[OPE-L:2446] Re: Dynamical systems

Paul Cockshott (wpc@cs.strath.ac.uk)
Mon, 3 Jun 1996 02:45:54 -0700

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In a posting with which I generally agreed, Alan Freeman raised
two points that I would like to question.

>This leads to a problematic in which the most important thing
>to establish are the transition rules governing the passage
>from one set of values, prices and quantities at one point in
>time, to another set of values, prices and quantities at
>another point in time. In such a problematic, as with any
>normal system of lagged or differential equations, there is a
>temporal sequence defined by an axiom which asserts (this is
>best specified in the language of mathematical State Theory)
>that the state at time T depends, and only depends, on states
>at times t<T. A simultaneous formulation on the contrary
>asserts either that the state at time T is self-determined
>without reference to previous times, or worse still that it
>depends on states at times t>T.

Paul C:

I think Alan overstates the requirements for a consistent treatment
of a dynamical system. He raises, later, the example of a system
evolving under the constraint of a Hamiltonian. One can use this
to illustrate that the simultaneous definition of certain properties
is quite consistent with a dynamical system.

Consider a non-relativistic classical particle system whose state
is characterised by

m(i) a set of particle masses
p(i) a set of particle momentum vectors
x(i) a set of particle position vectors

Given m(i), p(i) for each particle we can determine dx(i)/dt for
each particle, and, given m(i) and x(i) we can determine
dp(i)/dt for each particle.

Given these derivatives, one can integrate them over time to
obtain the future values of p and x.

Thus the values of the variables at time t(0) are determined by
their values at t(-1), t(-2) etc, much as Alan argues should be
the case.

However, such a dynamical system also has the, initially unintuitive,
effect that the state of the system at time t(0) is determined by
the state of the system at time t(1), t(2) etc. This is the famous
'time reversal paradox' of physical systems. So when Alan says:

>A simultaneous formulation on the contrary
>asserts either that the state at time T is self-determined
>without reference to previous times, or worse still that it
>depends on states at times t>T.

I think that he is failing to recognise a necessary consequence
of constructing a deterministic dynamical theory. If it is deterministic,
it necessarily works both ways - forwards and backwards in time.

Moreover, such dynamical models do not preclude one from defining
certain derived functions in a snapshot fashion. In classical dynamics
one derives the forces from a snapshot of the current positions and
momenta, similarly one can construct dynamical models using what
Mirowski calls a field interpretation of the labour theory of value
in which the value field is an instantaneous function of the current

> (b) we can define what in dynamical theory is known as
> 'constants of motion' of the system for each such broad
> category. Thus systems that have a potential function will
> exhibit the dynamical conservation of value (analogous to the
> Hamiltonian), which I consider a central and indispensible
> requirement of a valid economic theory.

Paul C:
I think it is invalid to make an analogy between the Hamiltonian,
which is concerned with energy conservation, and the principle of
conservation of value in exchange. The Hamiltonian defines a
time invariant of the system. In value theory, value is not
a time invariant since total value in the system can change
due to capital accumulation, and also due to technological

In his axioms below, Alan is more cautious:

>(1)true temporality (see above)
>(2)a distinct category of value
>(3)linear values and prices
>(4)objective values; the value and price vectors are the same
> for everyone
>(5)value cannot be created or destroyed in exchange
>(6)new value added in any period is directly proportional to
> total hours worked
>(7)physical stock conservation: total use value of each type
> = previous total, less consumption, plus production.
Paul Cockshott