Samenvatting met voorbeelden van situaties van Klassieke Mechanica 2
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Course
Klassieke Mechanica 2
Institution
Universiteit Van Amsterdam (UvA)
Overzichtelijke samenvatting met voorbeelden van situaties van het vak Klassieke Mechanica 2. Ook de bijbehorende vergelijkingen van de verschillende situaties staan bij de voorbeelden.
Calculuovovestinding theminimum
example of using polar coordinates Momenta and
Ignorable Coordinates
or maxm se In polar coordinates the components of the velocity are :
Vr r i Vp r
foranySystemWith generalized cordinatesiwerefertohenquantitiesOogfi
= =
as
generalee
> Shortest path
Short
segment :
between two
dS =
dx2 +
points : (With path y(x)
dya =
1 + y'(x)dx
Skinecenegimmick Fi
= pi
Pi
Generalized forces and momenta are not the same as the usual momenta and forces.
dy = dx h
<mrmimi corresponding generalized momentum" is conserved
=
y'(x)dx J) 2 is invariant under variations of a coordinate gi , then the .
The
requation or
Nowhetlengthofthepais
st
↳ The unknown is the function y y(x) = that defines the path
since Wecanrewrite Conservation of momentum
:
Tarwegraminimum
The
dequation m
Isolated System <translationally invariant : if we transport all N particles through the same displacement E ,
-
lo interpret this
equation we need to relate nothing physically significant about the system should change .
ith a min. , max or neit , a the left side to the appropriate component
The effect of moving
of the force : F U = -
> the whole
system through the fixed displacement is to replace every position r
...S Te
byradThe potentienergymustbeunaffectedbythisdisplacementscha
Urte . . . re,t) =
Ur
=U
The componentof theforce is just
en
=
OL
is thex component of momentum of paricehoe
Pax wherepox
Ee that the lef side is
just the corete *
Pxx = Px =
0 Where 4x is the component of total momentum . P = P
Euber-Lagrange equations repandontherightside mrh
can be recognit en > Provided that the
Lagrangian is unchanged under the translation , the total momentum of the
N-particle System is .
conserved
T =
&Horque equals rate of change in momentum
Of
If the ith
logi
=
*
component of the
generalized Conservation of energy
Integral the form S (y(x1 y(x) x)ax
momentumisconstan thentheihcomponentof the
of :
, , , en
(where
y(x) is the
yet unknown curve
joining the two points (X y ,, ,
1 and (x2 ,
y2) : As time advances the C 1q,
function ...., Quig , .... quit) changes ,
because t is
changing and
q's and
's change with the evolving system
7 and
y(x y, y(xz) y2
= =
.
Lagrange for constrained
systems
By the chaintule we
get : 219 ..... ni ...
quitga +
Among all the possible curves satisfying y(x . 1 =
y , and
y(x2) ya
=
,
we have to find the one that
makes the integral S a minimum.
Lagrangian approach Can
handesystemsbareconstrainese
:
Using Lagrange's equation , we can replace by :
hi =
pi (innetboek
gebruiken
pe is
In function of variables , but because the that
they occupy
the
integral f-fly ,
y : x) is a 3
integral follows the .
X ↳ Bead on a wire ; plane pendulum
pig) pigWhen
integrand f[y(X) y'(x) X] actually just function of (Sincey only depends on
path y y(x) the is a X
derivative
=
C
, ,
,
The momentum and thus : piq
=
is the
generalized
Consider of N particles , N with
an
arbitrary system X = 1
, ...,
the
Lagrangian does no
positions
The Euler-Lagrange equation (s) lets us find a path for which the integral S is stationary r . We
say that the parametersq , .... an are a set of gene. depend explicity on
time o
(H- pigi-2
zernorthesemmocanbeereseen pigi-2 Lagrangian
We see that =
0 if the does not depend explicity on the time ,
then the Hamiltonian is conserveda
Euler-Zagrange : gi =
gi(r , Nit) [i = 1
, ..., H]
b
TheHamiltonianisinmanysituationsthe energy theSystemisprovidedthat
.
..,
dan is
the relation between the
generalized coordinate
1 The number of the generalized coordinates (n) is the smallest number
in this
system to be parametrized
that allows the way .
Procedure of using :
Setupthe problemsothatthequantitywhosestationarypahyouetot
7
For
. simple pendulum there is the generalized coordinated and so
a :
H = T+ U
where fl ] ... is the function appropriate to
your problem
r = (X
y) Ilsing ecosid
,
=
,
4) We expressed the two Cartesian coordinates in terms of the one
C .Writedowntheueragrangequationsintermoffyy on generalized coordinate o
1 The number of degrees of freedom of is the number of
you want to know
(y(x)
a
system
coordinates that can be varied in
independently a small displacement
↳ small displacement number of independent "directions" in which
=
the
system can move from any given initial configuration
Euler-Lagrange equation only guarantees
Lagrangian Multipliers
* The to
give a path for which the integral is stationary
Whenthe numberofdegreesoffreedomofanatiesystem
* s
** When we have more than two variables we write the path in parametric form as :
and constrained
X X(u)
y y(u) and we
get :
= = .
S =
ff(x(4) ,
y(u) ,
X(4) , y'(u) u] ,
du Use when
you want to know the constraint forces.
We also two E 2
Instead of
choosing generalized coordinates 9 ,....
an ,
we use a larger number of coordinates and use
Lagrange multipliers to handle the constraints
.
now
get . .
equations :
& and Constraint equation equation of : the form f(x y) ,
= constant
. Now instead of
We consider a simple pendulum the
generalized coordinate of We use the
original coordinates X
andy that are not independent .
The constraint equation becomes : f(x y) l X- + The length is a constant
y2.
= =
, .
Lagrange multiplier : XIt) ,
i s an
arbitrary function of t
Lagrange's equations
·of with flycos
Lagrange's equations take the same form in
any coordinate system and
eliminates forces of constraint (normaalkrachten)
We now find that ofconstraint f constraint
<Lagrange Multiplier gives the corresponding components of the constraint forest
Lagrange's
Forunconstrained
mon te
eq .
We consider a particle moving unconstrained in 3 dimensions with a conservative net force
=ins on the particle .
Hamiltonian mechanics
The kinetic energy then is : T = mr =
[mr = m(x + y2+ 24
In
The potential energy is : U =
Uir) = U(X y z), ,
Lagrangian formalism the n coordinates (91 ..... qn) Specify a position ("configuration") of the system
defining a point in an n-dimensional configuration space
<
The Lagrangian is defined as 2 T U The 21 coordinates
qniq
:
qn)
= -
(91 define point in state and of initial conditions that determine
..... ...., a
space ,
specify a set a
unique solution of the n
notenthesameastaeneg (x , y , 2) and its
,
velocity (xy
Second order
↳ For each set
differential equations of motion :
of initial conditions these equations of
Lagrange's equations .
2 = 2(x y z, X , z)
motion determine a
unique path through state space .
,
y
pi
, ,
Generalized momentum : lalso called canonical momentum or momentum
conjugate zo gils
We consider the two derivatives : Px) when
respect
we differentiate
to time and we use
is one winhet
Newton's 2 law,
*
In
And ,
the Jamiltonian approach the Central role of the
Lagrangian 2 is taken over
by the Hamiltonian <J
=, pigi-
we shall use coordinates 191 ..., ani Pr , ..., Pn) instead of 191 ..., quid , ..., qn)
Fx = px =
mix = ma ,
we get :
·) We can
regard the 2n coordinates of the Jamiltonian approach as
defining a
point in a 2n-dimensional
space which is called the phase space
Newton's2 * law applies to the 3
Lagrange equations
: 2. assume that all the
* forces of interest are conservative <for conservative systems the
potential energy only depends on q
--
These equations have the exact same form independent of time >
asthefuer -Lagrange equatoseral s f2dt is
The
Lagrangian for a conservative system with "natural coordinates" has the
general form : 2 =
2(q q),
=
T. U = AlgiqUqi
stationary
=
↳
Lagrange's equation for this Lagrangian is
automatically a second-order differential equation for q .
Define the Hamiltonian as J piq 2 with the generalized moment as
pi
=
&2digit (or in one-dimension :
=A1g(g)
The integratiscalled heactionintegra
↳ the
gives generalized momentum
article's path is based on Hamilton's prieen
in terms of q and q
taken along the actual path We can solve the
generalized momentum so that q is in terms of p and g : q =
a)
=
&(g p) ,
> We can now replace 9 in I ,
so that I becomes a function of g and p:
J ((q p) ,
=
pq(q p) 2(q q(q p)
,
-
, ,
For single particle
a the
following three statements are equivalent :
A particle's path is determined
by Newton's 2ndlaw : F ma
Now find Hamilton's equations of
=
.
1
,
we want to motion > evaluate the derivatives of J1g p) , With respect to gand p .
. The
2 path is determined by the three Lagrange equations Euler
Lagrange
. The
3 path is determined
by Hamilton's principle
-
The Lagrange equations hold in almost
any coordinate
system
The second
↳
Any set of "generalized coordinates" 91 92 93 With the each position specifies
one
,
we find by differentiating with respect to p .
, ,
property that
& [q +-
a unique value of 19192 93) ,
and rice versa :
gi(r) for
!
qi = i = 1 , 2, 3
, ... These equations guarantee that for
r any value of (X y 2) there is a Unique
=
, ,
=
(9 ,, 92 , 93)
191 92 93)
, ,
and 191 92 93)
, ,
and vice versa .
amilon'sequaions for onedimensionsystem: po
We rewrite the Lagrangian in these new variables :
2 =
219 92 93 giga 93)
, . ,
<
521992 gi d , , 93) one for q and one for p .
(
=
Solving in more than 1 dimension pigila pit)-hqggpt) t) differentiating Hamilton's
Thecorrectpathmustsatisfythefueragrangeequations withrespecto
the new variables : =
leads o
equainen
ga en , ,
There for any choice of the 3N coordinates 93N needed to describe
7 are 3N
Lagrange equations valid g, .....
the N
particles .
When the Hamiltonian is time-independent : o ,
but when it is not time-independent : + +
We call
d and generalized force and
generelazid momentum respectivelyo
<generalized force-Irate of change of generalized momentum is
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