Howdy, readers!
Because I am in a highly-focused preparation for some tests, this blog is gonna be down for a while. Surely I still may write some words or more, but in a less frequency than my natural one. Just wait, I'll come back ^^In classical mechanics, there are two approaches that we can use to describe the details of particle’s motion (note: since we are talking about classical theory of nature, we should expect that this particle behaves like a macroscopic thing, in a low speed region). These two approaches are Newton’s law and variational principle. In Newtonian theory, every single details of particle’s motion can be described by the ultimate formula (called Newton’s law II), which relates the force of interaction and the acceleration of particle. If we write down the appropriate data of particle’s motion in a certain time , for instance its position, velocity, or its normal acceleration, in principle we can obtain all we want to know before, and after time
. In other words, sufficient information in a single time dictates what will happen in the next time, and records what has happened in the previous one. This is the root of the idea which says that we can predict future by using present data. It is also the thing that makes Newton’s theory become overwhelming for several centuries. At least this theory can make anyone who is contra becomes popular all over the world (just call it Einstein).
On the other hand, the variational principle has a contrast difference with the former theory, although its goal is the same. This principle says that if a particle moves from one point to another, then the actual path of that particle (the path that is really observed in experiment) is the one that has an extremum value of a certain functional. From the last statement, it’s very clear that our first strategy to find the actual path (and also another information related to the particle’s motion) is to compare. By comparing the values of that functional (next we call it action) for all possibilities of paths the particle can undergo (which is infinitely many), and finding out which one of them that has the extremum value, it’s done. Of course, this is not an easy stuff to find. In a complex case the only thing we can work out is just finding the differential equation that this path satisfies, not the equation which describes the curve explicitly.
The big difference between Newtonian theory and variational principle is in our view to tackle the nature. In Newtonian view, we are the follower of particle; everywhere and everytime this particle moves, we are watching and we are following wherever it is. But in variational view, we are watching also, but from a high hill; we are watching the motion for all time at a glance, and picking the suitable path among infinitely many possible ones. Despite of this difference, what we will get is the same: all informations and details related to particle’s motion in space.
Suppose we have a complex manifold endowed with the Riemannian metric
. We know that this metric is positive-definite simmetric bilinear tensor.
is called Hermitian metric if
for every , and
is a complex structure of
. Componentwise, we can state the Hermitian metric
as
Define a Kahler form of as
for every , and
is a complex structure of
. Hence we can write the componentwise version of Kahler form as
in equation (2) above is Hermitian. The complex manifold endowed with a Hermitian metric
is called Hermitian manifold.
The Riemann curvature tensor of a Hermitian manifold is defined as
for . From Riemann curvature tensor we can get the Ricci form defined as
where is the contraction of Riemann tensor:
where .
The metric of a Hermitian manifold
is called Kahler metric if the Kahler form is a closed
-form, i.e.
, i.e.
is called the Kahler potential. For if the metric
is Kahler, then we can get the expression for Kahler form as
The Hermitian manifold endowed with a Kahler metric is called Kahler manifold.
Kahler manifold is torsion free. The Ricci form of Kahler manifold is defined as before,
where we have used the notation instead of
since in Kahler manifold, the component of Ricci form,
, is same as Ricci curvature
, due to the additional simmetry of the components of Riemann tensor results from the Kahler condition.
Suppose you have the Kronecker delta defined for as
if , and
if . Then define a new tensor that invert the duty of Kronecker,
if , and
if . Obviously, we have this relation
, and also
, and
. Clearly the relation (5) is just the very special case of relation (6).
By taking in equation (6), and by substituting
from equation (5), we will have
in the equation (8) above, we will get
If is even, then
and if is odd, then
Take as an even positive number, we are clearly led to this relation
for .
Here are some intricate questions:
- By knowing that
and
is it right that
- By knowing that
and
can you write another description of
by considering that
?
The complex projective plane, , is the set of lines originated at the origin of
. Define the coordinate system
on
, where
. Define the equivalence relation
on
such that
belong to the same equivalence class if there exists a number
such that
. Then the class
is just merely a line originated at the origin of
. Hence
is just the set of
, for
.
Take a point , where
,
. Define a chart
such that
. Define
, where
.
acts as the coordinate of point in
, in the chart
. If we introduce another chart
, then the coordinate in this new chart is
, such that in the intersection
the transformation function between coordinates
and
must be holomorphic.
The coordinate system is called homogeneous coordinate, while
is called inhomogeneous coordinate.
Define a new notation for
, and
for
. It’s just the renaming program of the coordinate notation for
.
We need to find a function of these ’s that can represent the Kahler potential in
. The existence of this function will be the first sign for us to ensure that
is Kahler manifold. Let’s define the function on the chart
,
then on another chart we have another function
defined similarly with the former, and this relation must hold in the intersection of those two charts,
Then,
Since is a holomorphic function, then
. Also,
. Then,
Define a closed two-form locally by
will guarantee that the metric is Kahler.
Take , and define
by
, where
. The metric
is Hermitian. Indeed,
On a chart , we obtain
and from equation (3) we will have
Define a real vector field , such that
. Hence,
Suppose you have a Mobius strip.
- Draw the line from an initial point on the Mobius strip down the middle of that strip, and return to the initial point to form a loop. By definition, if you walk along that loop you will end up on the initial point you start to walk, with your feet have touched the two sides of the strip. This behavior will be the ultimate difference between Mobius and ordinary strip; in ordinary strip you just walk on one side of strip, and cannot touch another side.
If you cut the strip along the loop, what object you will have after your scissor returns to the initial point? - Suppose you also have the second identical Mobius strip. Place those two strips in front of you with their sides touch each other along the edge-to-edge line, such that at a glance it seems like a number 8. Cut along that line but immediately glue the corresponding sides such that they still stick together, merging two Mobius strips into one larger object, and making that number 8 like a big O. Yes, at a glance it’s just a big O, but precisely what kind of object it is? Anticipate all possibilities.
Compare the answers of question 1 and 2.
Misal kita mempunyai persamaan differensial orde-1 berbentuk
dan
adalah fungsi dua variabel
dan
.
— 1. Separable differential equations —
Definition 1 Persamaan differential (1) kita sebut separable jika kita dapat mengubahnya menjadi bentuk
di mana
dan
adalah fungsi 1 variabel.
Jika kita dapat mengubah persamaan differential (1) menjadi (2), maka kita dapat mengintegralkan suku demi suku pada persamaan (2) untuk mendapatkan solusi dari persamaan differential (1). Contoh dari persamaan differensial yang separable adalah
sedangkan contoh untuk persamaan differensial yang tidak bisa dipisahkan variabelnya adalah
Dengan mengintegralkan suku demi suku pada persamaan ini kita dapat dengan mudah mencari solusi dari persamaan differensial tersebut.
— 2. Homogeneous differential equations —
Definition 2 Suatu fungsi 2 variabel
disebut homogen orde-n jika kita dapat menyatakan fungsi
tersebut dalam bentuk
di mana
,
adalah fungsi dalam
, dan
adalah sebarang bilangan bulat. Atau dalam bentuk lain,
homogen (orde-n) jika kita dapat menyatakannya dalam bentuk
di mana
,
adalah fungsi dalam
, dan
adalah sebarang bilangan bulat.
Secara ekivalen kita juga dapat mengatakan bahwa homogen jika
untuk sebarang bilangan bulat .
Example 1 Fungsi
bukan fungsi homogen karena
Kita juga tahu bahwa fungsi
tidak homogen karena kita tidak bisa mencari
sehingga berlaku
.
Example 2 Fungsi
adalah fungsi homogen orde-5, karena
Definition 3 Persamaan differensial parsial (1) disebut persamaan differensial homogen jika koefisien
dan
adalah fungsi homogen berorde sama.
Example 3 Persamaan differensial ini,
adalah homogen, karena koefisien dari
dan
adalah fungsi yang homogen dan sama-sama berorde 1.
Metode untuk menyelesaikan persamaan differensial homogen adalah dengan mensubtitusi
ke dalam persamaan differensial, sehingga nantinya kita akan mendapat persamaan differensial yang separable. Setelah itu, tinggal mencari solusi untuk persamaan differensial separable seperti biasa.
Example 4 Cari solusi dari
Karena persamaan differensial di atas adalah homogen (just check it), maka kita substitusi
, dengan catatan bahwa
agar akar pada persamaan di atas tetap terdefinisi. Sehingga,
Jika kita mengeset
, maka kita dapat membagi kedua suku dengan
, dan akan kita dapat
Kemudian jika kita membatasi
agar
, akan kita dapat
Kita telah mendapat bentuk persamaan differensial yang separable, jadi untuk mencari solusinya adalah dengan mengintegral setiap sukunya. An easy task!





