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


Literature

• 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


What is Green’s theorem?
• 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,
I ZZ ✓ ◆
@M @L
(Ldx + M dy) = dxdy
@⌃ ⌃ @x @y

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

Green’s theorem vs. Green’s theorems

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

• where r = (@/@x, @/@y, @/@z) should be
understood as a symbolic vector operator
• in electrodynamics books one will ﬁnd
‘electrodynamic Green’s theorem’ (EGT),


Other Green’s theorems
• 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,


remark:
• 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.


The Green’s theorem (GT)
• Consider a two-dimensional domain D
with one-dimensional boundary @D then
for smooth functions M (x, y) and L(x, y)
we have the integral relation:
I ZZ ✓ ◆
@M @L
(Ldx + M dy) = dxdy
@⌃ ⌃ @x @y

@⌃
⌃


An intuitive example
• Consider a domain with boundary
described by piecewise smooth function y(x)
• Then, choosing L(x, y) = y, M (x, y) = 0
we have
y y1 (x)

y2 (x)

x1 x2 x

• Q: why did I put the minus sign?

The proof of GT
• Let us consider slowly varying functions L(x, y)
I and M (x, y) on a rectangular contour
L(x, y + dy)dx
(Ldx + M dy) =

M (x + dx, y)dy
M (x, y)dy
L(x, y)dx + M (x, +dx, y)dy
L(x, y + dy)dx M (x, y)dy
✓ ◆
@M @L
= dxdy L(x, y)dx
@x @y

• Q: Generalize this to an arbitrary polygon


General contour
• 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:

Z Z
(Ldx + M dy) (Ldx + M dy) = 0


2. Applications to classical mechanics

• Calculation of mass/area and momenta
• Criterion for a conservative force
• Kepler’s second law
• Other applications


Mechanics
• A (rather trivial) application of GT is the
calculation of various momenta of two-
dimensional shapes and axial symmetric
bodies
• IUse GT: ZZ ✓
@M @L
◆
(Ldx + M dy) = dxdy
@⌃ ⌃ @x @y


Mass & Center of mass
• 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

Moment of inertia
• 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 ?


Is F a conservative force?
• 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?
• Consider two paths and in xy-plane

• Similar arguments can be applied to any
plane.
• In vector calculus language, a conservative
is equivalent to,


Kepler’s second law

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


Kepler’s second law
• 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


Another applications in classical
mechanics

• 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)


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

• Connection between integral and
differential Maxwell equations
• The energy of steady currents
• Other applications


Maxwell equations
Maxwell equations (Integral form)

Integral form Differential form

I
~ · dA= qS
E ~ (Electric) Gauss’s law
S ✏0
I
~ ~
B · dA= 0 Magnetic Gauss’s law
IS
~ s d B
E · d~ = Faraday’s law
dt
IC
~ s d E
B · d~ = µ0 IC + ✏0 µ0 Amp`re–Maxwell law
e
C dt

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


Faraday’s law
• 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
• 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…


Other Maxwell equations
• 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


The energy of currents
• 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 steady currents
• 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,


Summary
• 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!


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