Planetary Motions and Lorentz Transformations

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1 Planetary Motions and Lorentz Transformations Kepler Problem and Future Light Cone Guowu Meng Department of Mathematics Hong Kong Univ. of Sci. & Tech. International Conference Symmetry Methods, Applications, and Related Fields celebrating the work of Professor George W. Bluman University of British Columbia, Vancouver, Canada, May 13-16, 2014 May 15, 2014

2 Think deeply about simple things! Arnold Ross Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

3 Kepler Problem The Kepler problem is a dynamical problem with configuration space R 3 := R 3 \ {0} and equation of motion r = r r 3. (1) In this talk I shall convince you that the Kepler problem is intimately related to the future light cone and Lorentz transformations. In view of the fact that the Kepler problem is non-relativistic, this seems to be odd or surprising. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

4 Kepler Problem The Kepler problem is a dynamical problem with configuration space R 3 := R 3 \ {0} and equation of motion r = r r 3. (1) In this talk I shall convince you that the Kepler problem is intimately related to the future light cone and Lorentz transformations. In view of the fact that the Kepler problem is non-relativistic, this seems to be odd or surprising. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

5 Future Light Cone Here is a picture in the Lorentz space R 1,2 : The future light cone in this talk is the one in the Minkowski space R 1,3. Although it cannot be visualized, it can be described as the solution set of x 2 0 x 2 1 x 2 2 x 2 3 = 0, x 0 > 0 (2) and is diffeomorphic to R 3 under the projection: (x 0, r) R 1,3 r R 3. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

Although it cannot be visualized, it can be described as the solution set of x 2 0 x 2 1 x 2 2 x 2 3 =

6 Future Light Cone Here is a picture in the Lorentz space R 1,2 : The future light cone in this talk is the one in the Minkowski space R 1,3. Although it cannot be visualized, it can be described as the solution set of x 2 0 x 2 1 x 2 2 x 2 3 = 0, x 0 > 0 (2) and is diffeomorphic to R 3 under the projection: (x 0, r) R 1,3 r R 3. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

7 Basic Facts on the Kepler Problem The angular momentum L := r r and the Lenz vector A := r L + r r are 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 := 1 2 r 2 1 r can be expressed in terms of L and A provided that the orbit is non-colliding (i.e., L 0): E = 1 A 2 2 L 2. (4) Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14 Note that this intrinsic formulation works in any dimension! So, in any

8 Remark about the Kepler Problem In any dimension, there exists an apparent analogue of the Kepler problem, whose non-colliding orbits are always conics. The Kepler problem is extremely important in the development of the fundamental physics in both Newton s time and 1920s. Its simplicity leads many people to believe that everything about it is already known. Its history indicates repeatedly that there is always a surprise lying ahead. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

9 Remark about the Kepler Problem In any dimension, there exists an apparent analogue of the Kepler problem, whose non-colliding orbits are always conics. The Kepler problem is extremely important in the development of the fundamental physics in both Newton s time and 1920s. Its simplicity leads many people to believe that everything about it is already known. Its history indicates repeatedly that there is always a surprise lying ahead. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

10 Remark about the Kepler Problem In any dimension, there exists an apparent analogue of the Kepler problem, whose non-colliding orbits are always conics. The Kepler problem is extremely important in the development of the fundamental physics in both Newton s time and 1920s. Its simplicity leads many people to believe that everything about it is already known. Its history indicates repeatedly that there is always a surprise lying ahead. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

11 Remark about the Kepler Problem In any dimension, there exists an apparent analogue of the Kepler problem, whose non-colliding orbits are always conics. The Kepler problem is extremely important in the development of the fundamental physics in both Newton s time and 1920s. Its simplicity leads many people to believe that everything about it is already known. Its history indicates repeatedly that there is always a surprise lying ahead. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

12 MICZ-Kepler Problems Here is a surprise about the Kepler problem. Towards the end of 1960s, D. Zwanziger, independently H. McIntosh and A. Cisneros, discovered a family of magnetized companions for the Kepler problem the MICZ-Kepler problems. Definition Let µ R. The MICZ-Kepler problem with magnetic charge µ is a dynamic problem with configuration space R 3 and equation of motion Remark: r = r r 3 +µ2 r r 4 r µ r r 3. (5) The Kepler problem is the one with µ = 0. A physics system corresponding to µ 0 has not been found yet. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

13 MICZ-Kepler Problems Here is a surprise about the Kepler problem. Towards the end of 1960s, D. Zwanziger, independently H. McIntosh and A. Cisneros, discovered a family of magnetized companions for the Kepler problem the MICZ-Kepler problems. Definition Let µ R. The MICZ-Kepler problem with magnetic charge µ is a dynamic problem with configuration space R 3 and equation of motion Remark: r = r r 3 +µ2 r r 4 r µ r r 3. (5) The Kepler problem is the one with µ = 0. A physics system corresponding to µ 0 has not been found yet. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

14 MICZ-Kepler Problems Here is a surprise about the Kepler problem. Towards the end of 1960s, D. Zwanziger, independently H. McIntosh and A. Cisneros, discovered a family of magnetized companions for the Kepler problem the MICZ-Kepler problems. Definition Let µ R. The MICZ-Kepler problem with magnetic charge µ is a dynamic problem with configuration space R 3 and equation of motion Remark: r = r r 3 +µ2 r r 4 r µ r r 3. (5) The Kepler problem is the one with µ = 0. A physics system corresponding to µ 0 has not been found yet. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

15 MICZ-Kepler Problems Here is a surprise about the Kepler problem. Towards the end of 1960s, D. Zwanziger, independently H. McIntosh and A. Cisneros, discovered a family of magnetized companions for the Kepler problem the MICZ-Kepler problems. Definition Let µ R. The MICZ-Kepler problem with magnetic charge µ is a dynamic problem with configuration space R 3 and equation of motion Remark: r = r r 3 +µ2 r r 4 r µ r r 3. (5) The Kepler problem is the one with µ = 0. A physics system corresponding to µ 0 has not been found yet. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

16 Key Features angular momentum L := r r + µ r r. So L 2 = r r 2 + µ 2. Lenz vector A := L r + r r. energy. For non-colliding orbit (i.e., L 2 > µ 2 ), we have orbits. One can show that L A = µ and E = 1 A 2 2( L 2 µ 2 ). (6) L r = µr, r A r = L 2 µ 2. (7) The non-colliding orbits are again conics: elliptic, parabolic, and hyperbolic according as the total energy E is negative, zero, and positive. That is because the eccentricity e of the conic orbit satisfies relation 1 e 2 = L 2 µ 2 L µa 2 (1 A 2 ). Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

17 Key Features angular momentum L := r r + µ r r. So L 2 = r r 2 + µ 2. Lenz vector A := L r + r r. energy. For non-colliding orbit (i.e., L 2 > µ 2 ), we have orbits. One can show that L A = µ and E = 1 A 2 2( L 2 µ 2 ). (6) L r = µr, r A r = L 2 µ 2. (7) The non-colliding orbits are again conics: elliptic, parabolic, and hyperbolic according as the total energy E is negative, zero, and positive. That is because the eccentricity e of the conic orbit satisfies relation 1 e 2 = L 2 µ 2 L µa 2 (1 A 2 ). Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

18 Key Features angular momentum L := r r + µ r r. So L 2 = r r 2 + µ 2. Lenz vector A := L r + r r. energy. For non-colliding orbit (i.e., L 2 > µ 2 ), we have orbits. One can show that L A = µ and E = 1 A 2 2( L 2 µ 2 ). (6) L r = µr, r A r = L 2 µ 2. (7) The non-colliding orbits are again conics: elliptic, parabolic, and hyperbolic according as the total energy E is negative, zero, and positive. That is because the eccentricity e of the conic orbit satisfies relation 1 e 2 = L 2 µ 2 L µa 2 (1 A 2 ). Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

19 Key Features angular momentum L := r r + µ r r. So L 2 = r r 2 + µ 2. Lenz vector A := L r + r r. energy. For non-colliding orbit (i.e., L 2 > µ 2 ), we have orbits. One can show that L A = µ and E = 1 A 2 2( L 2 µ 2 ). (6) L r = µr, r A r = L 2 µ 2. (7) The non-colliding orbits are again conics: elliptic, parabolic, and hyperbolic according as the total energy E is negative, zero, and positive. That is because the eccentricity e of the conic orbit satisfies relation 1 e 2 = L 2 µ 2 L µa 2 (1 A 2 ). Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

20 Light Cone Formulation of the Non-colliding Orbits (G.W. Meng, 2011) We shall lift orbits from R 3 to the future light cone in the Minkowski space. Let x = (x 0, r) R 1,3 and l = 1 L 2 µ (µ, L), a = 1 2 L 2 (1, A) (8) µ 2 where µ = L A. Note that l 2 = 1, l a = 0, a 0 > 0. The (lifted) orbit is the intersection of the affine plane with the future light cone l x = 0, a x = 1 (9) x 2 = 0, x 0 > 0. (10) The energy is E = a2 2a 0. Remark. The significance of this formulation is that a 2nd temporal dimension (i.e. x 0 ) appears naturally. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

21 Light Cone Formulation of the Non-colliding Orbits (G.W. Meng, 2011) We shall lift orbits from R 3 to the future light cone in the Minkowski space. Let x = (x 0, r) R 1,3 and l = 1 L 2 µ (µ, L), a = 1 2 L 2 (1, A) (8) µ 2 where µ = L A. Note that l 2 = 1, l a = 0, a 0 > 0. The (lifted) orbit is the intersection of the affine plane with the future light cone l x = 0, a x = 1 (9) x 2 = 0, x 0 > 0. (10) The energy is E = a2 2a 0. Remark. The significance of this formulation is that a 2nd temporal dimension (i.e. x 0 ) appears naturally. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

22 Light Cone Formulation of the Non-colliding Orbits (G.W. Meng, 2011) We shall lift orbits from R 3 to the future light cone in the Minkowski space. Let x = (x 0, r) R 1,3 and l = 1 L 2 µ (µ, L), a = 1 2 L 2 (1, A) (8) µ 2 where µ = L A. Note that l 2 = 1, l a = 0, a 0 > 0. The (lifted) orbit is the intersection of the affine plane with the future light cone l x = 0, a x = 1 (9) x 2 = 0, x 0 > 0. (10) The energy is E = a2 2a 0. Remark. The significance of this formulation is that a 2nd temporal dimension (i.e. x 0 ) appears naturally. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

23 Light Cone Formulation of the Non-colliding Orbits (G.W. Meng, 2011) We shall lift orbits from R 3 to the future light cone in the Minkowski space. Let x = (x 0, r) R 1,3 and l = 1 L 2 µ (µ, L), a = 1 2 L 2 (1, A) (8) µ 2 where µ = L A. Note that l 2 = 1, l a = 0, a 0 > 0. The (lifted) orbit is the intersection of the affine plane with the future light cone l x = 0, a x = 1 (9) x 2 = 0, x 0 > 0. (10) The energy is E = a2 2a 0. Remark. The significance of this formulation is that a 2nd temporal dimension (i.e. x 0 ) appears naturally. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

24 A Picture of Conics Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

25 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. We assume 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 shall be 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 of oriented elliptic MICZ-Kepler orbits and the set of oriented parabolic MICZ-Kepler orbits. Proof. Let O be the set of oriented MICZ-Kepler orbits, then we have a bijection between O and M := {(A, L) R 3 R 3 L 0} = R 3 R 3, hence, in view of Eq. (8), a bijection between O and M := {a, l)r 1,3 R 1,3 l 2 = 1, l a = 0, a 0 > 0}. The rest is almost clear. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

26 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. We assume 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 shall be 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 of oriented elliptic MICZ-Kepler orbits and the set of oriented parabolic MICZ-Kepler orbits. Proof. Let O be the set of oriented MICZ-Kepler orbits, then we have a bijection between O and M := {(A, L) R 3 R 3 L 0} = R 3 R 3, hence, in view of Eq. (8), a bijection between O and M := {a, l)r 1,3 R 1,3 l 2 = 1, l a = 0, a 0 > 0}. The rest is almost clear. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

27 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. We assume 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 shall be 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 of oriented elliptic MICZ-Kepler orbits and the set of oriented parabolic MICZ-Kepler orbits. Proof. Let O be the set of oriented MICZ-Kepler orbits, then we have a bijection between O and M := {(A, L) R 3 R 3 L 0} = R 3 R 3, hence, in view of Eq. (8), a bijection between O and M := {a, l)r 1,3 R 1,3 l 2 = 1, l a = 0, a 0 > 0}. The rest is almost clear. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

28 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. We assume 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 shall be 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 of oriented elliptic MICZ-Kepler orbits and the set of oriented parabolic MICZ-Kepler orbits. Proof. Let O be the set of oriented MICZ-Kepler orbits, then we have a bijection between O and M := {(A, L) R 3 R 3 L 0} = R 3 R 3, hence, in view of Eq. (8), a bijection between O and M := {a, l)r 1,3 R 1,3 l 2 = 1, l a = 0, a 0 > 0}. The rest is almost clear. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

29 Summary The light cone formulation of orbits is attractive and has yielded new insight. Does that mean that it is mathematically more advantageous to reformulate the Kepler problem on the future light cone? The answer is yes. This new formulation leads to a general theory based on Euclidean Jordan algebras, in which both the Kepler problem and the isotropic oscillator problems are special examples. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

30 Summary The light cone formulation of orbits is attractive and has yielded new insight. Does that mean that it is mathematically more advantageous to reformulate the Kepler problem on the future light cone? The answer is yes. This new formulation leads to a general theory based on Euclidean Jordan algebras, in which both the Kepler problem and the isotropic oscillator problems are special examples. Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

31 Let us conclude this talk with a diagram. Kepler Problem Intrinsic Formulation Higher Dim. KP Lorentz Formulation MICZ KP Intrinsic Formulation Lorentz Formulation Higher Dim. Magnetized KP Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

32 Thanks! Guowu Meng (HKUST) Planetary Motions and Lorentz Transformations May 15, / 14

Kepler problem and Lorentz transformations

Kepler problem and Lorentz transformations Based on [G. Meng, J. Math. Phys. 53, 052901(2012)] Guowu Meng Department of Mathematics Hong Kong University of Science and Technology God is a mathematician