Vol.2009, Issue No. 12, Dated 31st December 2009
9:02:06 PM
A PARTICLE CONSTRAINED TO MOVE ON A CONE UNDER A GRAVITATIONAL FIELD AND THE RENORMALIZATION PROCESS FOR THE TIME DEPENDENCE
By
Professor Kotcherlakota Lakshmi Narayana,
(Retd.Prof of Physics, SU) 171-11-10 Narasimha Ashram, Official Colony,
Maharanipeta.P.O. Visakhapatnam-530002.
Mobile: 9491902867
ABSTRACT:
The orbital motion of a particle on the surface of a cone has been examined to obtain the time dependence behaviour under a renormalization procedure. The Fourier series solution involves only the terms to the third order accuracy. The asymmetry of the time dependent potential allows the even and odd harmonics of the Fourier series solution. The new differential equation of motion has been formulated to involve the cubic and square terms apart from similar inverse square power of time variable. It’s very interesting that the time dependent particle motion constrained to move on the surface of a cone may be realized within certain limits of apsides. The specific choice of the time variable leads to the renormalization situation of the particles motion. The finding of a new angular momentum l' associated with the time dependent motion is of far reaching significance. The factor λ also plays as an additional frequency factor of the particles motion on the surface of a cone.
Keywords: Cone, Particle motion, Renormalization technique, Fourier series, orbital motion, Orbital angular momentum
PREAMBLE:
Two circles on the surface of a cone act as the limiting boundaries of the orbital motion of a particle constrained to move on the surface of the cone. The angle through which the article moves from an outer to an inner turning point has been given earlier by Bradbury [1]. The cylindrical co-ordinates (r, θ, z) solution asserts the existence of an angular momentum, the canonically conjugate momentum l of the angle variable θ and is a characteristic constant of the motion. The effective potential revealed the single minimum at r1= (l^2/ (m^2*g*a) ^1/3 where g is gravitational constant, a is a positive constant. The later specifies the equation of constraint as z=a*r. The energy equation HAS BEEN GIVEN BY Bradbury [1] as
E= T+V= 0.5*m*v^2*(1+a^2) +l^2/ (2*m*r^2) +m*g*a*r
where v is the velocity. The angle variable θ turns out to be a cyclic co-ordinate. The significance of the minimum at r1 implies that the particle moves in a stable orbit circular of radius r1. Fourier series solution has been obtained by finding the effective potential function about its minimum. The other possible solution given by Bradbury is the one that does not involve the time variable. The change of variables adopted as x=1/r gave the equation of motion as (1+a^2)*d^2x/d θ^2 +x – m^2*g*a/ (lx) ^2=0. The x being a periodic function of θ the Fourier solution of the form
x=a0+Σ bn*cos (n*λ* θ) where bn and a0 are the Fourier series coefficients and Σ is the symbol of Fourier series summation over n=1 to 3. Here λ plays the role similar to that of frequency. The minimum for x is
x1= (m^2*g*a/l^2) ^1/3.
When expanded about x1 the result obtained may be identified with the equation
(1+a^2)+ α* x- β*x^2 + γ*x^3=0
with α = 3/ (1+a^2), β= 3/(x1*(1+a^2)), γ= 4/(x1^2*(1+a^2))
and λ^2 = α*(1+ (1/6)*(b1* r1) ^2) respectively.
The two circles of radii r/ r1= 0.5 and r/ r1= 1.5 restricts the motion of the particle between the specified apsides. The angle between the apsides has been reported by Bradbury [1] as λΔ θ= 180 0 and Δ θ=125.5 0. The diagram below is obtained by using MATHLAB 7.0 version software. The values of b1* r1 =0.4 and λ= 1.434 have been used to generate this polar diagram. The orbit would close on itself if proper values of 2*Δ θ=2* π, π, etc and multiples of (2* π) are chosen.
PRESENT WORK:
The book by Bradbury also describes the time-dependent constraints, dissipative forces and the classification of constraints. In the present work I have chosen an additional term for the consideration of the affects of time dependence of the constrained orbital motion of the particle on the surface of the cone. The term newly introduced is given by 0.5* r^2*t^2*w^2. This allows to obtain an additional frequency variable w which leads in the Lagrangian formulation to the additional angular momentum constant l’. This is canonically conjugate to the new variable w. The effective potential now becomes
V(r) = (1/ (2*m*r^2)*(l^2+ l’^2/t^2) +m*g*a*r.
The minimum r1= (l^2+ l’^2/t^2)/ m^2*g*a) ^1/3. Here the additional term may be considered as either ± i.e. either as an addition or a subtraction. I have found that the + sign leads to reasonable values. The equation of motion is written using y instead of x, in order to emphasize the new orbital time dependent constrained motion of the particle on the surface of the cone.
The new equation of motion is
(1+a^2) d^2y/d θ^2 +y*(1-W^2)-m^2*g*a/ (l*y) ^2=0
where W^2= (1-l’^2/t^2) that yields new expressions as
α = (3-W^2)/ (1+a^2), β= 3/(x1*(1+a^2)), γ= 4/(x1^2*(1+a^2)) and consequently the expression for λ^2 will be
λ^2= (1/ (1+a^2)*((3-W^2) + b1*(3- 15/2*(1/ (3-W^2)))
y=a0+Σ bn*cos (n*λ* θ) and λ plays a role like the frequency.
Two numerical samples have been worked out with r1=.4 and b1=1.0.
I case: a0=3.8; b2= -3.5625 b3=1.3663 lamd=1.7078
ra=1+a0*r1+b1*r1*cos (lamd*teta)-b2*cos (2*lamd*teta) +b3*cos (3*lamd*teta)
where ra =r1/r. Here x1=1/r1 and x=1/r.
9:02:06 PM
A PARTICLE CONSTRAINED TO MOVE ON A CONE UNDER A GRAVITATIONAL FIELD AND THE RENORMALIZATION PROCESS FOR THE TIME DEPENDENCE
By
Professor Kotcherlakota Lakshmi Narayana,
(Retd.Prof of Physics, SU) 171-11-10 Narasimha Ashram, Official Colony,
Maharanipeta.P.O. Visakhapatnam-530002.
Mobile: 9491902867
ABSTRACT:
The orbital motion of a particle on the surface of a cone has been examined to obtain the time dependence behaviour under a renormalization procedure. The Fourier series solution involves only the terms to the third order accuracy. The asymmetry of the time dependent potential allows the even and odd harmonics of the Fourier series solution. The new differential equation of motion has been formulated to involve the cubic and square terms apart from similar inverse square power of time variable. It’s very interesting that the time dependent particle motion constrained to move on the surface of a cone may be realized within certain limits of apsides. The specific choice of the time variable leads to the renormalization situation of the particles motion. The finding of a new angular momentum l' associated with the time dependent motion is of far reaching significance. The factor λ also plays as an additional frequency factor of the particles motion on the surface of a cone.
Keywords: Cone, Particle motion, Renormalization technique, Fourier series, orbital motion, Orbital angular momentum
PREAMBLE:
Two circles on the surface of a cone act as the limiting boundaries of the orbital motion of a particle constrained to move on the surface of the cone. The angle through which the article moves from an outer to an inner turning point has been given earlier by Bradbury [1]. The cylindrical co-ordinates (r, θ, z) solution asserts the existence of an angular momentum, the canonically conjugate momentum l of the angle variable θ and is a characteristic constant of the motion. The effective potential revealed the single minimum at r1= (l^2/ (m^2*g*a) ^1/3 where g is gravitational constant, a is a positive constant. The later specifies the equation of constraint as z=a*r. The energy equation HAS BEEN GIVEN BY Bradbury [1] as
E= T+V= 0.5*m*v^2*(1+a^2) +l^2/ (2*m*r^2) +m*g*a*r
where v is the velocity. The angle variable θ turns out to be a cyclic co-ordinate. The significance of the minimum at r1 implies that the particle moves in a stable orbit circular of radius r1. Fourier series solution has been obtained by finding the effective potential function about its minimum. The other possible solution given by Bradbury is the one that does not involve the time variable. The change of variables adopted as x=1/r gave the equation of motion as (1+a^2)*d^2x/d θ^2 +x – m^2*g*a/ (lx) ^2=0. The x being a periodic function of θ the Fourier solution of the form
x=a0+Σ bn*cos (n*λ* θ) where bn and a0 are the Fourier series coefficients and Σ is the symbol of Fourier series summation over n=1 to 3. Here λ plays the role similar to that of frequency. The minimum for x is
x1= (m^2*g*a/l^2) ^1/3.
When expanded about x1 the result obtained may be identified with the equation
(1+a^2)+ α* x- β*x^2 + γ*x^3=0
with α = 3/ (1+a^2), β= 3/(x1*(1+a^2)), γ= 4/(x1^2*(1+a^2))
and λ^2 = α*(1+ (1/6)*(b1* r1) ^2) respectively.
The two circles of radii r/ r1= 0.5 and r/ r1= 1.5 restricts the motion of the particle between the specified apsides. The angle between the apsides has been reported by Bradbury [1] as λΔ θ= 180 0 and Δ θ=125.5 0. The diagram below is obtained by using MATHLAB 7.0 version software. The values of b1* r1 =0.4 and λ= 1.434 have been used to generate this polar diagram. The orbit would close on itself if proper values of 2*Δ θ=2* π, π, etc and multiples of (2* π) are chosen.
PRESENT WORK:
The book by Bradbury also describes the time-dependent constraints, dissipative forces and the classification of constraints. In the present work I have chosen an additional term for the consideration of the affects of time dependence of the constrained orbital motion of the particle on the surface of the cone. The term newly introduced is given by 0.5* r^2*t^2*w^2. This allows to obtain an additional frequency variable w which leads in the Lagrangian formulation to the additional angular momentum constant l’. This is canonically conjugate to the new variable w. The effective potential now becomes
V(r) = (1/ (2*m*r^2)*(l^2+ l’^2/t^2) +m*g*a*r.
The minimum r1= (l^2+ l’^2/t^2)/ m^2*g*a) ^1/3. Here the additional term may be considered as either ± i.e. either as an addition or a subtraction. I have found that the + sign leads to reasonable values. The equation of motion is written using y instead of x, in order to emphasize the new orbital time dependent constrained motion of the particle on the surface of the cone.
The new equation of motion is
(1+a^2) d^2y/d θ^2 +y*(1-W^2)-m^2*g*a/ (l*y) ^2=0
where W^2= (1-l’^2/t^2) that yields new expressions as
α = (3-W^2)/ (1+a^2), β= 3/(x1*(1+a^2)), γ= 4/(x1^2*(1+a^2)) and consequently the expression for λ^2 will be
λ^2= (1/ (1+a^2)*((3-W^2) + b1*(3- 15/2*(1/ (3-W^2)))
y=a0+Σ bn*cos (n*λ* θ) and λ plays a role like the frequency.
Two numerical samples have been worked out with r1=.4 and b1=1.0.
I case: a0=3.8; b2= -3.5625 b3=1.3663 lamd=1.7078
ra=1+a0*r1+b1*r1*cos (lamd*teta)-b2*cos (2*lamd*teta) +b3*cos (3*lamd*teta)
where ra =r1/r. Here x1=1/r1 and x=1/r.
II case: t=3.6741 alpha = 0.8333 beta= 7.3334 gamma=5.1359
a0=4.4003 b2=-3.6667 b3= 1.8062 lamd=2.0685;
ra=1+4.4*r1+1.0*r1*cos (lamd*teta) +3.6667*cos (2*lamd*teta) +1.8062*cos (3*lamd*teta)
This yields the Orbit as Illustrated in the Fig 2.
The complete 3Dimensional orbital motion within the limits of apsides (singular,apse) set by the time variable t and the angle variable θ is given in the Fig.3
SUMMARY:
It’s very interesting that the time dependent particle motion constrained to move on the surface of a cone may be realized within certain limits of apsides. The specific choice of the time variable leads to the renormalization situation of the particles motion. The finding of a new angular momentum l' associated with the time dependent motion is of far reaching significance. The factor λ also plays as an additional frequency factor of the particles motion on the surface of a cone.
ACKNOWELEDGMENT:
I am indebted to late Professor K. Rangadhama Rao D.Sc. (Madras) D.Sc. (London) for his guidance, inspiration and constant encouragement throughout my academic endeavour. I am also Indebted to Mrs. K. Peramma Garu. See a photograph of them attached.
It’s very interesting that the time dependent particle motion constrained to move on the surface of a cone may be realized within certain limits of apsides. The specific choice of the time variable leads to the renormalization situation of the particles motion. The finding of a new angular momentum l' associated with the time dependent motion is of far reaching significance. The factor λ also plays as an additional frequency factor of the particles motion on the surface of a cone.
ACKNOWELEDGMENT:
I am indebted to late Professor K. Rangadhama Rao D.Sc. (Madras) D.Sc. (London) for his guidance, inspiration and constant encouragement throughout my academic endeavour. I am also Indebted to Mrs. K. Peramma Garu. See a photograph of them attached.
REFERENCES:
1. T.C.Brdbury, book on" Theoretical Mechanics", John Wiley & Sons,Inc. London,p.255 (1968)
ADDENDUM:
Lakshmikshirasasamudhraraajathanayam Sri Rangadhameswari
Dasibhutasamasthadevedavanithamlokikadeepanokuram
Srimanyandhakatakshalabdhavibhavadaheysmandhragangadharam
Thvamthithralokkyakutumeneemsarasijamvandemukundapriya
Ramayamma mother of The Professor has christened her son as RANGADHAMA during a divine event at Visakhapatnam Kotcherlakota house when he was four years old.
Many people are speculating the probable colossal loss of life and natural disasters to occur during the turn of a few years. Also the global warming is threatening, the very existence of human race of its luxurious living on the PLANET EARTH. If it’s true that the PLANET EARTH would have an eventful death of human race, in a super volcano upsurge of water subversion, then many a mineral formations would happen of unprecedented nature. This formation is what is implied partly in the Sanskrit (hymn) sloka of ancient Veda.
1. T.C.Brdbury, book on" Theoretical Mechanics", John Wiley & Sons,Inc. London,p.255 (1968)
ADDENDUM:
Lakshmikshirasasamudhraraajathanayam Sri Rangadhameswari
Dasibhutasamasthadevedavanithamlokikadeepanokuram
Srimanyandhakatakshalabdhavibhavadaheysmandhragangadharam
Thvamthithralokkyakutumeneemsarasijamvandemukundapriya
Ramayamma mother of The Professor has christened her son as RANGADHAMA during a divine event at Visakhapatnam Kotcherlakota house when he was four years old.
Many people are speculating the probable colossal loss of life and natural disasters to occur during the turn of a few years. Also the global warming is threatening, the very existence of human race of its luxurious living on the PLANET EARTH. If it’s true that the PLANET EARTH would have an eventful death of human race, in a super volcano upsurge of water subversion, then many a mineral formations would happen of unprecedented nature. This formation is what is implied partly in the Sanskrit (hymn) sloka of ancient Veda.
No comments:
Post a Comment