Klassieke Mechanica 2 - Overview
Klassickemechanica 2 overview
Generalized
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
Solving inonedimensionsubu
e
Hamilton's
:Theacuapath whichartifollowbetween
opinangiven timstane
principle
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
