Introduction | 3 | Mathematical Methods in Physics and Engineering wit

ABSTRACT

An equation containing two variables at most and derivatives of the rst

or higher order of one of the variables with respect to the other is called a

dierential equation The order n of such an ordinary dierential equation

is the order of the highest derivative that appears To solve a dierential

equation of nth order n integrations are necessary Each integration delivers

an integration constant These arbitrary constants can be used to adapt the

general solution to the particular solution of the problem in question

Many dierential equations represent mathematical models describing a

physical or technical problem The free fall of a parachutist is described by

the equation of motion

x t

mg a

Herem is the mass of the parachutist mg is the gravitational attraction and

the last term on the righthand side describes aerodynamic drag By dening

the velocity v of fall

v t

dx t

the equation which is of second order can be reduced to a dierential

equation of rst order

dv t

This equation is of the form

P tdtQ vdv

and is called dierential equation with variables separable separable equation

since can be written in the form

lnC

avm

agm

avm

agm

This integration has been executed using the Mathematica command

Integrate[1/(a*vˆ2/m-g),v] and observe ln x x arctanh x

If we now assume the initial condition v v

then we obtain the

integration constant C from Thus our solution of for

v t reads

avm

agm

avm

agm

exp

For t this gives the nal falling speed

gma v

Inserting into yields a dierential equation for x t and a second

integration constant after integration Instead of solving for

x t we may derive a dierential equation for x v Substituting dt from

into we have

dx v

g av

and after integration using the substitution u v

one has

x C

In order to determine the second integration constant C let us consider that

the parachutist leaves the airplane at time t with zero velocity v

at a

certain height x

above the earths surface x Thus we have

ln v

as the second initial condition and the solution of reads

x x

In this example of a dierential equation of second order we used

two initial conditions ie two conditions imposed on the solution at the

same location x x

We could however impose two conditions at dierent

Then we can assume that the parachutist should contact the earths surface

x with a velocity v of fall equivalent to one tenth of v

We then have

the nal condition boundary condition that for x v v

or from

Then the general solution becomes the particular solution

If we solve using the command

x t C

cosh

g atmC

Here C

and C

are integration constants Using the two initial conditions

v and x x

one obtains the particular solution

x t x

cosh

For m kg a x

m and g ms

this function

is shown in Figure This gure has been produced with the help of the

commands

m=70; a=0.08; x0=5000; g=9.81; x[t_]=x0-m*Log[Cosh[t*Sqrt[g*a]/Sqrt[m]]]/a; Plot[x[t],{t,0,60}] In this expression we have used a semicolon This allows more than one

command to be written in one line The command to plot a function needs

a function that can be evaluated at any arbitrary value of the independent

variable t To obtain such a variable we replaced t by t_

We now treat an example to nd the force producing a given trajectory Let

us consider the relativistic onedimensional equation of motion of an astronaut

traveling into space along a straight line and returning to earth Neglecting

gravitation such a motion is described by three equations

relativistic equation of motion

Numbers in brackets designate bibliographical references see the list at the end of the

Figure

Trajectory of a parachutist

conservation of energy

addition theorem of velocities

w v

In these equations we used the following designations m

and m

are the

rest masses of the space rocket and the exhausted gases respectively v

and v

are the respective velocities and w is the exhaust velocity relative to the space

ship The repulsive force which we shall later designate by F is represented

by the rhs term of The quantities m

and v

are functions

of time t c is the constant velocity of light

We now turn to the integration of the equation of motion Dier

entiation delivers

whereas the energy theorem results in

Here A

and A

are abbreviations for the lhs terms Using now the addition

theorem we build the expressions

and

w v

These abbreviations allow the rewriting of the equations and

in the simple forms

and A

This allows the elimination of dm

The resulting equation is A

The four abbreviations only contain the timedependent variables

t and m

t After some straightforward algebra the resulting equation

takes the form

w c

or simplied

w t

When treating the motion of the parachutist we assumed a given force and we

derived his motion Now we assume a given trajectory x

t of the space ship

and we are determining from the repulsive force F producing this tra

jectory During space travel the space ship leaving earth will rst accelerate

then slow down to zero velocity make a degree turn and accelerate again

to return to earth Slowing down is again necessary prior to landing Since

always dm

dt the slowing down force F must be realized by a change

of sign of the exhaust speed w Therefore w will change its sign and must

t Let be the total duration of the trip into space then we can assume

t x

sin

t x

cos

Here x

is the largest distance from earth x

The trajectory satises

the following conditions x

surface of the earth x

maximum distance travelled v

initial speed and v

landing speed Inserting into we obtain for v

fc f

w t

sin

cos

G t

There seem to be several possibilities for the choice of m

t and w t

Whereas w t is quite arbitrary the function m

t has to satisfy the two

boundary conditions m

where M

and M

are the

start rest mass of the space rocket andM

the landing rest mass respectively

Equations involving more than one independent variable and partial deriva

tives with respect to these variables are called partial dierential equations

The order of a partial dierential equation is the order of the partial deriva

tive of highest order that occurs in the equation The problem of nding a

solution to a given partial dierential equation that will meet certain specied

requirements for a given set of values of the independent variables boundary

conditions constitutes now a boundary value problem The given set of val

ues is then given on a twodimensional curve or on a threedimensional

surface

If the values given by the boundary condition are all zero the boundary

condition is called homogeneous If the values do not vanish the boundary

condition is called inhomogeneous

For instance the vibrations of a membrane are described in cartesian coor

dinates by the socalled twodimensional wave equation

u x y t

where the Laplacian is given by

and c is a constant speed describing the material properties of the membrane

Trying to separate the independent variable t we use the setup

Since both sides of this equation depend on dierent independent variables

the lefthand side and the righthand side must be equal to the same con

stant that we called

We thus have to solve the ordinary dierential

equation

together with some initial or two point condition and the partial dierential

equation

where we used k

The dimension of k is given by m

Equation

is usually called the Helmholtz equation

One may look for the general solution of or one may be interested

in the vibrations of a rectangular membrane clamped along its boundary

Considering a rectangle as shown in Figure the homogeneous boundary

conditions for a membrane clamped at the boundaries read

b b

Figure

Rectangular boundary

va y for b y b

vx b for a x a

vxb for a x a

To nd a solution to we make the ansatz

Since both second derivatives divided by X and Y respectively must be

constants like

we now have the following choice

either a

or b

which we will discuss later in detail The constants are usually called

separation constants They are dependent on the boundary conditions

From Figure one can see that a solution satisfying the boundary condi

tions will be symmetric in both independent variables x and y We

hence take the solutions

Xx A cos x

Y y B cos y

where A and B are integration constants partial amplitudes

If the solutions are used the boundary conditions

determine the separation constants This is the usual method From

and we have

vx a y A cos a B cos y

vx y b A cos x B cos b

These boundary conditions are satised by

m b

n m n

Since there appear several even an innite number of solutions for the sep

aration constants we should write

membrane From we then have

Apparently our boundary value problem with has solutions

only for special discrete values of k If a dierential equation like

or has solutions only for special values of a parameter the prob

lem is called an eigenvalue problem k is the eigenvalue and the solutions

x A

cos

x are called eigenfunctions In contrast to this solu

tions of will be called modes or particular solutions

Let us now consider One possible solution is

T t cost

Due to and the denition used for k in the constant angular

frequency has the dimension sec

and assumes the discrete values

The

are called eigenfrequencies

A partial dierential equation is called linear if it is of the rst degree in

the dependent variable ux y ux y t and their partial derivatives u

ie if each term either consists of the product of a known function of the inde

pendent variables and the dependent variable or one of its partial derivatives

The equation may also contain a known function of the independent variables

only inhomogeneous equation If this term does not appear the equation is

called homogeneous It is a general property of all linear dierential equations

that two particular solutions can be superposed which means that the sum of

two solutions is again a new solution superposition principle So the gen

eral solution is a superposition of all particular solutions Hence the general

symmetric solution of may be written as

ux y t

cos

where the

are given by and the partial amplitudes A

may be

determined from an initial condition like

ux y fx y

where fx y is a given function ie the deection of the membrane at the

time t

Using now the solutions we have

vx y A cos

cos y

the general solution reads

vx y

cos

In order to satisfy boundary conditions the unknowns A

and k have to

be determined We will discuss this problem later is the starting

point for a boundary point collocation method

Problems

InMathematica a function fx is represented by f1[x] We now dene f1[x]=1/(1-a*xˆ2) and integrate it by the command

F1[x]=Integrate[f1[x],x] which results in ArcTanh p

a x

Now we verify by dierentiation D[F1[x],x] which gives f1[x]. Be careful the letter l looks very much like the number

If we want to dene an expression like

R[x]=a+b*x+c*xˆ2; f2[x]=x/Sqrt[R[x]]; we may end the ex pression or the line with a semicolon Then Mathematica allows to put

several commands on the same line If there is a last semicolon in the

line as above no outprint will be given

Integration F2[x]=Integrate[f2[x],x] results in

a bx cx

b Log

b ax

a bx cx

To verify again one may give the command D[F2[x],x] yielding

b cx

a bx cx

b cx

a bx cx

b cx

a bx cx

Apparently this can be simplied We give the command Simplify[%]

to obtain

a bx cx

Here the symbol means use the last result generated and means

the nexttolast result and ktimes indicates the kth previous

Mathematica knows how to solve dierential equations

DSolve[x[t]+aˆ2*x[t]==0, x[t],t] This solves x t a

xt with respect to the independent variable

t and gives the solution xt

xt C Cosa t C Sina t

As usual Mathematica drops the sign ! for the multiplication Do not

use quotation marks for the second derivative use two apostrophes

Specifying the integration constants we can plot the result in the

interval x by the command Plot[2 Sin[x],{x,0,2 Pi}]

ffyx

C e

Cgg

ffyx

C e

Cgg

Integrate[(x+2)/(x+1),x] x Log x

x x

ArcSin

Tan x

Tanx

D[Cosh[xˆ2-3*x+1],x] x Sinh x x

ffyx e

C e

Erfx

ffxt

g m t

a t

Cgg

Classication of partial dierential equations

Ordinary dierential equations containing only one independent variable and

two point boundary conditions are of minor importance for problems of physics

and engineering Hence we mainly concentrate on partial dierential equa

tions

Partial dierential equations possess a large manifold of solutions Instead

of integration constants arbitrary functions appear in the solution As an

example we consider in a specialization of two independent variables

x t u

x t

By inserting into it is easy to prove that

ux t fx ct gx ct

is a solution of Here f and g are arbitrary functions Such a solu

tion of a partial dierential equation of order n with n arbitrary functions

is called a general solution If a partial dierential equation contains p inde

pendent variables x y one can nd a complete solution which contains p

integration constants If a function satises the partial dierential equation

and the accompanying boundary conditions and has no arbitrary function or

constants it is called a particular solution If a solution is not obtainable by

assigning particular values to the parameters in the general complete solu

tion it is called a singular solution It describes an envelope of the family of

curves represented by the general solution But the expression is also used for

a solution containing a singular point a singularity We will later consider

such solutions containing singular points where the solution tends to innity

There are other essential properties characterizing various types of partial

dierential equations We give some examples below

x y u

x y

This Laplace equation is linear homogeneous and has constant coe"cients

The equation

x t x

x t

is of the rst order linear homogeneous and has variable coe"cients For a

rstorder partial dierential equation a boundary problem cannot be formu

lated An example of a nonlinear homogeneous partial dierential equation

is given by

x y u

x y

and the equation

is given by the linear Poisson equation

x y u

x y x y

where is a given function The most general linear partial dierential equa

tion of two independent variables has the form

ax yu

x y bx yu

x y cx yu

x y dx yu

x y

ex yu

x y gx yux y hx y

Now equations satisfying

x y ax ycx y

in a certain domain in the x y plane are called hyperbolic equations

x y ax ycx y

characterizes elliptic equations and if

x y ax ycx y

the equation is called parabolic Thus an equation is hyperbolic elliptic or

parabolic within a certain domain in the x y plane This is an important

distinctive mark determining the solvability of a boundary problem

Boundary curves or surfaces may be open or closed A closed boundary

surface is one that surrounds the domain everywhere conning it to a nite

surface or volume A simple closed smooth curve is called Jordan curve An

open surface is one that does not completely enclose the domain but lets it

extend to innity in at least one direction Then the Dirichlet boundary

conditions rst boundary value problem x the value ux y on the bound

ary Neumann boundary conditions second boundary value problem x the

value of the normal derivative u n on the boundary and a Cauchy condi

tion xes both value and normal derivative at the same place The Cauchy

condition actually represents an initial condition The normal derivative is

the directional derivative of a function ux y in the direction of the normal

at the point of the boundary where the derivative is taken A generalized

Neumann boundary condition third boundary value problem xes

kx y

ux y

lx yux y mx y

on the boundary This condition is of importance in heat ow and uid

mechanics It is possible to prove the solvability of boundary problems

see Table

Table Solvability of boundary problem

Boundary Equation

condition hyperbolic elliptic parabolic

Cauchy

open boundary solvable indeterminate overdeterminate

one closed boundary overdeterminate overdeterminate overdeterminate

Dirichlet

open boundary indeterminate indeterminate solvable

one closed boundary indeterminate solvable overdeterminate

Neumann

open boundary indeterminate indeterminate solvable

one closed boundary indeterminate solvable overdeterminate

In this connection the term solvable means solvable by an analytic solution If

an elliptic boundary problem has two closed boundaries an analytic solution

is no longer possible and singularities have to be accepted

If the value ux y or its derivatives or mx y in vanish on the

boundary the boundary condition is said to be homogeneous If the values on

the boundary do not vanish the boundary condition is called inhomogeneous

A boundary problem is called homogeneous if the dierential equation and

the boundary condition are both homogeneous If the dierential equation

or the boundary condition or both are inhomogeneous then the boundary

problem is said to be inhomogeneous Inhomogeneous boundary conditions

of a homogeneous equation can be transformed into homogeneous conditions

of an inhomogeneous equation This is made possible by the following fact

The general solution ux y of a linear inhomogeneous dierential equation

consists of the superposition of the general solution wx y of the matching

homogeneous equation and a particular solution vx y of the inhomogeneous

equation We consider an example Let

ux y x y

be an inhomogeneous equation with the homogeneous condition uboundary

If we insert the ansatz

ux y wx y vx y

into we obtain

wx y vx y x y

Putting

vx y x y

we obtain an inhomogeneous equation for v and a homogeneous equation for

w which reads

vboundary wboundary

We now see that the inhomogeneous condition that belongs to the

homogeneous equation has been converted into a homogeneous con

dition uboundary and an inhomogeneous equation We thus

have to construct a function vx y satisfying the inhomogeneous condition

for w and producing the term x y by application of on v as in

We now give an example of the conversion of an inhomogeneous boundary

condition for a homogeneous equation into a homogeneous condition matching

an inhomogeneous equation As the homogeneous equation we choose the

Laplacian and write it now in the form

x y w

x y

We consider again the rectangle of Figure but instead of the homogeneous

boundary conditions we now use inhomogeneous conditions

wa y for b y b

wx b fx for a x a

wxb fx for a x a

Then the corresponding inhomogeneous equation is given by and its

homogeneous boundary conditions are written for ux y To nd a

solution of we have thus to solve

x y u

x y x y

together with the homogeneous boundary conditions

ua y for b y b

uxb for a x a

The solution of the inhomogeneous equation is now a superposition

of a general solution u

x y of the homogeneous equation and a particular

solution vx y u

x y A setup u

x y Xx Y y together with the

symmetry expressed by and delivers

x y

cos

x cosh

To make the problem a little easier we assume x y p const This

tive must give the constant p we use the setup

x y c

xy c

x c

y c

which seems to be a general possibility where the c

are constants We obtain

from

Since the particular solution u

has to satisfy separately the boundary condi

tions we have from

ay c

a c

y c

for b y b hence also for y this gives c

and

p from From we get

x c

or with

is given by the homogeneous solution satises too

From we have

cos

cosh

This indicates that we have to expand the lefthand side for a x a

into a cosFourier series to satisfy the boundary condition This

procedure gives the A

So the solution ux y u

x y u

x y reads

ux y

cos

cosh

It satises the inhomogeneous equation for x y p and also the

associated boundary conditions and

The particular solution u

vx y has now to satisfy or

x y v

x y p

For vx y we have

x y vx y

Then gives an identity Due to the solution satises

the boundary conditions and fx

The inhomogeneous boundary conditions have been homogenized and

according to the solution of reads

wx y ux y vx y

cos

cosh

which satises due to the relation It is clear that the method

described can be used for x y const too

Problems

Determine if the following partial dierential equations are hyperbolic

h elliptic e parabolic p or of a mixed type m which means the

type depends on the domain in the x y plane

e u

p u

m u

x p x

u u

Using Mathematica calculate

dx x

dx x logx

x x

arcsin

arctan

tanx

sec

tan

coshx

x sinhx

x x

What happens if reads

x c

results in c

x c

y ax by c so that c

a c

b c

f1[x]=(xˆ2+2)ˆ2*3*xˆ2 F1[x]=Integrate[f1[x],x] D[F1[x],x]; Simplify[%] Explanations of these commands will be given later

Types of boundary conditions and the collocation

method

If a boundary curve or surface can be described by coordinate lines or sur

faces and if the partial dierential equation in question is separable into ordi

nary dierential equations in this coordinate system by a setup like

the boundary problem can be solved quite easily compare the calculations

through To express boundary conditions in a simple way

one must have coordinate surfaces that t the physical boundary of the prob

lem Very often however the situation is more complicated even for partial

dierential equations that are separable in only a few coordinate systems

if the boundary cannot be described in the corresponding coordinate system

On the other hand there are problems that belong to partial dierential equa

tions that cannot be separated at all into ordinary dierential equations As

an example we mention equation

x y fx yu

x y

If the coe"cient function fx y cannot be represented by a product fx y

gx hy then a separation of into ordinary dierential equations is

rarely possible However in a problem of plasma physics or

r z u

r z

ur z

ur z a br cr

is an example demonstrating the opposite but has to be solved In this

case the ansatz ur z Rr Zz leads to a separation into two ordinary

dierential equations

a br cr

arability in general terms On the other hand it is well known that the

Helmholtz equation u k

u is separable in coordinate systems

only These coordinate systems are

rectangular coordinates x y z circular cylinder coord r z

elliptic cylinder coord z parabolic cylinder coord z

spherical coordinates r prolate spheroidal coord

oblate spheroidal coord parabolic coordinates

conical coordinates r ellipsoidal coordinates

paraboloid coordinates

These coordinate systems are formed from rst and seconddegree sur

faces There are also systems built from fourthdegree surfaces that may

have practical applications

However the theory of separability of partial dierential equations is

of no great interest to us since we will be discussing methods to solve non

separable problems in this book In principle each boundary problem has a

solution The problem is though how to nd the solution In this book

we will discuss a method to solve boundary problems of various kinds see

Table In this table the term boundary tted means that the boundary

can be described by coordinate lines of the coordinate system in which the

partial dierential equation is separable

Table Various boundary problems

di equ boundary example solution

separable tted

yes yes rectangular membrane classical

yes yes circular ring membrane singularity

yes no circ membrane cartes coordinates possible

no no Cassini curve membrane numerical

yes yes boundaries from coordsystems sing nonuniform

no no toroidal problems singularity

no corners nonJordan curve special solution

We will discuss some examples of boundary problems that are mentioned in

Table A membrane described by the Helmholtz equation and bound by

a Cassini curve is such an example Membranes with holes and exhibiting

two closed boundaries and toroidal problems or boundaries with corners will

be treated If there are two boundaries it may happen that they belong

to two dierent coordinate systems This type of problem can be called a

nonuniform boundary problem It can be solved by special methods

In principle each reasonable boundary problem can be solved using the

fact that the general solution of a partial dierential equation contains one or

reasonable function can be expanded into an innite series of partial solutions

Thus an innite set of constants like partial amplitudes A

is equivalent to

an arbitrary function

In the case of dierent boundary problems one has two possibilities one

can choose an expression that satises either the dierential equation or the

boundary conditions exactly Coe"cients contained in the expression can

then be used to satisfy the boundary condition or the dierential equation

respectively We give an example We consider the problem

with the boundary condition on the square jxj jyj

u x y

u y x

Instead of using the method that we used to solve we rst write

down another expression satisfying the dierential equation For the

general solution of we require a particular solution of it together with a

solution of the homogeneous equation A particular solution u

is apparently

given by

compare

As the solution u

of the homogeneous equation satisfying the given symme

try conditions we could use or the real parts of xiy

which deliver the socalled harmonic polynomials Multiplying them by coef

cients we have

u u

This is a solution of for We now must determine the coe"

cients in such a way that the solution satises the boundary conditions

To do this we use a collocation method This means that we choose a set of n

socalled collocation points x

i n on the boundary curve on which

the boundary conditions must be satised The coe"cients a

will then be

calculated from the boundary conditions Due to the double

symmetry we need only consider the part of the boundary along x where

In order to determine the three unknown coe"cients a

we need three

equations This means that we have to choose three collocation points y

along the boundary line x y We choose y

and obtain from the three equations i

The method to determine the unknown coe"cients a

and a

from the

boundary conditions is called boundary collocation

On the other hand we can use another ansatz satisfying the boundary

conditions from the start Then the coe"cients have to be determined in

such a way that the dierential equation is satised in the whole domain

interior collocation Since we now need more collocation points to cover the

whole square this method is more expensive

In order rst to satisfy the boundary conditions we make the ansatz

u a

Each term has to satisfy the boundary conditions separately To obtain

this we calculate the a

and so on from whereas the coe"cients

have to be determined in such a way that the dierential equation is

satised Inserting into we have for the rst term

which is solved by a

The second term yields

which delivers a

We now have the

ansatz

u a

satises the dierential equation Inserting we obtain

For this expression with two unknowns we have to choose two collocation

points within the square domain We can dene x

and

Inserting this into we can calculate a

and a

obtain an approximate solution

Since collocation will be discussed later on in detail we postpone problems

Dierential equations as models for nature

In the last sections we have discussed boundary conditions but where are

the dierential equations coming from There are apparently two methods

to derive dierential equations as models for phenomena in nature and en

gineering intuition and derivation from fundamental laws of nature like the

energy theorem etc We will give two examples the rst for intuition Let

us assume we want to study the spread of an epidemic disease By St we

designate the number of healthy persons It will be the number of persons

contracting the disease and let Rt be the number of persons being immune

against the disease Apparently one then has a theorem for the conservation

of the number of people if nobody dies or is born during the short time period

considered This number balance reads

St It Rt const

Now intuition and experience come into play Apparently the number of

persons newly infected will be proportional to the number St of healthy

persons and to the number It of sick persons

dSt

St It

The parameter can be called rate of infection On the other hand the

number of persons becoming immune after having recovered from the disease

would be

dRt

where can be called rate of immunization We thus have three equations

Calculation of I and dIdt from and inserting into gives a

nonlinear dierential equation of second order

d lnS

We now assume the initial condition St S

Thus the number of

healthy persons at t t

is given by S

Neglecting d lnSdt we obtain

near t

Thus the curvature S

depends on the conditions S

and S

respectively This represents exactly the empirical basic theorem of epidemi

ology An epidemic starts if the number S

of healthy but predisposed

persons exceeds a specic threshold

Another method to derive dierential equations is given by a derivation from

fundamental laws Whereas boundary conditions describe actual situations

and are used to specify an actual particular solution by determining integra

tion components dierential equations of order n deliver the general solution

containing n integration constants For example the dierential equation de

scribing transverse vibrations of a thin uniform plate can be derived from the

empirical law of Hookean deformation plus the energy theorem A more

elegant way of deriving dierential equations is variational calculus Let us

assume that the eigenfrequencies of transversal vibrations of plates of vary

ing thickness are suddenly of practical interest eg for the investigation of

ssures in an airplane wing

The fundamental problem of the calculus of variation is to determine the

minimum of the integral

Jux y t

ZZZ

F x y t ux y t u

x y t u

x y t

x y t u

x y tdxdydt

for a given functional F The minimum of the integral J delivers this function

ux y t which actually makes J a minimum This function is determined

by the Euler equations which will be derived later on in section

F x y t u u

with respect to its variables u u

etc

For a physical or engineering problem the functional F is given by the La

grange functional dened by the dierence kinetic energy T minus potential

energy # If we assume that ux y t is the local transversal deection of a

plate the kinetic energy of a plate with modestly varying thickness hx y is

given by

hx y

dxdy

Here

is the constant surface mass density per unit of thickness so that

hx y is the local surface mass density The surface integral T is taken

over the area G of the plate

In order to nd the minimum of we have to vary the integral

hx y

dxdy #

is the variational symbol and # is the total elastic energy ie the local

elastic potential integrated over the domain G

Using the designations E for Youngs modulus and for Poissons ratio

the local elastic free energy per unit volume of the plate is given by

fx y z u

The total elastic energy is then given by integration over the volume of the

plate Integration rst over z

dz alone from hx y to hx y delivers

x y Thus the total elastic energy # is then given by

#x y u

x yu

dxdy

If the plate has to carry a load px y then we have to add the term

px y ux ydxdy

This term describes the work done by the external forces when the points on

the plate are displaced by the displacement u Now the total functional is

Since then F

takes the form

From the rst two terms of we thus obtain

p F

Furthermore we then get the plate equation for varying thickness hx y in

the form

xxxx

xxyy

yyyy

fh h

xxx

xyy

yyy

xxy

g px y

This is the plate equation for weakly varying thickness hx y

This derivation does however not answer the question where the energy

theorem like or a consequence of Hookes law comes from

There are people believing that these fundamental laws are preexistent in

nature or have been originated by a creator Modern natural philosophy

tends to another view So mathematicians know that dierential equations

are invariant under special transformations of coordinates If for instance the

equation of motion is submitted to a simple translation along the x coordinate

axis then the momentum mv

remains constant it will be conserved Emmy

Noether has shown that the invariance of a dierential equation against a

transformation has the consequence of the existence of a conservation theorem

for a related physical quantity Thus the energy theorem is a consequence of

the invariance of the equation of motion under a translation along the time

axis Human beings assume that the laws of nature are independent against

a time translation But intelligent lizards as coldblooded animals would

ton But the results of the lizard physics would be the same as in human

physics Apparently human assumptions on coordinate transformations cre

ate the laws we nd in nature But how could we nd out if these laws are

correct and true In his Discours de la M$ethode Poincar

e has shown that

there are always several true models or theories describing natural phenom

ena As an example we can mention that Dives theory of elliptic waves

and the special relativity theory give exactly the same results up to the or

der vc

Why have we chosen special relativity to describe nature When

we have to decide between two fully equivalent theories we should take into

account

Aesthetic points of view

Machs principle of economic thinking

The extensibility of a theory to broader elds of applications like the

extension of special to general relativity respectively

Problems

Derive the equation for transverse vibrations ux y t for a plate with

constant thickness h const see section

Try to solve using Mathematica Not possible

Type the command

gives St

t S

and plot the result But there is now a problem the solution is not given

by S[t]= . So it is necessary to dene a new function ut which gives a value for any arbitrary t This is done by replacing t by t_ We rst

select the integration constants C[2]=0, C[1]1=S0*(S0* ) to obtain St S and write u[t_]=Exp[t*(S0 )]*S0 This ut may be plotted for given arbitrary values of S

Learn partial derivatives Dene

u[x,y]=xˆ2+a*xˆ3*yˆ4+yˆ3 D[u[x,y],x] x a x y D[u[x,y],{x,2}] a x y D[u[x,y],y] a x y y