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Mathematical model of inertia

  1. Feb 17, 2006 #1
    [SOLVED] Mathematical model of inertia

    Can someone provide a mathematical model of inertia?
     
  2. jcsd
  3. Feb 17, 2006 #2
    I= sum(mi*ri^2) ???? what kind of math model are you looking for?
     
  4. Feb 18, 2006 #3
    Look up Higgs field.
     
  5. Feb 19, 2006 #4
    F=ma => a=F/m

    Anything further and we're into philosophy.
     
  6. Feb 20, 2006 #5
    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...)
     
  7. Feb 21, 2006 #6
    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.
     
  8. Feb 24, 2006 #7
    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.
     
  9. Mar 1, 2006 #8
    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".
     
  10. Mar 5, 2006 #9
    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 ======================================================
     
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