green's theorem in classical mechanics and electrodynamics

Applications of Green’s theorem in classical mechanics

and electrodynamics

C.Sochichiu

Wednesday, January 23, 13

Plan

1. Green’s theorem(s)

2. Applications in classical mechanics

3. Applications in electrodynamics

Wednesday, January 23, 13

• Goldstein, Poole & Safko, Classical mechanics

• Arnol’d, Mathematical methods of classical mechanics

• R.P.Feynman, Lectures on physics, vol.2 (Mainly electricity and magnetism)

• Jackson, Electrodynamics

Literature

Wednesday, January 23, 13

• There are several integral identities claiming the name “Green’s theorem” or “Green’s theorems”

• First there is a most basic identity proposed by George Green,

• We will call this ‘Green’s theorem’ (GT)

What is Green’s theorem?

I

@⌃(Ldx+Mdy) =

ZZ

⌃

✓@M

@x

� @L

@y

◆dxdy

Wednesday, January 23, 13

• Although, generalization to higher dimension of GT is called (Kelvin-)Stokes theorem (StT),

• where should be understood as a symbolic vector operator

• in electrodynamics books one will find ‘electrodynamic Green’s theorem’ (EGT),

Green’s theorem vs. Green’s theorems

r = (@/@x, @/@y, @/@z)

Wednesday, January 23, 13

• They are related to divergence (aka Gauss’, Ostrogradsky’s or Gauss-Ostrogradsky) theorem,

• All above are known as ‘Green’s theorems’ (GTs).

✴ They all can be obtained from general Stoke’s theorem, which in terms of differential forms is,

Other Green’s theorems

Wednesday, January 23, 13

• Here I used the standard notations: or for the line element, and , respectively, for area and volume elements

• and denote a space region and a surface, while and denote their boundaries

• As you might have noticed, all GTs, apart from GT require serious knowledge of vector calculus. GT requires only the knowledge of area and line integrals.

remark:

Wednesday, January 23, 13

• Consider a two-dimensional domain with one-dimensional boundary then for smooth functions and we have the integral relation:

The Green’s theorem (GT)

L(x, y)M(x, y)

D@D

I

@⌃(Ldx+Mdy) =

ZZ

⌃

✓@M

@x

� @L

@y

◆dxdy

@⌃⌃

Wednesday, January 23, 13

• Consider a domain with boundary described by piecewise smooth function

• Then, choosing we have

• Q: why did I put the minus sign?

An intuitive example

L(x, y) = �y, M(x, y) = 0

y(x)

y

x

x1 x2

y1(x)

y2(x)

Wednesday, January 23, 13

• Consider a domain with boundary described by piecewise smooth function

• Then, choosing we have

• Q: why did I put the minus sign?

An intuitive example

L(x, y) = �y, M(x, y) = 0

y(x)

y

x

x1 x2

y1(x)

y2(x)

Wednesday, January 23, 13

• Let us consider slowly varying functions and on a rectangular contour

• Q: Generalize this to an arbitrary polygon

The proof of GTL(x, y)

M(x, y)

M(x+dx,y)d

y

L(x, y)dx

�L(x, y + dy)dx

�M

(x,y)dy

I(Ldx+Mdy) =

L(x, y)dx+M(x,+dx, y)dy

� L(x, y + dy)dx�M(x, y)dy

=

✓@M

@x

� @L

@y

◆dxdy

Wednesday, January 23, 13

• The case of arbitrary contour and function can be obtained by dividing the domain in small parts and applying the argument from the previous slide

• Internal lines do not contribute:

General contour

Z(Ldx+Mdy)�

Z(Ldx+Mdy) = 0

Wednesday, January 23, 13

• Calculation of mass/area and momenta

• Criterion for a conservative force

• Kepler’s second law

• Other applications

2. Applications to classical mechanics

Wednesday, January 23, 13

• A (rather trivial) application of GT is the calculation of various momenta of two-dimensional shapes and axial symmetric bodies

• Use GT:

Mechanics

I

@⌃(Ldx+Mdy) =

ZZ

⌃

✓@M

@x

� @L

@y

◆dxdy

Wednesday, January 23, 13

• Choose and and parameterize the boundary as and

• Then the area or mass of uniform 2D object is,

• C.M.: and . Then the y-component of c.m. is given through

• Q: find similar formula for the x-component

Mass & Center of mass

Wednesday, January 23, 13

• In general case, the moment of inertia is a tensor quantity with three components in two-dimensions:

• To find , choose

• gives

• Exercise: Which choice gives ?

Moment of inertia

Wednesday, January 23, 13

• A force is conservative if its work does not depend on a chosen path

• For such a force we can define a potential energy such that

• How can we know if a given force is conservative?

Is F a conservative force?

Wednesday, January 23, 13

• Consider two paths and in xy-plane

• Similar arguments can be applied to any plane.

• In vector calculus language, a conservative is equivalent to,

Is F a conservative force?

Wednesday, January 23, 13

• “A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.”

Kepler’s second law

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• This law means angular momentum conservation. Indeed, expressing the area swept by the radius vector of the planet in the time interval and using GT, we get,

• Therefore, the quantity must be conserved

Kepler’s second law

Wednesday, January 23, 13

• There are many more applications of Green’s (Stokes) theorem in classical mechanics, like in the proof of the Liouville Theorem or in that of the Hydrodynamical Lemma (also known as Kelvin Hydrodynamical theorem)

Another applications in classical mechanics

Wednesday, January 23, 13

• Connection between integral and differential Maxwell equations

• The energy of steady currents

• Other applications

3. Applications in Electrodynamics

R.P. Feynman: “Electrostatics is Gauss’ law plus…”➪ “Electrodynamics is Green’s theorems plus…”

Wednesday, January 23, 13

Maxwell equationsMaxwell equations (Integral form)

I

S

~E · d~A= qS✏0

(Electric) Gauss’s law

I

S

~B · d~A= 0 Magnetic Gauss’s law

I

C

~E · d~s=� d�B

dtFaraday’s law

I

C

~B · d~s= µ0IC + ✏0µ0d�E

dtAmpere–Maxwell law

Integral form Differential form

• Let’s use Green’s theorem to derive the differential Faraday’s law from the integral form…

Wednesday, January 23, 13

Maxwell equationsMaxwell equations (Integral form)

I

S

~E · d~A= qS✏0

(Electric) Gauss’s law

I

S

~B · d~A= 0 Magnetic Gauss’s law

I

C

~E · d~s=� d�B

dtFaraday’s law

I

C

~B · d~s= µ0IC + ✏0µ0d�E

dtAmpere–Maxwell law

Integral form Differential form

• Let’s use Green’s theorem to derive the differential Faraday’s law from the integral form…

Wednesday, January 23, 13

• Consider a time-independent contour in the xy-plane. Faraday’s law for this contour,

• Use GT,

• Therefore, for any surface in xy plane,

Faraday’s law

Wednesday, January 23, 13

• The integral over an arbitrary surface vanishes iff,

• In a similar way consider xz- and zy-planes. Then, all three equations arrange into

• I used the original GT. Of course, better idea would be to use the Stokes theorem…

Faraday’s law

Wednesday, January 23, 13

• Differential form of the Ampère-Maxwell equation can be deduced in exactly the same way

• Differential forms of Gauss’ Law and Magnetic Gauss’ Law are best derived using the divergence theorem

• Green’s theorems are used also to derive the Maxwell term for the Ampère’s law

Other Maxwell equations

Wednesday, January 23, 13

• Consider a current loop and represent it as superposition of small loops

• The energy of a small loop is

• Therefore,

• Use the fact that , where is the vector potential.

The energy of currents

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• Then, using the GTs (StT), we obtain

• even more… We can take the circuit as consisting of interacting filaments with

• The total energy is sum of energies for every pair,

The energy of steady currents

Wednesday, January 23, 13

• Green’s theorems are integral identities an important toolkit in various areas of physics(≈all)

• In classical mechanics GT allows calculation of parameters like location of the center of mass, moment of inertia etc. As an example, it gives the criterion for the conservative nature of a force and relates Kepler’s second law to conservation of angular momentum

• Electrodynamics is entirely based on GTs. Examples include the relation between integral and differential forms of Maxwell equations and the energy of steady currents

• Disclaimer: The applications of GT(s) are not restricted by given examples. They are chosen basing on the taste of the Applicant!

Summary

Wednesday, January 23, 13