(Alfred Einstead) wrote in message . com...
(Patrick Reany) wrote:
(Patrick Reany) wrote:
Einstein told us that he was once an etherist. He (apparently)
accepted the Mechanical Program's dogma to explain all phenomena in
terms of mechanics? But what is mechanics? Mechanics is the science
that MODELS all matter as being aggregates of point mass particles in
motion. Newton's mechanics adds to that description that masses
interact via forces acting-at-a-distance. (Hertz's mechanics --
inspired by his positivist leanings -- does not use the force concept
as a primitive.) The logical extension to this rigid approach would be
to approximate matter as a continuum state. The mathematical
distinction between these two approaches was profound, for in the
former case one uses total differential equations and in the latter
one uses partial differential equations. Einstein told us that this
led the way for the development of field theories of electrodynamics.

It's possible to establish continuum mechanics as a 'principle'
theory too. One starts with the assertion that for a perfectly
cohesive system the discrete (Newtonian) laws apply, particularly:
d(mv^2/2)/dt = F.v

This is power...

().() denoting the dot product of vectors, and that this relation
pertain even in the smallest elements, so that if a system is
described by a continuum with
m_V = integral_V rho dV;
m_V = mass contained in volume V
and with
Perfect Cohesion:
integral_V rho f(x,v) dV = m f(x,v)
then
d/dt integral_{V(t)} rho v^2/2 dV = integral_{V(t)} rho/m F.v dV
over any "comoving" volume V(t) moving with.

Is there anyway to explain this simpler, maybe using a
quantum of power.

This requires a prior conception of the mass continuum being
associated with co-moving volumes (i.e., volumes that flow along
the streamlines of the velocity field v(t)). From the transport
theorem, one then derives the transport equations under the
assumption of perfect cohesion:
@(rho v^2/2) + del.(v rho v^2/2) = rho b.v
@ denotes in ASCII the curly partial derivative symbol
where b = F/m is the commonly used symbol to denote the force
per unit mass.

right, but why is "b" preferred above "a" for
acceleration (??).

That, then, is the starting point. One has a prior discrete
theory (he Newtonian mechanics) and from this arrives at
the idealized continuum form, as above.

Now, continuum mechanics, itself, follows by imposing two
invariance conditions:

(0) Correspondence Limit
In the limit of perfect cohesiveness, the equations of
motion are those derived from the corresponding discrete
mechanics.

(1) Scale Relativity or Additivity
The equations of motion be invariant with respect to
changes in level. In particular, if a system is
composed of parts
sum_a rho_a = rho
sum_a (rho_a v_a) = rho v
sum_a (rho_a b_a) = rho b
then the equations of motion satisfied by the system
as a whole are the same in form as those satisfied by
each of the parts.

I'm presuming rho is density...

(2) Galilean Relativity
The equations of motion are invariant with respect to
changes in the frame of reference.

Additionally, one can distinguish fundamentally continuous
systems from atomic systems by the assertion:

(3) Atomic Hypothesis
These exists a decomposition of a system into cohesive
subsystems.

But this assertion is immaterial for what follows.

Since the continuum equations arrived at by (0):

Where's (0)?

@(rho v^2/2)/@t + v.del(rho v^2/2) = rho b.v
are non-linear, and since non-linear combinations generally
do not preserve their form under summation of subsystems, these
equations are not scale invariant.

If I'm not mistaken, you're describing
turbulence, and in aeronautics the Reynolds
number is uded to scale?

This, then, requires
off-setting components which, if properly selected, will
also be invariant under changes in observer motion.

This leads to the definitions of quantities Tij, Wijk, qij
to respectively offset combinations of the form rho vi vj,
rho vi vj vk, rho vi bj, where v = (v1,v2,v3), b = (b1,b2,b3).
Then, one writes scale-invariant forms for the summations:

sum (rho_a vi_a vj_a + Tij_a) = rho vi vj + Tij

sum (rho_a vi_a vj_a vk_a + Tij_a vk_a + Tik_a vj_a + Tjk_a vi_a + Wijk_a)
= rho vi vj vk + Tij vk + Tik vj + Tjk vi + Wijk
and
sum (rho_a vi_a bj_a + qij_a) = rho vi bj + qij.

The quantities if interest in the equation of motion are
rho v^2/2, rho v v^2/2, and rho v.b.
The corresponding scale-invariant form for rho v^2/2 is then:
rho d_ij v^i v^j/2 + d_ij Tij/2
= rho (d_ij v^i v^j/2 + e)
= rho (v^2/2 + e)
where
e = trace(T)/(2 rho).
The summation convention is used above, and d_ij stands for the
Kroenecker Delta (in ASCII form).

For rho v.b, one has
rho d_ij v^i b^j + d_ij q^ij = rho v.b + q
where
q = trace(q).

And for rho v v^2/2, one has
1/2 rho d_jk v^i v^j v^k
= (rho d_jk v^i v^j v^k + d_jk (Tij v^k + Tik v^j + Tjk v^i) + d_jk Wijk)/2
= rho v^i (v^2/2 + e) + (T.v)^i + w^i
where
w^i = 1/2 d_jk Wijk = 1/2 Tr_{23}(W)
and
(T.v)^i = d_jk Tij v^k.

The equations of motion then assume the scale-invariant form:

@(rho (v^2/2 + e))/@t + del.(v rho (v^2/2 + e) + T.v + w) = rho b.v + q.

Under a change in frame of reference to an observe travelling at
a speed u relative to the original frame, one has the transformation:
v - u + v; @/@t - @/@t - u.del; del - del; rho - rho.

If F - F' under the transformation, then the combination
@F/@t + del.(v F)
transforms to:
@F'/@t - u.del F' + del.((u + v) F') = @F'/@t + del.(v F').

The particular forms chosen for the off-setting quantities,
T, W, q ensure that these are each invariant under Galilean
transformations. Therefore, the original equations transform to:

@(rho (|u+v|^2/2 + e)/@t +
del.(v rho (|u+v|^2/2 + e) + T.(u+v) + w) = rho b.(u+v) + q.

Subtracting out the original equation of motion, one gets:

@(rho (u^2/2 + u.v))/@t + del.(v rho (u^2/2 + u.v) + T.u) = rho b.u.

This split into a part linear in u, and a part quadratic in u,
which each must separately be equated:

@(rho u.v)/@t + del.(v rho (u.v) + T.u) = rho b.u
and
@(rho u^2/2)/@t + del.(v rho u^2/2) = 0.

From the first equation, factoring out the u.(), one gets:
@(rho v)/@t + del.(rho v v + T) = rho b
where
(rho v v + T) is the tensor with components (rho v^i v^j + Tij).
From the second equation, factoring out the u^2/2, one gets:
@rho/@t + del.(rho v) = 0.

The quantity T is identitied as the stress tensor of the system,
and e as the system's internal energy. They are related by:
1/2 trace(T) = rho e.

In fact, this is true for monoatomic gases. For diatomic and
polyatomic gases, there is an additional component to e that
does not arise from the 1/2 the trace of T.

Agreed, would you say these "additional components"
are antisymmetric and or nonorthogonal components
off the trace, where the trace T solves only mono's?

Ken S. Tucker