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Jun 2018
9:54am, 19 Jun 2018
103,642 posts
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GregP
Thus endeth round 2.
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Jun 2018
10:40am, 19 Jun 2018
103,644 posts
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GregP
Here's a thing. The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist)
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Jun 2018
10:44am, 19 Jun 2018
17,017 posts
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TRO Toddslayer
I quite liked the Darwin Lagrangian - here's a simple explanation. I'll test you all on it after lunch...;-)
When describing nature, we regard the fundamental interactions as those between point
particles. Thus here we turn to the question as to what potential functions V ( r1 − r2 )
between two point particles can be regarded as describing the nonrelativistic limit arising
from a fully Lorentz-invariant interaction between particles. We first try to solve this
problem by working backwards, trying to construct a relativistic theory which produces a
specific potential in the nonrelativistic limit.
The nonrelativistic mechanical behavior of two point particles interacting through a po-
tential V ( r1 − r2 ) can be written in terms of a Lagrangian
L(r1, r2, ˙r1, ˙r2) =
1
2
m1 ˙r2
1 +
1
2
m2 ˙r2
2 − V ( r1 − r2 ) (5)
The invariance of this Lagrangian under spacetime translations and spatial rotations leads to
the conservation laws for energy, linear momentum, and angular momentum. The system
is also invariant under Galilean transformations where the generator of proper Galilean
transformations is given by the system total mass times the system center of mass.[7] In
order to extend this system to a Lorentz-invariant system, we must preserve the invariance of
the Lagrangian under spacetime translations and spatial rotations while changing the system
invariance under Galilean transformations over to invariance under Lorentz transformations.
The generator of Lorentz transformations is the system total energy times the system center
of energy.[7] The first step in this transformation is the replacement of the nonrelativistic
expression for particle kinetic energy by the Lagrangian for a relativistic free particle
1
2
m˙r2 → −mc2(1 − ˙r2/c2)1/2 (6)
just as was done in moving from Eq. (1) to Eq. (3) above. With this replacement, the
nonrelativistic Lagrangian of Eq. (5) becomes now
L(r1, r2, ˙r1, ˙r2) = −m1c2(1 − ˙r2
1/c2)1/2 − m2c2(1 − ˙r2
2/c2)1/2 − V ( r1 − r2 ) (7)
This Lagrangian, which is of the sort given in the mechanics textbooks,[8] will lead to
relativistic expressions for particle kinetic energy and particle linear momentum. It is
invariant under spacetime translations and spatial rotations. However, this system is not
Lorentz invariant.
4
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Jun 2018
10:50am, 19 Jun 2018
16,274 posts
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Nicholls595
Numberwang!
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Jun 2018
10:56am, 19 Jun 2018
103,651 posts
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GregP
R3G6 is a tough matchup this early on.
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Jun 2018
11:00am, 19 Jun 2018
210 posts
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iRicey
I guess it might have already been said but surely we're heading for the big man himself to be beaten by the awards which prove his point....
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Jun 2018
11:03am, 19 Jun 2018
103,656 posts
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GregP
I don't think it *has* been said before, but I fear you're right. Still hoping the asteroid can sneak into the SuperDeath by way of the semis though.
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Jun 2018
1:55pm, 19 Jun 2018
37,594 posts
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.B.
I got to line 3 TRO
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Jun 2018
2:51pm, 19 Jun 2018
19,883 posts
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Maclennane
Gutted the bulldog was beaten by some Lycra clad fool
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Jun 2018
3:50pm, 19 Jun 2018
103,678 posts
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GregP
Not looking good for Jhon Darwin Atapuma Hurtado (born 15 January 1988), a Colombian cyclist who rides for the UAE Team Emirates professional cycling team to go much further though.
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SuperDeath XI
Dec 2024