Re: Did Einstein Repeal Newton's Second Law?

From: Dirk Van de moortel (dirkvandemoortel_at_ThankS-NO-SperM.hotmail.com)
Date: 09/27/04 

Date: Mon, 27 Sep 2004 16:57:58 GMT

"Donald Macnaughton" <donmac@matstat.com> wrote in message news:iTW5d.1930$tT2.384584@news20.bellglobal.com...
> I'm a statistician, not a physicist. In an essay I'm writing
> about scientific reasoning I would like to say whether Newton's
> second law of motion (F = ma) was "repealed" by one of Einstein's
> two theories of relativity. However, although I've done some
> digging, I haven't found a good discussion of this topic. So I
> have the following questions:
>
> 1. Does one of Einstein's theories repeal (or modify) Newton's
> second law? If so, how? If not, do Einstein's theories take
> Newton's second law as a given?

A partial answer:

Actually Newton's second law does not give force as mass
times acceleration,
        F = m a
but as the time derivative of momentum,
        F = dp/dt

Newton defined momentum p as mass times velocity
        p = m v
and velocity v as
        v = ds/dt
hence, when mass is invariant or constant, the force is given by
        F = m dv/dt
            = m a

Einstein treated mass M as dependent on the so-called rest-mass m
and the velocity v:
        M = gamma m
where
        gamma = 1 / sqrt( 1 - v^2/c^2 )
so he kept Newton's second law
        F = dp/dt
and still defined defined momentum p as mass times velocity,
        p = M v
           = gamma m v.

So we could say that Einstein did not change Newton's second law,
but he used another defintion of mass.

The force is now given by
       F = d(M v)/dt
           = m d(gamma v)/dt
           = m d(gamma v)/dt
           = m [ gamma dv/dt + v d(gamma)/dt ]
           = m gamma [ 1 + gamma^2 v^2 ] a
           = m gamma^3 a
This is only valid when the acceleration has the same direction
as the velocity. [For other direction, see article below].

Nowadays this "relativistic mass" M is a bit out of fashion, and
momentum is *directly* defined as
        p = gamma m v.

You will find a very good read in this FAQ article:
   http://hermes.physics.adelaide.edu.au/~dkoks/Faq/Relativity/SR/mass.html

Hth,
Dirk Vdm

>
> 2. If Newton's second law is repealed, does the repeal depend on
> the assumption of the equivalence of inertial and gravita-
> tional mass? If we no longer make this assumption, do things
> change?
>
> 3. Is it possible to build viable models in which Newton's second
> law always holds and viable models in which the law sometimes
> doesn't hold? (The type of model might be controlled by con-
> trolling the order in which physical concepts are introduced
> in a formal development.) If so, which type is preferred?
>
> Citations of books or articles about these ideas are also wel-
> come. Thank you.

Relevant Pages

  • Re: Did Einstein Repeal Newtons Second Law?
    ... >> Newton's second law as a given? ... > Actually Newton's second law does not give force as mass ... > Newton defined momentum p as mass times velocity ... > and still defined defined momentum p as mass times velocity, ...
    (sci.physics.relativity)
  • Re: Did Einstein Repeal Newtons Second Law?
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    (sci.physics)
  • Re: Did Einstein Repeal Newtons Second Law?
    ... >> Newton's second law as a given? ... > Actually Newton's second law does not give force as mass ... > and still defined defined momentum p as mass times velocity, ... Theory of Fields page 24 the most geranial lagrangian consistent with SR is ...
    (sci.physics)
  • Re: Did Einstein Repeal Newtons Second Law?
    ... >> Newton's second law as a given? ... > Actually Newton's second law does not give force as mass ... > and still defined defined momentum p as mass times velocity, ... Theory of Fields page 24 the most geranial lagrangian consistent with SR is ...
    (sci.physics.relativity)
  • Re: Did Einstein Repeal Newtons Second Law?
    ... >> Newton's second law as a given? ... > Actually Newton's second law does not give force as mass ... > Newton defined momentum p as mass times velocity ... > and still defined defined momentum p as mass times velocity, ...
    (sci.physics)