PETER PAZMANY CATHOLIC UNIVERSITYConsortiummembersSEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben

***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.

Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**

Consortium leader

PETER PAZMANY CATHOLIC UNIVERSITY

Consortium members

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***

PHYSICS FOR NANOBIO-TECHNOLOGY

Principles of Physics for Bionic Engineering

Appendix: Mathematical Appendix

(A nanobio-technológia fizikai alapjai )

(Matematikai függelékek)

Árpád I. CSURGAY,

Ádám FEKETE, Kristóf TAHY, Ildikó CSURGAYNÉ

Table of Contents

Appendix Mathematical Appendix 1. Calculus of Variations

2. Generalized Coordinates 3. Vector Analysis

4. Inverse Problems 5. Hilbert Space

6. Linear Operators in Hilbert Space

7. Eigenvalues and Eigenvectors

1. Calculus of Variations

Determine those

functions, for which

the integral („functional”) is extremum.

Let us assume that we know the y = y(x) function which

satisfies the conditions and fits on points P₁ (x₁, y₁), P₂ (x₂, y₂).

Let us „vary” the y (x) function as y* = y (x) +   (x). Where  is a small real parameter,  (x) is a continuously differentiable, otherwise arbitrary

function, for which  (x₁) =  (x₂) = 0.

 

y  y x

 

2

1

, , d .

x

x

I 



 

y x

x1 x₂

1( ,1 1) P x y

2( ,2 2) P x y y

The integral is extremum if for any  (x), thus the value of the integral as a function of  is extremum at  = 0.

     

1

, , d , ,

x

I 



  ^{ }

 

^{ }

1 1

fixed , , ' d , , d .

x x

x x

I y I F x y y x F x y y x

   

  





0

d 0

d I

  _ ^

2

1 2

0 1

d d ,

d

d d .

d

x

' x

x

' x

I F F

y y x

I F F

y y x



 



 



 



    

    

   

     





 

I 



  0

Partial integration:

Because the product of a function multiplied with an arbitrary function

integrated over a given interval should be zero for any one of the arbitrary functions, then the function itself should be zero:

2 2 2 2 2

1 1 1 1 1

d d

d d d d

d d

x x x x x

' ' ' ' '

x x x x x

F F F F F

x x x x

y  y   x y y   x y

                

           

       

   

2 2

1 1

0

d d

d d 0.

d d

x x

' '

x x

I F F F F

x x

y y y x y



  

 _

      







d 0.

d ^'

F F

y x y

   

 

Functionals to be extremized Euler equations This is the Euler–Lagrange equation of calculus of variations:

 

2

1

, , d 0 d 0.

d

x

' x

F F

I F x y y x

y x y

  ^   ^  ^ 

 



 

2

1

1 1

, , , , , , d

x

n n

x

I 



d ^'

i i

F F

i n

y x y

 

  

 

1 2 3 1 2 3

1 2 3

, , , , , , d d d

V

u u u

I F x x x u x x x

x x x

    





_{ }

i i

F F

u x u x

  

 





Descartes coordinates Generalized coordinates Spherical coordinates

2. Generalized Coordinates

Any set of parameters which can characterize the position of a mechanical system may be chosen as a suitable set of generalized coordinates.





r  ^{r }





cos ,

sin cos , sin sin . z r

x r y r



 

 











 r

 

 

 

1 2 3

1 2 3

1 2 3

, , , , , , , , . x x q q q y y q q q z z q q q





 z r

x

z

y

x y

r z

x y





2 2 2 2

ds drdr  dx dy d ,z

1 2 3

1 2 3

d d d

d d d d ,

d q d q d q

q q q

 r  r  r

r

2 2 2

2 2 2 2

1 2 3

1 2 3

1 2 1 3 2 3

1 2 1 3 2 3

d d d

d d d d d d

d d d

d d d d d d

2 d d 2 d d 2 d d .

d d d d d d

s q q q

q q q

q q q q q q

q q q q q q

 

   

        

     

  

r r r

r r

r r r r r r

r z

x

z

y

x

y dr

d r r

 

 

 

1 2 3

1 2 3

1 2 3

, , , , , , , , . x x q q q y y q q q z z q q q







Local Pythagoras theorem Descartes-coordinates

Cylindrical coordinates Spherical coordinates

If the coordinates locally are orthogonal :

2 2 2

2 2 2 2

1 2 3

1 2 3 1 2 3

d d d d d d

d d d d dq d ,

d d d s d q d d q

q q q q q q

 

   

           

     

r r r r r r

r r

2 2 2 2 2 2 2

1 1 2 2 3 3

ds drdr  g qd  g qd  g qd .

1 2 3

1 2 3

1 2 3

1, 1, 1,

1, , 1,

1, , sin .

g g g

g g r g

g g r g r 

  

  

   

Scalar field Vector field

3. Vector Analysis

 

 r  v r

 

 

 

 

Line-(path)- integral

Integral along a closed line

Surface- integral

Integral on a closed surface Volume-

integral

d 0

:

lim d d ,

Q Q

l





dlim0 d d ,

A





dlim0 d d ,

V V V

V V

 







d ,

L



A



‘Slope’ gradient Divergence Rotation

Sources

sinks

Whirls, curls

„NABLA VECTOR”:

Maxwell JC, On the Mathematical Classification of Physical Quantities, 1862 x y z .

  

   

 i  j  k

0 ⁰ ^d

dn  d cosl  l0

dl



n0

dn

0 grad

n

 

  n 

grad   divv  v rot v  v

Local changes of scalar fields

‘Slope’

gradient ^grad _{x y z} ^,

  

    ^  ^  ^

 i  j  k

Local changes of vector fields

Divergence

Rotation ^{rot ,}

x y z

x y z

v v v

  

  

  

i j k

v v

0

d

div lim ^A .

V V

 



 



div v^x v^y v^z , v

x y z

  

    

  

v v

 

0

d

rot lim ^L .

n ^{ }A A



 



v

 

0

( )

grad _n lim .

n

r n n

 

 

  



Gradient of a scalar field

Divergence of a vector field

   

0

( )

grad lim , grad .

n n n

r n

n x y z

   

   

 

    

       

  i  j  k

 

 

 

1 1 2 2 3 3

1 1 1

grad ; grad ; grad ;

g q g q g q

  

  ^   ^   ^

  

     

0

2 3 1 1 3 2 1 2 3

1 2 3 1 2 3

d

div lim , div ,

div 1 .

x y

A z

V

v v v

V x y z

g g v g g v g g v

g g g q q q

 

   

      

   

    

      



v v v

v

Rotation (curl) of a vector field

 

0

d

rot lim ^L , rot .

n A

x y z

A x y z

v v v

 

   

   

   



v v v

     

     

     

3 3 2 2

1

2 3 2 3

1 1 3 3

2

1 3 3 1

2 2 1 1

3

1 2 1 2

rot 1 ,

rot 1 ,

rot 1 .

g v g v

g g q q

g v g v

g g q q

g v g v

g g q q

   

    

   

    

   

    

v

v

v

Gauss theorem Stokes theorem

 

d 8, div d 8.

A V

  V 





div d d

V A

V  





L A

  





Compare Descartes and Spherical coordinates

2

d d d d

sin d d d ,

V x y z

r  r  

   

    

grad

1 1

sin ,

r

x y z

r r ^ r ^

  

 

  

  

  

     

  

  

  

  

i j k

e e e



 ^{ }

2

div

1 1 1

sin .

sin sin

x y z

r

A A A

x y z

r A A A

r r r r



 

   

  

     

  

  

  

  

A A

r z

x

y



 e

e

er

i j

k

 

       

rot

1 sin sin

1 1 1

sin .

x y z

r

r r

x y z

A A A

A A

r

A rA rA A

r r r r

 

 

 



  

  

  

   

  

   

 

  

   

 

       

 

           

i j k

A A

e

e e

 

grad div .

r sin

r r  ^ r   ^

  

       

  

A A A

A A A e e e

r z

x

y



 e

e

er

i j

k

Vector-analytic identities

   

div grad     

   

rot grad ^{  } ^ 0

   

div rot v   v  0

   

rot rot v  grad divv   v

2 2 2

2 2 2

2 3 1 3 1 2

1 2 3 1 1 1 2 2 2 3 3 3

div grad

1 .

x y z

g g g g g g

g g g q g q q g q q g q

  

  

  

  

       

  

           

            

 

div u   divu u grad

 

rot u   rotugradu

 

div v u  u rotv v rotu

4. Inverse Problems

Solution of the Maxwell equations from given sources of the field leads us to the following inverse problems:

 Solve for given

 Solve for given

 Solve for a given volume V.

The first problem is looking for a curl-free vector field with prescribed sources, thus the introduction of a scalar potential as reduces the problem to finding a scalar function , because

 

 

v r r

 

 r ^{rot grad}





     

divu r  0; rot u r  s r , ^{s r}

 

   

div r ^ 0; rot w r ^0,

     

div v r  g r ; rot v r 0, ^g

 

Substitute the gradient of the scalar function into the source equation

in Descartes coordinates we get a linear, second order partial differential equation, the LaplacePoisson equation

If the boundary is at infinity, and the solution goes to zero as R^ with positive

, where R is the distance between the sources and the boundaries, then the solution of can be given as

   

div grad ^{  } ^{   } g,





x y z

  

 ^{  }

     

  







  

 

  

 



1 ( , , )

, , d d d .

4π _V x y z g

x y z

  

   

  





The second problem is looking for a source-free vector field with prescribed curls, thus the introduction of a vector potential as

automatically satisfies the source-free requirement, because Substitute the curl of the vector potential into the equation

However

but we are interested only in the curl of the vector potential, thus its

divergence can be chosen arbitrarily, e.g. to zero , thus the problem is reduced to the solution of , in Descartes coordinates

 

 

v r A r

 

div rotA  0.

   

rot rotA r( ) s r .

   

rot rot A r( )  grad divA   A s.

 

div A r( ) 0

  A s

2 2 2

2 2 2 2 2 2

2^x 2^x 2^x _x, A2^y A2^y A2^y _y, 2^z 2^z 2^z _z,

A A A A A A

s s s

x y z x y z x y z

  

     

           

        

 

  

 



1 ( , , )

, , d d d .

4π _V x y z

x y z

     

  





A

In summary

divv  g rotv 0 ^{grad ( ) 1 d}

4π _V

g V

  r

  



v(r) r

divu  0 rotu s 1

rot ( ) d

4π _V V

 



u(r) A r A





5. The Hilbert Space

The Hilbert space is a generalization of the linear vector space of n-dimension.

It plays a central role in physical modeling.

A Hilbert space H is an abstract set of elements, called vectors, and so on, having the following properties:

1. The space H is a linear vector space over the field of complex numbers, such as  and , with the properties

a) for each pair of vectors there is determined a vector called the sum such that

commutative,

associative.

b) One vector is called null vector

, , a b c

a  b  b  a 









0 a  0  a ,

b) For each vector there is a vector such that c) For any complex numbers λ and μ

2. There is defined a scalar product in H denoted by This is a complex number such that

The norm of a vector is defined as 3. The space H is ‘separable’

4. The space H is ‘complete’

There exists a unique limit in H.

a a a   a 0 .

   

 

, ,

, 1 .

a b a b a a a

a a a a

      

  

     

 





; ; .

a b ^ b a ^ a b ^  a b a b c^{ } a b ^ a c

; a  a a

n ;

f  f 

lim_{n m}, _ f_m  f_n  0;

Definitions

Two vectors are ‘orthogonal’ if A set is an ‘orthonormal system’ if

A set is a ‘complete orthonormal system’ of H if for every there is a series

and

f g f g  0;

 

0, if .

n m mn

n m f f

n m

 ^{ }

   

 

n n

n

f 



n

n n

, .

m m n m n n

f f 





6. Linear Operators in Hilbert Space

A ‘linear operator’ A is defined as a mapping of H onto itself (or onto a subset of H), such that

A linear operator is said to be ‘bounded‘ if there is a constant C that

A bounded linear operator is ‘continuous‘ in the sense that if then

Two operators are called to be equal if for

 

A  f  g  A f   A g

for .

A f C f  f H

fn  f

n .

A f  A f

;

 f H .

A f  B f  A B

We define

The products AB and BA in general are not equal We define the ‘commutator‘ of A and B as

‘Adjoint‘ operator A⁺ of a bounded linear operator A is defined as

 

   

1 identity operator;

0 0 null operator;

sum of and ;

product of and for .

f f

f

A B f A f B f A B

A B f A B f A B f





  

  H

. AB BA





for , .

g A f  f A g^{ }  f g H

The adjoint has the properties

An operator is said to be ‘Hermitian‘ or ‘self-adjoint‘ if A = A⁺ which implies that

 

real.

f A f  f A^ f ^  f A f ^ 

7. Eigenvalues and Eigenvectors

There exists a vector such that where

is a complex number, we say that is an eigenvector of A, which corresponds to the eigenvalue

Hermitian (self-adjoint) operators have the following properties:

1. The eigenvalues of a Hermitian operator are real.

2. If two eigenvectors belong to two different eigenvalues, then

3. The eigenvectors of a bounded Hermitian operator after normalization form a denumerable complete orthonormal system (discrete spectrum).

0

n  A n  _n n ,

n n

n.



; ; 1 .

n n n

n n n n n n

 







,

n m _n  _m

0.

n m 