inverse problems of newton's laws

©Freund Publishing House Ltd.. International Journal of Nonlinear Sciences & Numerical Simulation. 10(9). 1087-1091.2009

Inverse Problems of Newton's Laws

Ji-Huan HeModern Textile Institute, Donghna University,1882 Yan'anXilu Road, Shanghai 200051, ChinaEmail: Jhhe(u),dhu. edu. en

Abstract

All motions or static matters are subjected to Nature Laws, which might be already known or still waitfor discovery. Motion trajectory or surface of a subject can be easily measured, and it is used to discover apossible law involved in the motion using a new concept called morph-force. A modified Newton'sgravitational law and a modified Newton's second law are suggested. This paper also rebuts the basicassumption of Newton's law of universal gravitation.PACS numbers : 02.30.Xx, 02.30.Zz, 04.20.Cv, 04.40.-b

Keywords: Least action principle, Lagrange multiplier, morph-force, gradient, capillary, pendulum,moving boundary, modified Newton's laws

1. Introduction

When we know the forces acting on astone and its initial speed and place, then weknow exactly where it will land, for the stone isexactly controlled by the Newton's law [1]. Theopposite is to determine the force when thetrajectory of the stone is known, this is theinverse problem of the Newton's law. In thispaper we assume that the Maupertuis-Lagrange's principle of least action is valid forall cases in our study.

2. Mauperfuis-Lagrange's Principle anda Modified Newton's Gravitational Law

The Maupertuis-Lagrange's principle ofleast kinetic potential action for a particle withmass /// can be expressed as follows [2,3]

Γ — m(xz + y~)dt -> min·*» 2 (1)

Now we consider the motion of the Eartharound the Sun. For simplicity, we assume thetrajectory is a circle with radius of R:

We consider Eq.(2) is the constraint of thefunctional, Eq.(l). Using the Lagrangemultiplier, we have[4j

(3)

where λ is a generalized Lagrange multiplier.Calculating the variation of the functional

(3) results in the following stationary conditions

mx = 2/Lx, (4)

my = 2Ay, (5)

Differentiating Eq.(2) with respect of timetwice, we have

xx + yy + x2 + y2 = 0 (6)

Using Kepler's second law, we obtain

V - -Jx- + y2 = constant (7)

Thus Eq.(6) reduces to

-V2 (8)

t ·> — ·»x~+y~=R- (2)Multiplying Eqs. (4) and (5) by,

respectively, x, and y, we obtain

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1088 Ji-Huan lie: Inverse Problems of Newton's Laws

mxx = 2λχ2,

and

myy = 2Ay2,

(9)

(10)

Adding the above two equations together,and considering Eqs.(2) and (8), we have

( ·* **\ o i / 2 2 \ i i i )" /1 ι \xx + yy) = 2A.(x +y ) = 2AR , (II)

from which the multiplier can be identified,which reads

m(xx + yy)2R- 2R-

We, therefore, obtain

mV2

(12)

(13)

(14)

which agree with Newton's gravitational law,and we now know that the force acting on theEarth is called gravitational force.

If the Earth's motion trajectory is anellipse, that is

a(15)

By a similar manipulation as illustratedabove, we obtain

ι? _ .„v _ 9 ·\ x (ι /ς\/ A — mx — LA.—- (.iv)

Fr=„,y = 2 ^ (17)

where the Lagrange multiplier can be identifiedas follows

— τνa~ b~

a b

(18)

Eqs.(16) and (17) with the determined λcan be considered as a modification ofNewton's gravitational law.

3. Morph-force

Now we consider a general trajectory

Similar to the Section 2, we have

_ Γ 1 / ' / - 2 - 2 - 2 - , Λ/γ χ] ι= 1\2"'(Χ +)' +Σ )+ J(X'y'=)]

(19)

(20)

where k is a Lagrange multiplier.We can, therefore, obtain Newton's law in

forms

x = k—

F. =,»:: =d:

or

(21)

(22)

(23)

(24)

The total resultant force can be \\ritten inthe form

F = k-dn

(25)

where;; denotes the normal to the surface /=0 ,which can be considered as an "equipotentialsurface". Vujicic called Vf as a "constraintforce" [5]. Considering that f=Q is not aconstraint, but a solution, thus we define theforce kVf as a morph-force .

As an illustrating example, we consider themotion of a pendulum as illustrated in Fig.l,where we assume that the rod is unseen, and themotion of the mass attached to the unseen rod isobservable and can be measured. According tothe above analysis, we know there is a resultantforce F = kdf Idn as marked in Fig.l.



ISSN: 1565-1339 International Journal of Nonlinear Sciences & Numerical Simulation. 10(9). 1189-1200.2009 1089

\

unseen rod

kVf

Fig. 1: Pendulum motion in one-dimensionalcurved space

The motion of the pendulum is in one-dimensional curved line, which results in amorph-force, k V f . That means any matters ina curved surface are subject to morph-forces,which might be gravitational force in celestialbodies; gravitation of all matters on the Earth'ssurface; atom-scale forces on the curved two-dimensional world sheet of string.

4. Modified Newton's Second Law

Hereby we suggest a modified Newton'ssecond law:

m— = (26)

where F is the sum acting vector forces exceptthe force acting on the surface ^=0. For staticcase, we haveF + -V/ = 0 (27)

Considering a capillary as illustrated inFig.2, the total resultant morph-force acting onthe surface is k V f , using Eq.(27), we canimmediately obtain

(28)

t\£7

Αν/-

Fig. 2: Capillary

5. Moving boundary(Surfacc)

If the surface is variable with time , i.e.,f(x,y,z,x,y,z,x,y,z) = Q (29)

then the morph-force can be deduced as follows

^\ · * ϊ ~clx c/r

*%·Ο)-1

dt<32)

The morph-force, Eqs.(30)-(32), is veryuseful for solving moving boundary problems.

6. Discussion and Conclusions

Surface gradient or concentration gradientor other subject gradient yields a "force", whichcan be called gradient force. In electricalconduction, we have the Ohm's law:dqc _ , dU ci')\— = — ft· 9 \J*·)

dl dn



1090 Ji-I luan I le: Inverse Problems of Newton's Laws

where U is the electrical potential. The gradientof the electrical potential results in electricalflow, i/(,, and we call kdU / dn the gradientforce of the electrical potential.

In heat conduction, we have a similarFourier's law:

—iln

(33)

where 7" is the thermal potential. The gradientof the thermal potential yields heat flo\v,i/A,and we call kdT/dn the gradient force of thethermal potential.

In diffusion, we have the Pick's first law

dt(34)

where φ the concentration, Jthe diffusion flux.The total resultant force resulting from the

curved surface, temperature difference andmass transfer can be written in the form

F ~ k V/" + &,V7" + k VA/ (35)

where ktV/ is the morph-force, k2VT is thetemperature gradient force, and £3VA/ is theforce due to mass transfer, and it generallyfollows the Pick's law

Fig. 3 illustrates the morph-force andgradient forces, useful for establishing thegoverning equations.

Fig. 3: Forces due to gradients

Everyone knows Newton's law, known asdeterminative mechanics-when initial speed ofa subject and forces acting on it are known, itstrajectory is definitely determinative, inverseproblems of Nature laws appear everywhereand everyday, to discover the hidden naturelaws is the main aim of scientific activity. Thispaper suggests a simple but effective concept,the morph-force and gradient forces, which aresimilar to the hierarchical level force discussedin Rcf. [6]. Any matter (in motion or staticstation) is subject to the morph-force when it isin a curved surface, the morph-force can alsoexplain why fatalness of virus depends upon itscell fractal geometry [7J. This paper also rebutsthe basic assumption of Newton's law ofuniversal gravitation that "every point massattracts c\ery other point mass by a forcepointing along the line intersecting bothpoints". A point mass is attracted only in thecase when it locates in a curved surface. I couldclaim that weight did not exist \\hen the Earthwas an absolutely smooth two-dimensionalplate, in such a case, all matters on the Earthwere not attracted by the Earth, and thegravitational acceleration vanishes (g=0). Anearthquake might change local gravity, andcurved time might lead to acceleration. Anirregular bent lithosphcrc might result in avolcano. At a depth between 300 and 700kilometers, the rock of the descending platemelts. A downward bent lithosphere with anupward morph-force might push the moltsupward to Earth's surface, producing a volcano.An earthquake or some special structure ofplate tectonics might also yield an upwardmorph-force, resulting in a strange phenomenonthat water will flow upwards. We will discussall these points in detail in forthcoming papers.

References

[1] M.S. El Naschie. Nanotechnology for thedeveloping world, Chaos Soliton. Fruct.,30(4)(2006): 769-773

[2] J.H. He, Generalised variutional principlesin fluids, Science and Culture PublishingI louse of China, I long Kong, China, 2003

[3] J.H. He, An elementary introduction torecently developed asymptotic methods and



ISSN: 1565-1339 International Journal of Nonlinear Sciences & Numerical Simulation. 10(9), 1189-1200,2009 1091

nanomechanics in textile engineering, Int. J.Mod. Phys. B, 22(21) (2008): 3487-3578

[4] J. H. He, A Variational Approach to the [6]problem of two bodies, Facia Universiiatis,Series Mechanics, Automatic Control &Robotics, 2( 10)(2000) 1049-1054 [7]

[5] V.A. Vujicic, One approach to the problemof two bodies, Facia Universitatis, Series

Mechanics, Automatic Control & Robotics,2(8)( 1998): 629-634J.H. He, Hubert cube model for fractalspacetime, Chaos, Solitons & Fract-als, 42(5)(2009) 2754-2759J.H. He, Fatalncss of virus depends upon itscell fractal geometry, Chaos Soli ton. Fract.,38(5)(2008) 1390-1393



inverse problems of newton's laws

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