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8/14/2019 Close to Kepler

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Generalization of Kepler's second and third law to the group ofsolutions linearly scalable from each other in the Newtonian N-bodygravitation problem

The plot of the derivation presented in this paper starts from defining a linearscaling of a differential equation and applying it via the chain rule for

derivatives to the Newtonian N-body system and solving for a condition ofinvariability of the equations. After that it is shown that the invariability of theequations corresponds to Kepler's third and second law of planetary motion.

Definition of Linear Scaling of a Differential Equation System

du

dt=f u , 1

where

u= u1 ,u2 ,. .. ,u2NT

, f= f1 , f2 , .. . ,f2N .

Linear scaling of a 2N dimensional differential equation system can be definedwith a 2N + 1 dimensional vector

= 0 ,1 , . .. ,2N 2

that is included in linear scaling in which one scales the time and functions oftime as follows

t0t 3a

ui i ui 3b

for i = 1,2,3,...,2N and >0 .

The quest for a linear scaling during which the Newtonian theNewtonian N-body differential equation system remains invariant

One condition for the system (1) to be invariant in the linear scaling definedabove (3) can be found with the chain rule. Marking for the scaled time and

functions of time ts=0 t and ui

s=i ui , one obtains via the chain rule

duis

dt s=

duis

dui

dui

dt

dt

dt s=

i

dui

dt

0= i0

dui

dt=

i0fi u . 4

For the differential equation system (1) to be invariant in the defined linearscaling the system is to be valid with the scaled time and functions of time

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duis

dts=f

ius 5a

which is written with (4) and (3b) as

i0

fi u =fi

1u1 ,. . . ,

2Nu2N , i=1,2, ...,2N. 5b In the Newtonian N-body gravitational problem the differential equation systemis of the form

dui

dt=fi u =uN+i , i=1,2, . .. ,N 6a

duN+i

dt=fN+i u=

mj

uj

ui

ujui , i=1,2 ,. . . ,N. 6b

Every ui is a point in a three dimensional space. For i = 1,...,N the point ui

describes the coordinates of the i:th particle and for i=N+1, ..., 2N its velocityvector.

Combining equations (6a) and (5b) one can derive

i

0uN+i= 6a =

i

0fi u = 5b =fi

1u1

,. . . ,2Nu2N= 6a =

N+i uN+i

from which the relation

N+i =10 7

is obtained.

Applying the chain rule condition (4) to the Newtonian equations (6b) combinedwith the result (7), one can derive

duN+is

dts=f

N+i

us = 4 =N+i0f

N+i

u= 7 = i20

mj

ujuiu

j

ui. 8

Taking into account the definition of the linearly scaled us

one can write (6b)as a scaled equation directly as

duN+is

dt s=

m j

j uj

i ui j uj

iui. 9

Therefore for the Newtonian N-body differential equation system to be

invariant with respect to linear scaling, the right hand sides of (8) and (9) are to

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equal

mj

j uj

i uij uj

iui =

i2

0mj

ujuiujui , i=1,2 ,. . . ,N. 10

A solution to the equation (10) can derived by setting i =C,i=1,2 ,. . . ,N whereC is an arbitrary constant. After a bit of algebra the sum factor cancels out of

the equation leavingC

3C=

C20which is equally 0=

3

2C . Using the condition

(7) one finds for the i=N+1,...,2N: N+i =i0=1

2C .The solved parameter

values can be more compacty represented with a 2N+1 dimensional vectorintroduced in (2) as

=C

3

2

,1,...,1 ,1

2,

. .. ,1

2 . 11

Discussion around Kepler's second and third laws

Kepler's third law is a statement derived from observations and states Thesquares of the orbital periods of planets are directly proportional to the cubesof the semi-major-exis of the orbits.

Assuming there is a periodical solution u t to an N-body Newtoniangravitational problem with a period time T.

With the linear transformation procedure, we can find an infinite amount of

solutions from one solution. For the scaled solution the period time Ts =

3

2C

T

and the position coordinates uis ts =Cui t . Therefore for all pairs i,j of the N-

body group analyzed the quantity appearing implicitly in the Kepler's third lawremains invariant as we move from one scaled solution to another

maxtuis t s uj

s t s

Ts 2

=3Cmaxtui t uj t

3C Ts 2

=maxtui t uj t

T2

. 12

These constants depending on selected i and j can be summed over all theparticle pairs to

max tui t uj t =ConstantT2. 13

which holds for all scaled solutions with coordinates u t and period time T.

This is somewhat similar to the Kepler's third law which holds for every

planetary orbit as it is law that holds remains constant between different

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solutions.

Keplers second law states that "A line joining a planet and the sun sweeps outequal areas during equal intervals of time. which can be represented withmathematical symbols as

ddt 12 r

2 14

The correspondence to this particular law can be shown to hold within thegroup of solutions linearly scalable from each other as follows. We study aquantity obtainable by an educated guess and look at its derivative withrespect to the scaled time and use the chain rule and the conditions forinvariability with respect to linear scaling. Marking the constant in timepresent in the Kepler's second law by

g u =ui t uj t 2u

i t uj t . 15

We can write its derivative with respect to the scaled time

d

dtsg us = d

dtg us dt

dts=

d

dtg us

3

2C

16

finishing the calculation with (3b) and (11)

ddt

g us = ddt

uis ts uj

s ts 2uis ts uj

s ts = ddt

2Cui t uj t 2

1

2Cui t uj t 17

which yields

d

dtsg us = d

dtg u . 18

.

We have thus shown that the constant present in Keplers second law appears

in similar fashion in every scaled solution. In a two body system with one largeobject the quanity is constant as Kepler derived from observations.

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