^
This
can
be
generalized
for
any
two
dynamical
variables.
•^
Hamilton’s
equations
are
the
Poisson
bracket
of
the
coordinates
with
the
Hamitonian.
(^1) S
^
^
H z
J
F z
H
F
^
^
B z
J
A z
B
A
i
i
i
i^
A q
B p
B p
A q
B
A
^
H
z
H z
J
z z
z^
^
Poisson
Properties
•^
In
addition
to
the
Lie
algebra
properties
there
are
two
other
properties.^ –
Product
rule
Chain
rule
•^
The
Poisson
bracket
acts
like
a
derivative.
,^
C A B C B A
BC
A
^
J
z z^
^
B z
J
A z
t
B t
A
B z J z A B A
^
^
^
^
B
A
B
A
Poisson
Bracket
Theorem
•^
Let
z
t )
describe
the
time
development
of
some
system.
This
is
generated
by
a
Hamiltonian
if
and
only
if
every
pair
of
dynamical
variables
satisfies
the
following
relation:
^
^
^
^
B A B A B A
d dt
^
,^
^
^
^
^
B A t H B A B A
d dt
,^
^
^
^
^
^
H B A B H A H B A
,^
^
B^ t
A
B A t
t z
B
J A z
B z
J t z
A
B z
J A z
t
B A t
,
, ,
2
2