Planetary Motions and Lorentz Transformationsmath.usask.ca/~cheviakov/bluman2014/talks/Meng.pdf ·...

Planetary Motions and Lorentz TransformationsKepler Problem and Future Light Cone

Guowu Meng

Department of MathematicsHong Kong Univ. of Sci. & Tech.

International ConferenceSymmetry Methods, Applications, and Related Fieldscelebrating the work of Professor George W. Bluman

University of British Columbia, Vancouver, Canada, May 13 - 16, 2014

May 15, 2014

Think deeply about simple things!

—Arnold Ross

Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, 2014 2 / 14

Kepler Problem

The Kepler problem is a dynamical problem with configuration spaceR3∗ := R3 \ {0} and equation of motion

r′′ = − rr3 . (1)

In this talk I shall convince you that the Kepler problem is intimatelyrelated to the future light cone and Lorentz transformations. In view ofthe fact that the Kepler problem is non-relativistic, this seems to be oddor surprising.

Kepler Problem

The Kepler problem is a dynamical problem with configuration spaceR3∗ := R3 \ {0} and equation of motion

r′′ = − rr3 . (1)

In this talk I shall convince you that the Kepler problem is intimatelyrelated to the future light cone and Lorentz transformations. In view ofthe fact that the Kepler problem is non-relativistic, this seems to be oddor surprising.

Future Light ConeHere is a picture in the Lorentz space R1,2:

The future light cone in this talk is the one in the Minkowski space R1,3.Although it cannot be visualized, it can be described as the solution setof

x20 − x2

1 − x22 − x2

3 = 0, x0 > 0 (2)

and is diffeomorphic to R3∗ under the projection:

(x0, r) ∈ R1,3 7→ r ∈ R3.Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, 2014 4 / 14

Future Light ConeHere is a picture in the Lorentz space R1,2:

The future light cone in this talk is the one in the Minkowski space R1,3.Although it cannot be visualized, it can be described as the solution setof

x20 − x2

1 − x22 − x2

3 = 0, x0 > 0 (2)

and is diffeomorphic to R3∗ under the projection:

(x0, r) ∈ R1,3 7→ r ∈ R3.Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, 2014 4 / 14

Basic Facts on the Kepler Problem• The angular momentum L := r ∧ r′ and the Lenz vector A := r′yL + r

rare constants of motion.• L ∧ A = 0 and

L ∧ r = 0, r − A · r = |L|2. (3)

So a non-colliding orbit is a conic with eccentricity e = |A| .

• The total energy E := 12 |r′|2 − 1

r can be expressed in terms of L andA provided that the orbit is non-colliding (i.e., L 6= 0):

E = −1− |A|2

2|L|2. (4)

Note that this intrinsic formulation works in any dimension! So, in anydimension, the Kepler problem exists and its non-colliding orbits areconics.

Remark about the Kepler Problem

In any dimension, there exists an apparent analogue of the Keplerproblem, whose non-colliding orbits are always conics.The Kepler problem is extremely important in the development ofthe fundamental physics in both Newton’s time and 1920s.Its simplicity leads many people to believe that everything about itis already known.Its history indicates repeatedly that there is always a surprise lyingahead.

MICZ-Kepler Problems

Here is a surprise about the Kepler problem. Towards the end of1960s, D. Zwanziger, independently H. McIntosh and A. Cisneros,discovered a family of magnetized companions for the Kepler problem— the MICZ-Kepler problems.

DefinitionLet µ ∈ R. The MICZ-Kepler problem with magnetic charge µ is adynamic problem with configuration space R3

∗ and equation of motion

r′′ = − rr3+µ

2 rr4 − r′ × µ r

r3 . (5)

Remark:The Kepler problem is the one with µ = 0.A physics system corresponding to µ 6= 0 has not been found yet.

r′′ = − rr3+µ

2 rr4 − r′ × µ r

r3 . (5)

r′′ = − rr3+µ

2 rr4 − r′ × µ r

r3 . (5)

r′′ = − rr3+µ

2 rr4 − r′ × µ r

r3 . (5)

Key Featuresangular momentum L := r× r′ + µ r

r . So |L|2 = |r× r′|2 + µ2.Lenz vector A := L× r′ + r

r .energy. For non-colliding orbit (i.e., |L|2 > µ2), we have

E = − 1− |A|2

2(|L|2 − µ2). (6)

orbits. One can show that L · A = µ and

L · r = µr , r − A · r = |L|2 − µ2. (7)

The non-colliding orbits are again conics: elliptic, parabolic, andhyperbolic according as the total energy E is negative, zero, andpositive. That is because the eccentricity e of the conic orbitsatisfies relation

1− e2 =|L|2 − µ2

|L− µA|2(1− |A|2).

E = − 1− |A|2

2(|L|2 − µ2). (6)

L · r = µr , r − A · r = |L|2 − µ2. (7)

1− e2 =|L|2 − µ2

|L− µA|2(1− |A|2).

E = − 1− |A|2

2(|L|2 − µ2). (6)

L · r = µr , r − A · r = |L|2 − µ2. (7)

1− e2 =|L|2 − µ2

|L− µA|2(1− |A|2).

E = − 1− |A|2

2(|L|2 − µ2). (6)

L · r = µr , r − A · r = |L|2 − µ2. (7)

1− e2 =|L|2 − µ2

|L− µA|2(1− |A|2).

Light Cone Formulation of the Non-colliding Orbits(G.W. Meng, 2011)We shall lift orbits from R3

∗ to the future light cone in the Minkowskispace. Let x = (x0, r) ∈ R1,3 and

l =1√

L2 − µ2(µ,L), a =

1L2 − µ2 (1,A) (8)

where µ = L ·A. Note that l2 = −1, l · a = 0, a0 > 0. The (lifted) orbit isthe intersection of the affine plane

l · x = 0, a · x = 1 (9)

with the future light cone

x2 = 0, x0 > 0. (10)

The energy is E = − a2

Remark. The significance of this formulation is that a 2nd temporaldimension (i.e. x0) appears naturally.

l =1√

L2 − µ2(µ,L), a =

1L2 − µ2 (1,A) (8)

l · x = 0, a · x = 1 (9)

x2 = 0, x0 > 0. (10)

l =1√

L2 − µ2(µ,L), a =

1L2 − µ2 (1,A) (8)

l · x = 0, a · x = 1 (9)

x2 = 0, x0 > 0. (10)

l =1√

L2 − µ2(µ,L), a =

1L2 − µ2 (1,A) (8)

l · x = 0, a · x = 1 (9)

x2 = 0, x0 > 0. (10)

A Picture of Conics

A New Discovery (G. W. Meng, 2011)Let SO+(1,3) be the identity component of the Lorentz group SO(1,3)and R+ be the multiplicative group of positive real numbers. Weassume that the action of R+ on a is the scalar multiplication and R+

acts on l trivially. A non-colliding orbit of a MICZ-Kepler problem shallbe referred to as a MICZ-Kepler orbit.

Theorem (G. W. Meng, 2012)The Lie group SO+(1,3)× R+ acts transitively on both the set oforiented elliptic MICZ-Kepler orbits and the set of oriented parabolicMICZ-Kepler orbits.

Proof.Let O be the set of oriented MICZ-Kepler orbits, then we have abijection between O andM := {(A,L) ∈ R3 × R3 | L 6= 0} = R3 × R3

∗,hence, in view of Eq. (8), a bijection between O andM := {a, l)R1,3 × R1,3 | l2 = −1, l · a = 0,a0 > 0}. The rest is almostclear.

Remark. The significance of this theorem is that the magnetic chargeis relative.

Summary

The light cone formulation of orbits is attractive and has yieldednew insight. Does that mean that it is mathematically moreadvantageous to reformulate the Kepler problem on the future lightcone? The answer is yes.This new formulation leads to a general theory based onEuclidean Jordan algebras, in which both the Kepler problemand the isotropic oscillator problems are special examples.

Summary

The light cone formulation of orbits is attractive and has yieldednew insight. Does that mean that it is mathematically moreadvantageous to reformulate the Kepler problem on the future lightcone? The answer is yes.This new formulation leads to a general theory based onEuclidean Jordan algebras, in which both the Kepler problemand the isotropic oscillator problems are special examples.

Let us conclude this talk with a diagram.

Kepler Problem

IntrinsicFormulation−→ Higher Dim. KP

LorentzFormulation

y y LorentzFormulation

MICZ KP

IntrinsicFormulation−→ Higher Dim. Magnetized KP

Thanks!

Planetary Motions and Lorentz Transformationsmath.usask.ca/~cheviakov/bluman2014/talks/Meng.pdf ·...

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