www.ifi.unicamp.br/~assis
Relational Mechanics
A. K. T. Assis
University of Campinas, Brazil
1
Isaac Newton (1642 – 1727)
1687: Principia
2
2
21
r
mmGF
maF
=
=
Free fall in Newtonian mechanics
maF =
mamg =
22 8.9s
m
R
GMga ===
3
Free fall in Newtonian mechanics
maF =
mamg =
22 8.9s
m
R
GMga ===
4
However, this is not a two-body problem. There
are also stars and galaxies around the Earth.
2
21
r
mmGF =
5
Newton in the Principia
“Theorem 30: If to every point of a
spherical surface there tend equal
centripetal forces decreasing as
the square of the distances from
these points, I say, that a
corpuscle placed within that
surface will not be attracted by
these forces any way.”
0=F
2
21
r
mmGF =
6
“Theorem 31: The same things supposed as above,
I say, that a corpuscle placed without the spherical
surface is attracted towards the centre of the sphere
with a force inversely proportional to the square of
its distance from that centre.”
2r
mMGF
gg=
7
Free fall in Newtonian
mechanics:
amFF iE =+ *
amR
MmG i
E
gEg=+ 0
2 2
E
gE
i
g
R
GM
m
ma =
8
Free fall in Newtonian
mechanics:
amFF iE =+ *
amR
MmG i
E
gEg=+ 0
2
===
==
22
8.9
1
s
mg
R
GMa
m
m
m
m
E
gE
Leadi
Leadg
Corki
Corkg
Galileo: cork and
lead fall together.
Therefore:
2
E
gE
i
g
R
GM
m
ma =
9
All bodies fall freely with
the same acceleration
in a gravitational field.
An alpha particle moves with
half of the acceleration of a
proton in an electric field.
Therefore, inertia seems to be connected with
gravitation and not with electromagnetism. 10
amF i=Newton’s 2nd law of
motion:
11
The free fall acceleration of 9.8 m/s2 is the
acceleration of the apple relative to what?
amF i=Newton’s 2nd law of
motion:
Newton in the Principia:
“Absolute space, without relation to
anything external, remains always
similar and immovable.”
12
The free fall acceleration of 9.8 m/s2 is the
acceleration of the apple relative to what?
Newton’s bucket experiment
22
2r
gz
=
13
Newton’s bucket experiment
22
2r
gz
=
14
According to Newton, the concavity depends on the
angular rotation of the water relative to absolute
space, that is, relative to empty free space.
• “The principles of mechanics can be so
conceived, that even for relative rotations
centrifugal forces arise.”
• “Try to fix Newton’s bucket and rotate the
heaven of fixed stars, and then prove the
absence of centrifugal forces.”
Ernst Mach in The Science of Mechanics, 1883:
15
16
What would be the shape of
the water if it were possible to
let the bucket at rest on a table
while the distant stars and
galaxies rotated together once
a second around the axis of
the bucket?
17
What would be the shape of
the water if it were possible to
let the bucket at rest on a table
while the distant stars and
galaxies rotated together once
a second around the axis of
the bucket?
“What is to be expected
along the line of Mach’s
thought? A rotating
hollow body must
generate inside of itself
a Coriolis field, and a
radial centrifugal field
as well.”
Einstein, The Meaning of Relativity, 1922:
18
“What is to be expected
along the line of Mach’s
thought? A rotating
hollow body must
generate inside of itself
a Coriolis field, and a
radial centrifugal field
as well.”
Einstein, The Meaning of Relativity, 1922:
19
However, according to
general relativity
(Lense-Thirring effect):
Relational Mechanics
and Implementation of
Mach’s Principle with
Weber’s Gravitational
Force
A. K. T. Assis
(Apeiron, Montreal)
20
Available in PDF at:
www.ifi.unicamp.br/~assis
Postulates of Relational Mechanics:
+−−=
22
2
22 1
631
ˆ
c
rr
c
r
r
rmmHF ggg
The sum of all forces acting on any body
is always zero in all frames of reference.
21
Weber (1804-1891) in 1846:
Coulomb (1785):2
0
21ˆ
4 r
rqqF
=
Ampère (1822): ),,(ˆ
4 221
0
f
r
rIIF =
Faraday (1831):
dt
dIMemf −=
22
Weber (1804-1891) in 1846:
Coulomb (1785):2
0
21ˆ
4 r
rqqF
=
Ampère (1822): ),,(ˆ
4 221
0
f
r
rIIF =
Faraday (1831):
dt
dIMemf −=
Weber’s hypothesis: vqId
1222112
0
21 1ˆ
4akvvk
r
rqqF ++=
23
Weber’s Force:
+−=
22
2
2
0
21
21
ˆ
4 c
rr
c
r
r
rqqF
dt
drr =
2
2
dt
rdr =
s
mc 8
00
1031
==
24
Properties of Weber’s Electrodynamics
• In the static situation (dr/dt = 0 and d2r/dt2 = 0) we recover
the laws of Coulomb and Gauss.
• Action and reaction, conservation of linear momentum.
• Central force, conservation of angular momentum.
• It can be deduced from a velocity dependent potential
energy:
−=
2
2
0
21
21
1
4 c
r
r
qqU
• Conservation of energy: ( )
0=+
dt
UKd
25
• Ampère’s circuital law can be derivedfrom Weber’s force.
• Faraday’s law of induction can be derivedfrom Weber’s electrodynamics (seeA Treatise on Electricity and Magnetism,by J. C. Maxwell).
• It is completely relational. That is, itdepends only on r, dr/dt and d2r/dt2.Therefore, it has the same value for allobservers and in all frames of reference.It depends only on intrinsic magnitudes ofthe system, that is, on the relationsbetween the interacting bodies. 26
Weber’s Electrodynamics, A. K. T. Assis
(Kluwer, 1994, available at Springer) 27
Force exerted by a stationary spherical shell of mass M acting
on a test body m which is moving with acceleration a inside it:
+−−=
22
2
22 1
631
ˆ
c
rr
c
r
r
rmmHF ggg
28
Main calculation of Relational Mechanics:
0 Newton
=F
Force exerted by a stationary spherical shell of mass M acting
on a test body m which is moving with acceleration a inside it:
amF g
Mech. Relat. −=
2
2with
Rc
MH gg=
+−−=
22
2
22 1
631
ˆ
c
rr
c
r
r
rmmHF ggg
29
Main calculation of relational mechanics:
0 Newton
=F
Free fall in Relational Mechanics:
0* =+ FFE
0 2
=− amr
MmH g
gEg
g
2r
MHa
gEg
=
Gkg
NmHHog
==
−
2
211
*
2
107.6 4
with30
31
The previous slide presents the essence of Relational Mechanics.
We begin with the postulate that the sum of all forces acting on any
body is always zero. Then we deduce an expression analogous to
Newton’s second law of motion, namely:
0*
=− amF g
In this equation F represents the usual forces acting on the test
body. The expression –ma represents the gravitational force
exerted by the set of distant galaxies acting on the test body,
according to Weber’s law applied to gravitation. The mass
appearing here is the gravitational mass of the test body (that is, it
is not the inertial mass, as in Newtonian mechanics). The
acceleration appearing here is the acceleration of the test body
relative to the frame of distant galaxies (that is, it is not the
acceleration of the test body relative to absolute space, as in
Newtonian mechanics).
Weber’s gravitational
force exerted by a
spinning shell acting on a
particle moving inside it:
( )
++−= vramRc
MHF
g
gg2
22
32
Weber’s force has a real centrifugal
component and a real Coriolis component.
The bucket experiment according to
Newton and Relational Mechanics:
2
2
2r
gzNewton
=
33
The bucket experiment according to
Newton and Relational Mechanics:
2
2
2r
gzNewton
=
( ) 2
2
Mech. Relat.2
rg
z universewater −=
34
According to Relational Mechanics, the concavity
depends on the relative angular rotation between the
water and the set of distant galaxies.
Experimental test 1: What will be the acceleration of free fall when the test
body is surrounded by a stationary spherical shell of mass M?
35
Experimental test 1: What will be the acceleration of free fall when the test
body is surrounded by a stationary spherical shell of mass M?
24
2
3
2Mech. Relat.
2
102
then 10 and 1 If
:magnitude oforder
21
/ 8.9
−===
−=
===
Rc
GMkgMmR
Rc
GMga
smgaa EinsteinNewton
36
Experimental test 2: Consider a bucket at rest relative to the ground. We place a
spherical shell of mass M around the bucket. What will be the shape of the
water if only the shell rotates unformly around the axis of the bucket?
37
Experimental test 2: Consider a bucket at rest relative to the ground. We place a
spherical shell of mass M around the bucket. What will be the shape of the
water if only the shell rotates unformly around the axis of the bucket?
38
2
22
Mech. Relat.
2 with
2 Rc
GMkr
gkz ==
mzk
cmrsradmRkgM
2724
3
10 and 10then
10 ,/ 1 , 1 , 10 If
:magnitude ofOrder
−− ==
====
Conclusion
Postulates of Relational Mechanics:
+−−=
22
2
22 1
631
ˆ
c
rr
c
r
r
rmmHF ggg
= 0F
39
Main Results:
• Deduction of Newton’s 2nd law: F - m a = 0
• Deduction of the equivalence principle: mi = mg
• We show that the centrifugal and Coriolis forces
are real forces of gravitational origin exerted by
the set of distant galaxies.
• Quantitative implementation of Mach’s principle.
www.ifi.unicamp.br/~assis
40
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