Subject: This Week's Finds in Mathematical Physics (Week 292)
From: [email-masked] (John Baez)
Newsgroups: sci.physics.research,sci.physics,sci.math

Also available at http://math.ucr.edu/home/baez/week292.html

January 29, 2010
This Week's Finds in Mathematical Physics (Week 292)
John Baez

I've been telling a long tale about analogies between different
physical systems.  Now I finally want to tell you about "bond graphs"
- a technique engineers use to exploit these analogies.  I'll just say
a bit, but hopefully enough so you get the basic idea.  

Then I'll sketch a rough classification of physical systems, and
discuss the different kinds of math used to study these different
kinds of systems.  I'll only talk about systems from the realm of
classical mechanics!  To people who love classical field theory,
quantum mechanics and quantum field theory, this may seem odd.  Isn't
classical mechanics completely understood?  

Well, nothing is ever completely understood.  But there are some
reasons that mathematical physicists like myself tend to slip into
thinking classical mechanics is better understood than it actually is.  
We love conservation of energy!  Taking this seriously led to a
wonderful framework called Hamiltonian mechanics, which we have been
studying for over a century.  We know a lot about that.  We all studied
it in school.

But the Hamiltonian mechanics we learn in school needs to be 
generalized a bit to nicely handle "dissipative systems", with 
friction - or more general "open systems", where energy can flow 
in and out of the boundary between the system and its environment.  

(A dissipative system is really a special sort of open system, since
energy lost to friction is really energy that goes into the
environment.  But the study of dissipative systems has not been fully
integrated into the study of open systems!  So, people often treat
them separately, and I may do that too, now and then - even though it's
probably dumb.)

Anyway, while lovers of beauty have the freedom to neglect 
dissipative systems and open systems if they want, engineers don't!
Every machine interacts with its environment, and loses energy to
its environment thanks to friction.  Furthermore, machines are made
of pieces, or "components".  Each piece is an open system!  Each
component needs to be understood on its own... but then engineers 
need to understand how the components fit together and interact.

A lot of engineers do this with the help of "bond graphs".  These are
diagrams that describe systems made of various kinds of components:
electrical, mechanical, hydraulic, chemical, and so on.  The one thing
all these components have in common is *power*.  Energy can flow from
one component to another.  The rate of energy flow is called "power",
and bond graphs are designed to make this easy to keep track of.

The idea behind bond graphs is very simple.  I've been describing
various "n-ports" lately, and I've drawn pictures of them.  In my
pictures, a 3-port looked like this:

      |  |  |
      V  V  V
      |  |  |
    -----------
   |           |
   |           |
    -----------
      |  |  |
      V  V  V
      |  |  |

In the case of an electrical system, this means 3 wires coming in and
3 going out.  More generally, an n-port is a gadget with n inputs and
n outputs, where the flow into each input equals the flow out of the
corresponding output.

The idea of bond graphs is to draw these pictures differently.  Don't
draw individual wires!  Instead, draw each pair of wires - input and
output - as a single edge!

Such an edge is called a "bond".  So, an n-port has n "bonds" coming
out of it.

Take an electrical resistor, for example.  This is a kind of 1-port -
an example of what bond graph experts call a "resistance".
Mathematically, a resistance is specified by a function relating
effort to flow.  In the example of an electrical resistor, effort is
"voltage" and "flow" is current.

It's pretty natural to draw a resistor like this:

        |
        V
        |
      -----
     |     |
     |     |
      -----
        |
        V
        |

But in the world of bond graphs, people draw it more like this:

        |
        V
        |
      -----
     |     |
     |     |
      -----

One "bond" for two "wires"! 

Actually, to be a bit more honest, they draw it a bit more like this:

           p'       \
    ------------------ R
           q'

Now the arrow is pointing across instead of down.  There's a bond
coming in at the left, but nothing coming out at right.  The p' and
q' let us know that the resistance is relating effort to flow.  The
R stands for resistance.  

To be even more honest, I should admit that most bond graph people use
"e" for effort and "f" for flow.  So, they really draw something like
this:

           e        \
    ----------------- R
           f

But I want to stick with p' and q'.

Another famous 1-port is a capacitor.  Bond graph people draw it
like this:

           e        \
    ----------------- C
           f

A nice example of a 2-port is a transformer.  I explained this back in
"week292".  Bond graph people draw it like this:

          e_1     \         e_2     \
    --------------- TF -------------- 
          f_1              f_2

There's a bond coming in at left and a bond coming out at right: 2
bonds for a 2-port.  Similarly, a 3-port has 3 bonds coming out of it,
and so on.  

Bond graphs were invented by Henry Paynter.  You can read his story
here:

1) Henry M. Paynter, The gestation and birth of bond graphs,
http://www.me.utexas.edu/~longoria/paynter/hmp/Bondgraphs.html

It reminds me slightly of Hamilton's story about inventing quaternions
while walking down the river with his wife to a meeting of the Royal
Irish Academy.  Just slightly... but you can tell that bond graphs
thrilled Paynter as much as quaternions thrilled Hamilton.  I want to
quote a bit, and comment on it.  He begins:

   Bond graphs were born in their present form on April 24, 1959. 
   They were the direct outgrowth of my academic and professional
   experience during two previous decades.  My MIT undergraduate and
   graduate training was centered on hydroelectric engineering, as was
   my work at Puget Power and my 8 years teaching in the Civil
   Engineering Department at MIT.  This involved all aspects of the
   typical power plant indicated below. 

Here he shows a picture of a hydroelectric power plant, and the bond
graph that abstractly describes it.  A reservoir full of water acts as
an "effort source", since water pressure is a form of "effort".  Water
flows down a conduit and turns a turbine.  Here hydraulic power gets
converted to mechanical power.  Then the turbine turns a generator,
which produces electricity - so here, mechanical power is getting
converted to electrical power.

There are also some feedback loops, shown with dotted arrows.  Solid
arrows represent power, while dotted arrows represent "signals".  For
example, the turbine sends a signal about how fast it's turning to a a
gadget called a "speed governor".  If the turbine starts turning too
fast or too slow, this gadget reduces or increases the flow of water
to the turbine.  There's a similar feedback loop involving the generator.  

I haven't said anything about "signals" yet.  The idea here is that
information can be transmitted using negligible power.  For example,
you don't slow a turbine down much by attaching a little gadget that
measures how fast it's turning.  So, we can often get away with
pretending that a signal carries *no* power.  But this idealization
breaks down in quantum mechanics - so if we ever get to talking about
"quantum bond graphs", we'll have to rethink things.  In fact, the
idealization often breaks down long before quantum effects kick in!
I think this aspect of bond graphs deserves more mathematical study.

You can see in Paynter's picture that the reservoir is a 1-port.  It's
an example of an "effort source" - a kind of 1-port I explained back in
"week290".  The turbine and generator are 2-ports, since they have an
input and output.  These are both "transformers" - a kind of 2-port I
explained last week.  You'll also see that the feedback loops involve
some 3-ports.  I explained these too last week.  The 0 stands for a
"parallel junction", and the 1 stands for a "series junction".

Paynter continues:

   This training and experience in hydroelectric power actually forced
   certain insights upon me, most particularly an awareness of the
   strong analogies existing between:

          TRANSMISSION   (fluid pipes & electric lines);
          TRANSDUCTION   (turbines & generators);
          CONTROL        (speed governors & voltage regulators).

   When these analogous devices were reduced to equations for computer
   simulation, distinctions became completely blurred.

  Even before 1957 it was obvious that the above hydro+electric plant
  necessarily involved two energy-converting transduction multiports:
  the hydraulic turbine converting fluid power to rotary shaft power and
  the electrical generator converting this shaft power into polyphase AC
  power.  Moreover, the strict analogy between these two devices holds
  right down to the local field-continuum level.  Thus the fluid
  vorticity corresponds precisely to the current density and the fluid
  circulation to the magnetizing current, so that even the turbine
  blades correspond to the generator pole pieces!  In dynamic
  consequence, both these highly efficient components become 2-port
  gyrators, with parasitic losses.  Common sense dictated that
  such compelling analogies implied some underlying common
  generalization from which other beneficial specializations might
  ensue.  My efforts were also strongly motivated by a preoccupation
  with the logical philosophy underlying analogies in general.  Such
  concerns were much earlier formalized by the mathematician, Eliakim
  Hastings Moore, in the following dictum:
  
     "We lay down a fundamental principle of generalization by 
     abstraction: The existence of analogies between central
     features of various theories implies the existence of a 
     general theory which underlies the particular theories and 
     unifies them with respect to those central features... "

But Paynter only got the idea of bond graphs when he moved from the civil
engineering department to the mechanical engineering department at
MIT.  Then comes the part that reminds me of Hamilton's famous
description of inventing the quaternions.  In a letter to his pal
Tait, Hamilton wrote:

   To-morrow will be the 15th birthday of the Quaternions. They started
   into life, or light, full grown, on the 16th of October, 1843, as I
   was walking with Lady Hamilton to Dublin, and came up to Brougham
   Bridge, which my boys have since called the Quaternion Bridge. That
   is to say, I then and there felt the galvanic circuit of thought
   *close*; and the sparks which fell from it were the *fundamental
   equations between i, j, k*; exactly such as I have used them ever
   since. I pulled out on the spot a pocket-book, which still exists,
   and made an entry, on which, *at the very moment*, I felt that it
   might be worth my while to expend the labour of at least ten (or it
   might be fifteen) years to come. But then it is fair to say that
   this was because I felt a problem to have been at that moment
   solved - an intellectual want relieved - which had haunted me for
   at least fifteen years before.

Paynter writes:

   In 1954, I moved over to the MIT mechanical engineering department
   to establish the first systems engineering subjects at MIT.  It was
   this specific task which 5 years later produced bond graphs,
   drawing naturally upon all the attitudes and experience indicated
   above. So it was on April 24, 1959, when I was to deliver the
   lecture as posted below, I awoke that morning with the idea of the
   0,1-junctions somehow planted in my head overnight!  Moreover the
   very symbols (0,1) for Kirchoff's current law and Kirchoff's
   voltage law, respectively, made direct the correspondence between
   circuit duality and logical duality.  (The limited use of these
   3-ports in the hydro plant bond graph above hardly does justice to
   their role in rendering bond graphs a complete and formal discipline.)

The picture on the talk poster makes it clear that even without
knowing it, Henry Paynter was helping invent a branch of applied
*category theory* - a branch where physical systems that interact
with their neighbors are treated as *morphisms*.

(If you don't understand what Paynter means by Kirchoff's current law
and Kirchoff's voltage law, and "the correspondence between circuit
duality and logical duality", you can see a bit of explanation in 
the Addenda.)  

Paynter's book on bond graphs came out in 1961:

2) Henry M. Paynter, Analysis and Design of Engineering Systems, MIT
Press, Cambridge, Massachusetts, 1961.

About a decade later, bond graphs were taken up by many others
authors, notably Jean Thoma:

3) Jean U. Thoma, Introduction to Bond Graphs and Their Applications,
Pergamon Press, Oxford, 1975.

By now there is a vast literature on bond graphs.  This website is
a bit broken, but you can use it to get a huge bibliography:

4) Bondgraph.info, Journal articles, http://www.bondgraph.info/journal.html
Books, http://www.bondgraph.info/books.html

I've listed some of my favorite books in previous Weeks.  But if you want
a really quick online introduction to bond graphs, try this:

5) Wikipedia, Bond graph, http://en.wikipedia.org/wiki/Bond_graph

It covers a topic I haven't even mentioned, the "causal stroke".  And
it gives some examples of how to convert bond graphs into differential
equations.  If you read the talk page for this article, you'll see
that various people have found it confusing at various times.  But
it's gotten a lot clearer since then, and I hope people keep improving
it.  I'll probably work on it myself a bit.

Then, watch some of these:

6) Soumitro Banerjee, Dynamics of physical systems, lectures on
YouTube.  Lectures 13-19: The bond graph approach.
Available at http://www.youtube.com/view_play_list?p=D074EEC1EBEFAEA5

These lectures are very thoughtful and nice.  I thank C. J. Fearnley  
for pointing them out.

Now I'd like to veer off in a slightly different direction, and ponder
the various n-ports we've seen, and how they fit into different
branches of mathematical physics.  My goal is to dig a bit deeper into
the mathematics behind this big analogy chart:

                displacement    flow          momentum      effort
                     q           q'              p            p'

Mechanics       position       velocity       momentum      force
(translation)

Mechanics       angle          angular        angular       torque
(rotation)                     velocity       momentum

Electronics     charge         current        flux          voltage
                                              linkage

Hydraulics      volume         flow           pressure      pressure
                                              momentum

Thermodynamics  entropy        entropy        temperature   temperature
                               flow           momentum

Chemistry       moles          molar          chemical      chemical
                               flow           momentum      potential

But I won't be using the language of bond graphs!  The reason is 
that I want to talk about gizmos where the total number of inputs 
and outputs can be either even or odd, like this:

        |
        |
      -----
     |     |
      -----
      /   \
     /     \

Even though I'm talking about all sorts of physical systems, I'll use
the language of electronics, and call these gizmos "circuit elements".
We can stick these together to form "circuits", like this:
        
        |           |
        |           |
      -----         |
     |     |        |
      -----         |
      /   \         |
     /     \        |
  ------------      |
 |            |     |
  ------------      |
    |  |  |         |
    |  |   \_______/
    |  |
    |  |

Category theorists will instantly see that circuits are morphisms in
something like a compact closed symmetric monoidal category!  But the
rest of you shouldn't worry your pretty heads about that yet.  The main
thing to note is that we have "open" circuits that have wires coming 
in and out, as above, but also "closed" ones that don't, like this:

          _________
         /         \
        |           |
        |           |
      -----         |
     |     |        |
      -----         |
      /   \         |
     /     \        |
  ------------      |
 |            |     |
  ------------      |
    |  |  |         |
    |  |   \_______/
    |  |
   ------
  |      |
   ------

I will also call circuits "systems", since that's what physicists call
them.  And indeed, they often speak of "closed" systems, which don't
interact with their environment, or "open" ones, which do.

We've seen different kinds of circuit elements.  First there are
"active" circuit elements, which can absorb and emit energy, and for
which we cannot define a Hamiltonian that makes energy conserved.
Then there are "passive" ones, which come in two kinds:

  "conservative" circuit elements, which can absorb and emit 
  energy, but for which we can define a Hamiltonian that makes 
  energy conserved.

  "dissipative" circuit elements, which only absorb energy, and 
  for which we cannot define a Hamiltonian that makes energy 
  conserved.  

Not surprisingly, circuits made of different kinds of circuit elements
want to be studied in different ways!  We get pulled into all sorts of
nice mathematics this way - especially symplectic geometry, Hodge
theory, and complex analysis.  Here's a quick survey:

  1. If we have a circuit made of conservative circuit elements,
  we can study it using the PRINCIPLE OF LEAST ACTION.  So, we can
  define a Lagrangian for such a circuit, which is a function L(q,q') 
  of the displacements (q) and flows (q') of all its wires.  This
  function is a sum of terms, also called Lagrangians, one for each 
  circuit element.  The  "action" for the circuit is the integral 
  over time of the total Lagrangian.  The circuit will roughly do
  whatever minimizes this action.  It's lazy!  (Experts will know I'm
  lying slightly here.)

  By using a Legendre transform, we can compute p as a function of q'.
  Then we can work out the "Hamiltonian" of our circuit, as follows:

  H(p,q) = L(q,q') - p.q'

  Like the Lagrangian, this Hamiltonian will be the sum of Hamiltonians 
  for each piece - and I've told you what those Hamiltonians are for 
  all the conservative circuit elements I've mentioned.
 
  If the overall circuit is closed, no wires coming in or going out, 
  its Hamiltonian will be conserved in the strongest sense:

  dH/dt = 0

  There are elegant ways to study closed systems using Hamiltonian 
  mechanics - or in other words, symplectic geometry.  This is
  something mathematical physicists know well.

  We can also examine the special case of a conservative closed
  system in a static state, meaning that p and q don't depend on time.  
  The behavior of such systems is governed by the PRINCIPLE OF LEAST
  ENERGY: it will choose p and q that minimize the Hamiltonian H(q,p).

  If the circuit is open, we need a slight generalization of 
  Hamiltonian mechanics that can handle systems that interact with 
  their environment.  Open systems are less familiar in mathematical 
  physics - but as I explained in "week290", they're studied in
  "control theory".  Open systems obey a weaker form of energy 
  conservation, called the "power balance equation".

  2) If we have a circuit made of only dissipative circuit elements,
  we can study it using the PRINCIPLE OF LEAST POWER.  At least in
  a stationary state, where the efforts p' and flows q' don't depend
  on time, the system will act to minimize the power

  p'.q'
  
  Using this we can often solve for q' as a function of p' or vice 
  versa.  

  The principal of least power is closely related to other minimum
  principles in physics.  For example, if we build a network of
  resistors and fix the voltages on the wires coming in and out, the
  voltages on the network will obey a discretized version of the Laplace
  equation. This is the equation a function f satisfies when it 
  minimizes

  integral (grad f)^2

  So, circuits of this second kind are closely related to the Laplace
  equation, differential forms, Hodge theory and the like.  In fact 
  this is why Raoul Bott switched from electrical engineering to 
  differential topology!

  3. If we have a circuit made of only conservative and dissipative
  circuit elements, it's called "passive".  In a linear passive 
  circuit we can multiply the vector of efforts p' by an operator 
  called the "admittance matrix" to get the vector of flows q':

  q' = Ap'

  Or, we can take the inverse of this operator and get the "impedance 
  matrix", which tells us the flows as a function of the efforts:

  p' = Zq'

  Here both efforts and flows are functions of time.  Taking a Fourier
  transform in the time variable we get a version of the impedance 
  matrix that's a function of the dual variable: "frequency".  And if 
  the circuit is built from linear resistances and inertances, we'll 
  get a *rational* function of frequency!   The poles of this function 
  contain juicy information.  So, we can use complex analysis to study 
  such circuits.  This is very standard stuff in electrical engineering.

  4. If we have a circuit made of linear passive circuit elements
  together with effort and flow sources, we can still use the ideas
  that worked in case 3, but now we get an *inhomogeneous* linear
  equation like

  p' = Aq' + e

  where e comes from the effort sources.  This is called Norton's
  theorem.  Alternatively we can write

  q' = Zp' + f

  where f comes from the flow sources.  This is called Thévenin's
  theorem.  Again, these are standard results that electrical 
  engineers learn - but don't forget, they apply to *all* the systems
  in our chart of analogies!

  5. If we drop the linearity assumption and consider fully general
  circuits, things get more complicated.

I hope in future Weeks to say more about this stuff.  I hope you see
there are some strange and interesting patterns here - like this trio:

THE PRINCIPLE OF LEAST ACTION
THE PRINCIPLE OF LEAST ENERGY
THE PRINCIPLE OF LEAST POWER

We've seen the trio of action, energy and power before, back in
"week289".  Action has units of energy times time; power has units of
energy over time.  How do these three minimum principles fit together
in a unified whole?  I know how to derive the principal of least
energy from the principle of least action by starting with a
conservative system and imposing the assumption that it's static.  But
how about the principle of least power?  Where does this come from?

I don't know.  If you know, tell me!  

I'll tell you a bit about linear dissipative circuits and Hodge theory
next week.  But if you're impatient to learn circuit theory - or at
least know what books are lying next to my bed - let me give some
references!

This book is quite good:

7) Brian D. O. Anderson and Sumeth Vongpanitlerd, Network Analysis 
and Synthesis: A Modern Systems Theory Approach, Dover Publications,
Mineola, New York, 2006.

There's a lot of complex analysis in here!  Some is familiar, but 
there's also a lot we mathematicians don't usually learn: the Positive 
Real Lemma, the Bounded Real Lemma, and more.  

Speaking of Norton's and Thévenin's theorems, these articles demystify
those:

8) Wikipedia, Norton's theorem,
http://en.wikipedia.org/wiki/Norton%27s_theorem

9) Wikipedia, Thévenin's theorem,
http://en.wikipedia.org/wiki/Th%C3%A9venin%27s_theorem

These articles cover only circuits with one input and one output
which are made from flow sources, effort sources and linear resistances.
I know the results generalize to circuits where we also allow 
capacticances and inertances, and above I was willing to wager that
they apply to circuits with as many inputs and outputs as you like.

This book of classic papers is also good:

10) M. E. van Valkenburg, ed., Circuit Theory: Foundations and
Classical Contributions, Dowden, Hutchington and Ross, Stroudsburg,
Pennsylvania, 1974.

I mentioned Raoul Bott - mathematicians will be pleased and perhaps 
surprised to see a 1948 paper by him here!  It's 5 paragraphs long,
and it solved a basic problem.

-----------------------------------------------------------------------

Quote of the Week:

"I was born not knowing and have had only a little time to change that
here and there." - Richard Feynman

-----------------------------------------------------------------------

Addenda: Joris Vankerschaver writes:

   I've been following TWF the past few weeks extra carefully since
   I'm also interested in a systematic approach of electrical
   circuits, mechanical systems, and the like.  For this issue of TWF,
   I was wondering if you know whether there is a link between the
   *Hamilton-Pontryagin* (HP) variational principle and the action
   principles you mentioned.  I hope you don't mind me asking a direct
   question like this...

   The HP principle consists of taking a Lagrangian L(q, v), and
   thinking of v as an extra coordinate.  We then add the condition
   that q' = v as an extra constraint with a Lagrange multiplier
   p, to get a functional of the form

   S(q, v, p) = \int   p (q' - v) + L(q, v) dt

   where q, v, p are varied independently.  The result is the
   Euler-Lagrange equations in implicit form, together with Hamilton's
   equations and the Legendre transformation.

   I've added a PDF draft where these calculations are done in more
   detail.

   H. Yoshimura (who is a classical bond grapher) and J. Marsden have
   been working on this variational principle and apparently used it
   to great effect in circuit theory as well.  Circuits typically have
   degenerate Lagrangians and nonholonomic constraints, and the HP
   principle handles these very well.  But the HP variational
   principle has been re-discovered many times before.

   The equations of motion obtained from the HP principle can also be
   incorporated into a Dirac structure, which (according to van der
   Schaft and Maschke) is very well suited for interconnection
   purposes (where power is conserved).  So again, I was wondering if
   there was a link between the HP principle and what you are
   considering.

   I would be very interested in hearing your thoughts about this.  I
   am really looking forward to the next few issues of TWF!

Unfortunately I had to tell him that I've never heard of the
Hamilton-Pontryagin variational principle.  More to learn!

In "week292" I briefly mentioned the "dual" of a planar electrical
circuit, where we switch series junctions and parallel junctions,
switch efforts (voltages) and flows (currents), and so on.  You'll
note that in my quote of Paynter he was drawing a perhaps slightly
obscure analogy between this sort of duality and what he called
"logical" duality.  This is usually called "De Morgan duality": it's 
a symmetry of classical logic, which consists of switching true and
false, AND and OR, and so on.  In binary notation it consists of
switching 0 and 1.  This is why Paynter called a parallel junction a
"0-junction" and the series junction a "1-junction".  I didn't really
understand the connection until Chris Weed explained it:

   John, 

   The point is pretty trivial, but it's perhaps worth reminding the
   reader of the immediate connection to the dualities of Boolean
   algebra.

   More precisely, a series connection of two switches can be
   considered to implement the function AND(x,y) - defined by the
   usual truth table - if one encodes 'True' as a closed connection
   and 'False' as an open connection. Of course, this can be
   considered a convention. If 'True' is encoded by an open connection
   and 'False' is encoded by a closed connection, then a series
   connection of the switches implements OR(x,y).

   Of course, the "dual" of this little exposition applies to a
   parallel connection.

   I have a continuing interest in these simple observations in
   connection with an idea that I attempted to present in this post on
   Math Overflow:

   http://mathoverflow.net/questions/1078/boolean-network-as-a-gauge-field

   For understandable reasons, it didn't generate much of a
   response. Perhaps a few people were motivated to chew on it for a
   while.

   - Chris 

Francesco La Tella writes:

   Greetings John,

   I am having fun reading about bond graphs (in an attempt to stay
   awake during the graveyard shift at work) on your site.

   With regard to the principle of least power (PLP). I remember
   writing an assignment for the subject of Optimization II, in my
   senior year of an undergraduate maths degree. Basically we were
   asked to use mathematical optimization techniques to model an
   appropriate physical, industrial, financial, etc. system in order
   to determine optimal operational parameters or values. Most of my
   classmates chose typical, classic, textbook problems from one of
   the many fine textbooks available to us. However, having had a
   vague recollection, at the time, to a reference in a 1989 (circa?)
   issue of Electronics & Wireless World which touched on this very
   subject, I got to digging.

   In brief, it turns out that, the distributions of voltages and
   currents in electrical networks (circuits), containing both active
   and passive circuit elements, can be solved for by using a
   stationary-power-condition dictated by the principle of least
   power. Using this idea an objective function is formulated in which
   each term describes the power dissipated in each of the circuit
   elements comprising the network. Since the objective is bivariate,
   one only needs to find the stationary point in this "power
   manifold" to determine the real, physical values of currents and
   voltages in and around all circuit elements.

   The situation is only slightly complicated by the presence of
   active and reactive circuit elements, but is covered sufficiently
   by a generalized version of this concept.

   Over the years I've had occasion to mention this alternative
   technique for circuit analysis to many of my electrical engineer
   colleagues and friends, only to be surprised that ALL were
   blissfully unaware of this very elegant yet very useful
   solution. It's unfortunate that engineers today are taught nodal
   analysis (Kirchhoff's current law & Kirchhoff's voltage law), a
   little linear algebra and perhaps some physics, certainly lots of
   experience using circuit-CAD packages, but no time exploring
   alternative possibilities. In contrast ALL my physicist friends are
   intuitively, if not explicitly attuned to the existence of the
   unifying power of the three principles PLA, PLE and PLP, and all their
   possibilities.

   Thank you for helping to keep my brain cells active.

   Kind regards,

   Francesco La Tella

For more discussion visit the n-Category Cafe:

http://golem.ph.utexas.edu/category/2010/01/this_weeks_finds_in_mathematic_53.html

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Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at

http://math.ucr.edu/home/baez/

For a table of contents of all the issues of This Week's Finds, try

http://math.ucr.edu/home/baez/twfcontents.html

A simple jumping-off point to the old issues is available at

http://math.ucr.edu/home/baez/twfshort.html

If you just want the latest issue, go to
 
http://math.ucr.edu/home/baez/this.week.html