Re: On ArXiv: Natural selection maximizes Fisher information

Perplexed in Peoria <jimmenegay@xxxxxxxxxxxxx> wrote:

Well, I've looked at Frieden's book now. Really cool, but the math
is just at the borderline of my abilities. I wish I understood the
Lagrangian formulation of dynamics better. I recently watched some
of Susskind's video lectures on classical mechanics as an ipod video -
those helped a lot. But I still don't understand it well enough to explain
it with the confidence that I am not making mistakes. But here goes
anyways.

The usual (undergrad) formulation of dynamics uses force laws. For
example, the trajectory of a weight-on-a-spring harmonic oscillator is
explained as the solution to a (simple in this case) system of differential
equations - one of which would be Hooke's law of elasticity for the spring.
A more complicated example might trace the dynamics of fields and
charged particles with the key diffy-Q force laws being Maxwell's.

However it has been known for centuries that there is an alternate
formulation based upon the principle of 'least action'. In this approach,
you set up a space called a phase-space in which each point represents
some possible state of your dynamical system. Any curve connecting
two points in phase space represents a conceivable trajectory which
a system might take in changing from one state to another. But only
one of these curves is 'physical', i.e. only one is the trajectory that
would actually be taken in accordance with the dynamical force laws
of physics.

Q: So, what is special about the physical trajectory among
all of the conceivable trajectories? A: The physical trajectory is the
one which minimizes the 'action'. Q: So, what is an 'action'?
A: It is a property of a trajectory - specifically, it is the line
integral (along the trajectory) of a particular scalar 'field' in phase
space called the 'Lagrangian'. The Lagrangian captures all of the
physics involved in a physical situation being modeled.

An example may help. In the weight-on-a-spring system, the phase
space is two dimensional, with the coordinates being the (vertical)
position of the weight as one axis and the vertical (momentum) of
the weight as the other axis. A point in this space represents a
state of the system. At any such point we can compute some
properties of that particular physical state, for example the kinetic
energy or the potential energy (gravitational + elastic).

Now comes the part that still seems almost magical to me. Define
the Lagrangian of a point in this phase space as the difference
(KE - PE) between the kinetic energy and the potential energy
in that state of the system. Think of this Lagrangian as a kind of
temperature. Take any trajectory connecting an initial state and
final state in this phase space. As a point (representing the
system state at a point in time) travels along this trajectory, it
will 'soak up' this Lagrangian 'temperature' at a rate which depends
upon both the value of the Lagrangian at the points of the trajectory
and on how much time the point spends in that region of the trajectory.
The total amount of Lagrangian stuff soaked up will be given by
a line integral. It is called the 'action' because it has the physical
units of action (energy times time, same as Planck's quantum of
action). And the 'action' along the physical trajectory is lower
than the action would be along some other (conceivable, but
physically impossible) trajectory. Principle of Least Action. Magic.

There is a lot of math at this stage - you need to use calculus of
variations to actually find out which trajectory has the least action,
and there is quite a bit more complicated math used to switch back
and forth between the force law formulations of physics and the
Lagrangian formulations. The Lagrangian formulation is preferred
by theoretical physicists because it changes form the least when you
switch from classical to quantum mechanics.

But here is the weird thing. No one can give an explanation from
first principles why the Lagrangian of a system takes the form it
does. For example, in our weight-on-a-spring example, the
Lagrangian is KE - PE. But why? What is the significance of
the quantity KE - PE? (KE + PE might make some sense - it
is the total energy and is conserved). This is where Frieden
comes in. Frieden seeks to *explain* physical Lagrangians as
informational entities. It isn't a completely mysterious 'action'
which Nature is minimizing, it is instead a kind of uncertainty.
After all, Heisenberg showed that the physical unit of uncertainty
has the physical units of action and now Frieden explains
Lagrangian action as the difference between two particular
kinds of Fisher information - which are determined by the physical
situation. This may not be the best of all possible worlds, but
it is, in some sense, the least confusing.

The Lagrangian *is* pulled out of the air.

Newton's Laws are really postulates. They can not be derived
(except that one of the normal laws can be derived from the others.)
His "Law" of gravity is exactly the same.

They were and are used because they work.

There is an alternate way of inventing mechanics. That is to
assume that the phase space path that a system actually follows
is the path involving the least action. This approach is actually
slightly more general because you can derive Newton's Laws from
it. And you can get the Lagrangian.

What the Lagrangian *is* is a formualation of Newton's Laws in
a way that is coordinate independent.

It works with "Lagrange's Equation". If you use cartesian
coordinates Lagrange's Equation reduces to F = ma.

If you use other coordinate systems you have to do a transformation
of variables to take F = ma into that new coordinate system. Using
Lagrange's equation you get that automatically.

To see this all laid out in many many words (and a few equations)
go to

http://scholar.chem.nyu.edu/2600/notes.html

and open the pdf. Then go to pdf page 176 (which will be page
160 of the actual text. You can rapidly skim over the Newtonian
mechanics through to section 13.5 (pdf page 182). What follows
is an alternate approach to the Lagrangian. Note that the
Lagrangian is pulled out of the air and no mention is made of
the principle of least action. Physicists think this is cheating,
but it isn't. One would have to pull the principle of least action
out of the air doing it their way, and that is just as painful.

--
--- Paul J. Gans

.

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