# Mathematical model of inertia

1. Nov 4, 2006

### Mike Helland

Can someone provide a mathematical model of inertia?

2. Nov 4, 2006

### Igor

Look up Higgs field.

3. Nov 4, 2006

### jamesahart79@gmail.com

F=ma => a=F/m

Anything further and we're into philosophy.

4. Nov 4, 2006

### Chris Rohr

Nice one jamesahar...
Nobody could have put it so simply.
Newton's second law is *the* mathematical model of inertia...
If you're in a good mood you can even find Kepler's Law with only
SumF=ma and v=a^2/r (centripetal acceleration...)

5. Nov 4, 2006

### Mike Helland

jamesahart79@gmail.com wrote:
> F=ma => a=F/m
>
> Anything further and we're into philosophy.

Ok, can you show the work that uses "F=ma => a=F/m" to answer the
following question:

There is a ball with a mass of 20kg moving toward a 5kg ball at 1 m/s.
At 0 seconds, the balls are 2 meters apart.

What is the state of the system at 3 seconds.

Show all the math involved to make that prediction.

6. Nov 4, 2006

### markwh04@yahoo.com

Mike Helland wrote:
> Can someone provide a mathematical model of inertia?

Quite simply, the geodesic equation
x^m'' + Gamma^m_{nr} x^n' x^r' = 0.
This has, underlying it, a representation of a spacetime as a manifold
equipped with a connection. And those are the essential ingredients
needed to mathematically encapsulate the concept of inertia.

Though not well-known, this is not specific to General Relativity but
to representations of spacetime in general, including Galilean
spacetime and -- possibly -- even Aristotlean spacetime, though I don't
know of anyone who's ever written out the ancient theory in the
language of modern geometry.

For Lorentzian spacetimes, the connection is that having the property
of being torsion-free and metric-preserving -- the Levi-Civita
connections. In the presence of torsion, the connection defines
autoparallels, which differ from geodesics. The definition of geodesic,
however, can be recovered intrinsically. The resulting equation is
x^m'' + Gamma^m_{nr} x^n' x^r' = g^{ms}
tau^r_{sn} x^n' p_r
where p_r = g_{rn} x^n' is the "momentum" and tau^r_{sn} the torsion.
Torsion comes into the geodesic equation as an effective Lorentz force
coupled to momentum.

For Galilean spacetimes, the analogue of "Levi-Civita" connections are
the torsion-free connections that preserve the (singular) contravariant
metric and the time-like flow field. In coordinates adapted to the
latter, with x^0 parallel to the time-flow, the connection coefficients
Gamma^0_{mn} will all be 0; the coeffiicents Gamma^i_{00} will give you
minus the gravitational force and Gamma^i_{0j} will involve a
"gravito-magnetic" contribution B^i_j.

(Yes, that's right. Graveto-magnetism is NOT purely relativistic, but
is present in the Newtonian theory, as well).

An Aristotlean spacetime would be more interesting. There, you're not
just given a time-like field but also a covariant vector field
representing the "absolute velocity". That actually allows you to write
down a general covariant metric, with the velocity field playing the
role the shift field. In place of Levi-Civita, you'll probably want
invariance with respect to the contravariant metric, the time-like
field and the velocity field.

7. Nov 4, 2006

### a student

markwh04@yahoo.com wrote:
> Mike Helland wrote:
> > Can someone provide a mathematical model of inertia?

>
> Quite simply, the geodesic equation
> x^m'' + Gamma^m_{nr} x^n' x^r' = 0.
> This has, underlying it, a representation of a spacetime as a manifold
> equipped with a connection. And those are the essential ingredients
> needed to mathematically encapsulate the concept of inertia.

This doesn't seem enough by itself - (i) doesn't any (relatively low
energy) body, in the absence of external forces, satisfies this
equation *independently* of its "inertia" ? (equivalence principle);
and (ii) many phenomenological accelerations are not a quadratic
function of the velocity. Einsteins field equation for the metric
would seem to do better here, as it allows one to model/attribute
inertial effects as arising from the local curvature of spacetime.

> For Lorentzian spacetimes, the connection is that having the property
> of being torsion-free and metric-preserving -- the Levi-Civita
> connections. In the presence of torsion, the connection defines
> autoparallels, which differ from geodesics. The definition of geodesic,
> however, can be recovered intrinsically. The resulting equation is
> x^m'' + Gamma^m_{nr} x^n' x^r' = g^{ms}
> tau^r_{sn} x^n' p_r
> where p_r = g_{rn} x^n' is the "momentum" and tau^r_{sn} the torsion.
> Torsion comes into the geodesic equation as an effective Lorentz force
> coupled to momentum.

For a non-curved space (so that Gamma vanishes), wouldn't this imply
that acceleration is always proportional to momentum? This conflicts
with most phenomenological forces.

I would have thought that a "model of inertia" should at least model
the fact that in many physical situations the dynamics of different
bodies only vary according to an "inertial" parameter, m, associated
with the body [eg, the dynamics described by the Hamiltonian p^2/(2m) +
V(x)].
But I admit to quite possibly not having understood in what sense you
suggest the geodesic equation as a "model".

8. Nov 4, 2006

### Mike Helland

Mike Helland wrote:
> Can someone provide a mathematical model of inertia?

Here's a computer algorithm that attempts to model three objects in
Newtonian inertia and gravity.

The first has a mass of 20kg, is located at 0,0,0, and is moving at
sqrt(2) m/s: 1 along the x-axis and 1 along the y-axis.

The second has a mass of 5kg, is located at 24,24,0, and is at rest.

The last has a mass of 20kg, is located at 76,20,0, and is moving at
sqrt(2) m/s: -1 along the x-axis and 1 along the y-axis.

Is it a mathematical model?

* Set the initial conditions =======================================

* The parameters are X, Y, Z, dX, dY, dZ, and mass
local aObjects[3]
aObjects[1] = create("Matter", 0, 0, 0, 1, 1, 0, 20)
aObjects[2] = create("Matter", 24, 24, 0, 0, 0, 0, 5)
aObjects[3] = create("Matter", 76, 20, 0, -1, 1, 0, 20)
t = 0

* The laws of motion ==============================================

do while .t.

* Look at every object
for each oP in aObjects

* In motion stays in motion
* At rest stays at rest
oP.X = oP.X + oP.dX
oP.Y = oP.Y + oP.dY
oP.Z = oP.Z + oP.dZ

* Look at every other object
for each oP2 in aObjects
if oP <> oP2

* Find its distance
nX = oP.X - oP2.X
nY = oP.Y - oP2.Y
nZ = oP.Z - oP2.Z
nD = SQRT( nX^2 + nY^2 + nZ^2 )

* See if we rammed into it
if nD < 1

* If so, swap momenta
nMX = oP.Mass * oP.dX
nMY = oP.Mass * oP.dY
nMZ = oP.Mass * oP.dZ

oP.dX = (oP2.Mass * oP2.dX) / oP.Mass
oP.dY = (oP2.Mass * oP2.dY) / oP.Mass
oP.dZ = (oP2.Mass * oP2.dZ) / oP.Mass

oP2.dX = nMX / oP2.Mass
oP2.dY = nMY / oP2.Mass
oP2.dZ = nMZ / oP2.Mass

else

* Find the force of gravity, then accelerate
nG = 6.67300 * 10^-11 * ((oP.Mass * oP2.Mass) / nD^2)
nG = nG / oP2.Mass

nX = nX * (nG / nD)
nY = nY * (nG / nD)
nZ = nZ * (nG / nD)

oP2.dX = oP2.dX + nX
oP2.dY = oP2.dY + nY
oP2.dZ = oP2.dZ + nZ

endif
endif
endfor
endfor

t = t + 1
if t = 40
exit
endif
enddo

* Print the final state of the model ===============================

for ni = 1 to alen(aObjects)
?"Object " + tran(ni)
?"X = " + tran(aObjects[ni].X) + space(5) + ;
"Y = " + tran(aObjects[ni].Y) + space(5) + ;
"Z = " + tran(aObjects[ni].Z)
?"dX = " + tran(aObjects[ni].dX) + space(5) + ;
"dY = " + tran(aObjects[ni].dY) + space(5) + ;
"dZ = " + tran(aObjects[ni].dZ)
endfor

* Object Structure for Matter ======================================

define class Matter as Custom
X = 0
Y = 0
Z = 0
dX = 0
dY = 0
dZ = 0
Mass = 0

function init
lparameters x, y, z, dx, dy, dz, mass
with this
.X = x
.Y = y
.Z = z
.dX = dx
.dY = dy
.dZ = dz
.Mass = mass
endwith
return

enddefine

* End of File ======================================================