Chaotic Motion in Problem of Dumbell Satellite
|
|
- Lee Jenkins
- 5 years ago
- Views:
Transcription
1 Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 7, Chaotic Motion in Problem of Dumbell Satellite Ayub Khan Department of Mathematics, Zakir Hussain College University of Delhi, Delhi, India Neeti Goel Research Scholar, Department of Mathematics University of Delhi, Delhi , India neeti Abstract In our present problem we have analytically determined the chaotic parameter in the problem of a dumbell satellite. Computational studies reveal that the chaotic parameter is effective in making the regular behavior of the system chaotic. Mathematics Subject Classification: 70F15 Keywords: Chaos, Hamiltonian, Dynamical System 1 Introduction There is a great revolution in the world of celestial Mechanics due to the necessity of including the relativistic effects in the motion of celestial bodies and the discovery of new dynamical situations in the solar system, satellites etc. Investigations on chaos is the most attracting feature which is widely studied in variety of problems of Satellites. In our present paper, we consider the model of a dumbell satellite and investigate it by using the control theory of Hamiltonian systems based on [1, 2, 3]. We observe that the control parameter (as appeared in the mathematical literature obtained by the method employed is rather behaving as a chaotic parameter in our studies which has not been experienced in the earlier studies, allow us to introduce the analytically estimated control parameter as a chaotic parameter. The regular behaviour of the system under consideration at certain initial conditions is caught under chaotic situation with the inclusion of chaotic parameter which is a very interesting phenomenon in the present manuscript.
2 300 A. Khan and N. Goel 2 Theory of Hamiltonian Systems Let A be the Lie algebra of real functions defined on phase space. For H A, let {H} be the linear operator action on A such that {H}H = {H, H }, for any H Awhere {.,.} is the Poisson bracket. The time-evolution of a function V Afollowing the flow of H is given by dv dt = {H}V, which is formally solved as V (t =e t{h} V (0, if H is time independent, where e t{h} = n=0 t n n! {H}n. Any element V Asuch that {H}V = 0, is constant under the flow of H, i.e. t R, e t{h} V = V. Let us now fix a Hamiltonian H 0 A. The vector space Ker{H 0 } is the set of constants of motion and it is a sub-algebra of A. The operator {H 0 } is not invertible since a derivation has always a non-trivial kernel. For instance {H 0 }(H0 α = 0 for any α such that Hα 0 A. Hence we consider a pseudoinverse of {H 0 }. We define a linear operator Γ on A such that i.e. {H 0 } 2 Γ={H 0 }, (2.1 V A, {H 0, {H 0, ΓV }} = {H 0,V}. If the operator Γ exists, it is not unique in general. Any other choice Γ satisfies Rg(Γ Γ Ker({H 0 } 2. We define the non-resonant operator N and the resonant operator R as N = {H 0 }Γ R = 1 N, where the operator 1 is the identity in the algebra of linear operators acting on A. We notice that Equation (2.1 becomes {H 0 }R =0
3 Chaotic motion in problem of dumbell satellite 301 which means that the range Rg R of the operator R is included in Ker{H 0 }. A consequence is that any element RV is constant under the flow of H 0, i.e. e t{h0} RV = RV. We notice that when {H 0 } and Γ commute, R and N are projectors i.e. R 2 = R and N 2 = N. Moreover, in this we have RgR = Ker{H 0 }, i.e. the constant of motion are the elements RV where V A. Let us now assume that H 0 is integrable with action-angle variables (A, ϕ B T n where B is an open set of R n and T n is the n-dimensional torus, so that H 0 = H 0 (A and the Poisson bracket {H, H } between two Hamiltonians is {H, H } = H A H ϕ H ϕ H A The operator {H 0 } acts on V given by as V = k Z n V k (Ac ik ϕ {H 0 }V (A, ϕ = k iω(a.kv k (Ae ik ϕ where the frequency vector is given by ω(a = H 0 A. A possible choice of Γ is ΓV (A, ϕ = k Z n V k (A iω(a k eik ϕ ω(a k 0 We notice that this choice of Γ commutes with {H 0 }. For a given V A, RV is the resonant part of V and N V is the nonresonant part: RV = k N V = k V k (Aχ(ω(A k =0e ik ϕ (2.2 V k (Aχ(ω(A k 0e ik ϕ (2.3 where χ(α vanishes when proposition α is wrong and it is equal to 1 when α is true. From these operators defined for the integrable part H 0, we construct a control term for the perturbed Hamiltonian H 0 + V where V A, i.e. we construct f such that H 0 + V + f is canonically conjugate to H 0 + RV.
4 302 A. Khan and N. Goel If H 0 is resonant and RV = 0, the controlled Hamiltonian H = H 0 + V + f is conjugate to H 0. In the case RV = 0, the series (6 which gives the expansion of the control term f, can be written as f(v = f s, (2.4 s=2 where f s is of order ε s and given by the recursion formula f s = 1 s {ΓV,f s 1} (2.5 where f 1 = V. 3 Application to the problem of a Satellite The equation of planer oscillation of a dumbell satellite in the central gravitational field of the Earth under the influence of the solar radiation pressure together with the effects of the Earth s shadow and some phenomenological factor [4] is given by (1 + e cos vψ 2e sin vψ + 3 sin ψ cos ψ + ρ 3 k cos ε sin(ψ + α =2esin v + E sin nv (3.1 where ψ is the angular deviation of the line joining the satellites with stable position of equilibrium, e is eccentricity of the orbit of the centre of mass, ρ is variable radius of circular orbit, ε is the Inclination of the osculating plane of the orbit of the centre of mass of the system with the plane of ecliptic, α is the angular separation of the solar position vector projected on the orbital plane, v is the true [ anomaly] of the centre of mass of the system in elliptical orbit and k = ρ3 B 1 πμ m 1 B 2 m 2 δ r sin θ, where μ is the product of gravitational constant and mass of the Earth, B i(i=1,2 are the absolute values of the forces due to direct solar radiation pressure exerted on masses of satellites m 1 and m 2 respectively, δ r is the Earth s shadow function, θ is the angle between the axis of cylinder and line joining the Earth s centre and the end point of the orbit of the centre of mass, E is the phenomenological parameter characterizing the periodic term, n is the frequency of the external periodic force. Substituting 1 ρ = 1+ecos v, 2ψ = q, kcos ε = ek 1, E = E 1 e, (3.2 the equation (3.1 reduces to (1 + e cos v d2 q dq 2e sin v dv2 dv + 3 sin q +2k 1e(1 + e cos v 3 sin 2 + α =4esin v +2E 1 e sin nv. (3.3
5 Chaotic motion in problem of dumbell satellite 303 The Hamiltonian for the above equation is H = 2p p2 3 cos q { +e p 2 cos v 2E ( 1 q } n p cos nv 3 cos q cos v 4k 1 cos 2 + α (3.4 In order to apply the theory developed by Vittot [5], we need to put the Hamiltonian in an autonomous form. We consider v as an additional angle whose conjugate action is E. Then in the autonomous form Hamiltonian can be perceived as H(p,q,E,v= 2p p2 3 cos q + E { + e p 2 cos v 2E ( 1 q } n p cos nv 3 cos q cos v 4k 1 cos 2 + α (3.5 where the actions are A =(p, E and the angles are φ =(q, v. The unperturbed Hamiltonian to be used for constructing the operator Γ is H 0 = 2p + p2 2 3 cos q + E (3.6 The action of {H 0 } and Γ on { V = e p 2 cos v 2E ( 1 q } n p cos nv 3 cos q cos v 4k 1 cos 2 + α,v A are expressed as: [ {H 0 }V = e 3(p 2 sin q cos v +2(p 2k 1 sin 2 + α + p 2 sin v ] +2E 1 p sin nv (3.7 [ 3 sin q cos v ΓV = e + 2k 1 sin + α 2 + sin v + 2E ] 1 sin nv (3.8 p 2 p 2 p 2 p for p 0, 2. The term f is given by f = 1 2 {ΓV,V } = 1 { ΓV 2 p V q ΓV q } V. p
6 304 A. Khan and N. Goel The explicit expression of f for p = 1 is given by f = 1 [9 2 e2 sin 2 q cos 2 v +12k 1 sin 2 + α sin q cos v + 6 sin v cos v sin q +6E 1 sin nv cos v sin q +4k1 2 sin2 2 + α +4k 1 sin v sin 2 + α +4E 1 k 1 sin nv sin 2 + α + 6 cos 2 v cos q +2k 1 cos 2 + α cos v + 6E 1 cos v cos q cos nv n + 2E ( 1k 1 q ] cos n 2 + α cos nv. (3.9 By introducing a central parameter β in the expression of f we get f = 1 [9 2 βe2 sin 2 q cos 2 +12k 1 sin 2 + α sin q cos v + 6 sin v cos v sin q +6E 1 sin nv cos v sin q +4k1 2 sin2 2 + α +4k 1 sin v sin 2 + α +4E 1 k 1 sin nv sin 2 + α + 6 cos 2 v cos q +2k 1 cos 2 + α cos v + 6E 1 n cos v cos q cos nv + 2E 1k 1 cos n 2 + α cos nv ]. ( Results and Discussion Figure 1(a,b and Figure 2(a,b depict the Poincare surface of section and Poincare map of the Hamiltonian given by (3.4 without and with the inclusion of (3.9 respectively for e =0.2, k 1 =0.01, α =0.001, E 1 =0.09, n =0.009, β = 0.6. Figure 3(a,b and Figure 4(a,b depict the Poincare surface of section and Poincare Map of (3.4 without and with with inclusion of (3.9 respectively for e =0.07, k 1 =0.01, α =0.001, E 1 =0.09, n =0.009, β =1.50. From these figures we observe that for e =0.2 and e =0.07 the system exhibits regular behavior without the expression of f while with the addition of the term f the system becomes chaotic for a particular value of the parameter β.
7 Chaotic motion in problem of dumbell satellite 305 Figure 1 (a Poincare surface of section for e =0.2 without the control term. (b Poincare Map for e =0.2 without the control term. Figure 2 (a Poincare surface of section for e =0.2 with the control term. (b Poincare Map for e =0.2 with the control term. Figure 3 (a Poincare surface of section for e = 0.07 without the control term. (b Poincare Map for e =0.07 without the control term.
8 306 A. Khan and N. Goel Figure 4 (a Poincare surface of section for e =0.07 with the control term. (b Poincare Map for e =0.07 with the control term. 5 Conclusion We observed that the system started to behave in chaotic manner for a particular value of the parameter β which asserts that the term obtained analytically is effective to make the system under consideration chaotic which is completely opposite to the research studies that are made in this direction so far now; and gives the new dimension to the analytical studies of the chaotic phenomenon. References [1] Ciraolo, G., Chandre, C., Lima, R., Vittot, M., Pettini, M., 2004, Control of Chaos in Hamiltonian Systems, Celestial Mechanics and Dynamics Astronomy, 90:3 12. [2] Ciraolo, G., Chandre, C., Lima, R., Vittot, M., Pettini, M. Figarella, C. and Ghendrih, Ph., 2004, Controlling chaotic transport in a Hamiltoman model of interest to magetized plasmas, J. Phys. A. Math. Gen., 37, [3] Ciraolo, G., Chandre, C., Lima, R., Vittot, M., Pettini, M. Figarella, C. and Ghendrih, Ph., 2004b, Contol of Hamiltonian chaos as a possible tool to control anomalous transport in fusion plasmas. Phy. Rev. E 69(4, [4] Sharma, S. and Narayan, A., 2001, Non-Linear Oscillation of interconnected satellites system under the combined influence of the solar radiation pressure and dissipative force of general nature, Bull. Astr. Soc., India, 29, [5] Vittot, M., 2004, Perturbation Theory and Control in Classical or quantum Mechanics by an inversion formula, J. Phys. A. Math. Gen., 37:
9 Chaotic motion in problem of dumbell satellite 307 Received: January, 2010
Control of chaos in Hamiltonian systems
Control of chaos in Hamiltonian systems G. Ciraolo, C. Chandre, R. Lima, M. Vittot Centre de Physique Théorique CNRS, Marseille M. Pettini Osservatorio Astrofisico di Arcetri, Università di Firenze Ph.
More informationControlling chaotic transport in Hamiltonian systems
Controlling chaotic transport in Hamiltonian systems Guido Ciraolo Facoltà di Ingegneria, Università di Firenze via S. Marta, I-50129 Firenze, Italy Cristel Chandre, Ricardo Lima, Michel Vittot CPT-CNRS,
More informationThe in-plane Motion of a Geosynchronous Satellite under the Gravitational Attraction of the Sun, the Moon and the Oblate Earth
J Astrophys Astr (1990) 11, 1 10 The in-plane Motion of a Geosynchronous Satellite under the Gravitational Attraction of the Sun, the Moon and the Oblate Earth Κ Β Bhatnagar Department of Mathematics,
More informationarxiv:nlin/ v1 [nlin.cd] 17 Dec 2003
Control of Hamiltonian chaos as a possible tool to control anomalous transport in fusion plasmas Guido Ciraolo Facoltà di Ingegneria, Università di Firenze, via S. Marta, I-529 Firenze, Italy, and I.N.F.M.
More informationRegular n-gon as a model of discrete gravitational system. Rosaev A.E. OAO NPC NEDRA, Jaroslavl Russia,
Regular n-gon as a model of discrete gravitational system Rosaev A.E. OAO NPC NEDRA, Jaroslavl Russia, E-mail: hegem@mail.ru Introduction A system of N points, each having mass m, forming a planar regular
More informationANNEX 1. DEFINITION OF ORBITAL PARAMETERS AND IMPORTANT CONCEPTS OF CELESTIAL MECHANICS
ANNEX 1. DEFINITION OF ORBITAL PARAMETERS AND IMPORTANT CONCEPTS OF CELESTIAL MECHANICS A1.1. Kepler s laws Johannes Kepler (1571-1630) discovered the laws of orbital motion, now called Kepler's laws.
More informationM2A2 Problem Sheet 3 - Hamiltonian Mechanics
MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian,
More informationA Lie Algebra version of the Classical or Quantum Hamiltonian Perturbation Theory and Hamiltonian Control with Examples in Plasma Physics
A Lie Algebra version of the Classical or Quantum Hamiltonian Perturbation Theory and Hamiltonian Control with Examples in Plasma Physics Michel VITTOT Center for Theoretical Physics (CPT), CNRS - Luminy
More informationOrbit Characteristics
Orbit Characteristics We have shown that the in the two body problem, the orbit of the satellite about the primary (or vice-versa) is a conic section, with the primary located at the focus of the conic
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationGeneral classifications:
General classifications: Physics is perceived as fundamental basis for study of the universe Chemistry is perceived as fundamental basis for study of life Physics consists of concepts, principles and notions,
More informationTheory of mean motion resonances.
Theory of mean motion resonances. Mean motion resonances are ubiquitous in space. They can be found between planets and asteroids, planets and rings in gaseous disks or satellites and planetary rings.
More informationTheoretical physics. Deterministic chaos in classical physics. Martin Scholtz
Theoretical physics Deterministic chaos in classical physics Martin Scholtz scholtzzz@gmail.com Fundamental physical theories and role of classical mechanics. Intuitive characteristics of chaos. Newton
More informationLecture 2c: Satellite Orbits
Lecture 2c: Satellite Orbits Outline 1. Newton s Laws of Mo3on 2. Newton s Law of Universal Gravita3on 3. Kepler s Laws 4. Pu>ng Newton and Kepler s Laws together and applying them to the Earth-satellite
More informationAnalysis of Lunisolar Resonances. in an Artificial Satellite Orbits
Applied Mathematical Sciences, Vol., 008, no., 0 0 Analysis of Lunisolar Resonances in an Artificial Satellite Orbits F. A. Abd El-Salam, Yehia A. Abdel-Aziz,*, M. El-Saftawy, and M. Radwan Cairo university,
More informationThe restricted, circular, planar three-body problem
The restricted, circular, planar three-body problem Luigi Chierchia Dipartimento di Matematica Università Roma Tre Largo S L Murialdo 1, I-00146 Roma (Italy) (luigi@matuniroma3it) March, 2005 1 The restricted
More informationLunisolar Secular Resonances
Lunisolar Secular Resonances Jessica Pillow Supervisor: Dr. Aaron J. Rosengren December 15, 2017 1 Introduction The study of the dynamics of objects in Earth s orbit has recently become very popular in
More informationDIFFUSION OF ASTEROIDS IN MEAN MOTION RESONANCES
DIFFUSION OF ASTEROIDS IN MEAN MOTION RESONANCES KLEOMENIS TSIGANIS and HARRY VARVOGLIS Section of Astrophysics, Astronomy and Mechanics, Department of Physics, University of Thessaloniki, 540 06 Thessaloniki,
More informationChaotic motion. Phys 750 Lecture 9
Chaotic motion Phys 750 Lecture 9 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t =0to
More informationSymplectic Correctors for Canonical Heliocentric N-Body Maps
Symplectic Correctors for Canonical Heliocentric N-Body Maps J. Wisdom Massachusetts Institute of Technology, Cambridge, MA 02139 wisdom@poincare.mit.edu Received ; accepted 2 ABSTRACT Symplectic correctors
More informationANALYTICAL MECHANICS. LOUIS N. HAND and JANET D. FINCH CAMBRIDGE UNIVERSITY PRESS
ANALYTICAL MECHANICS LOUIS N. HAND and JANET D. FINCH CAMBRIDGE UNIVERSITY PRESS Preface xi 1 LAGRANGIAN MECHANICS l 1.1 Example and Review of Newton's Mechanics: A Block Sliding on an Inclined Plane 1
More informationTHIRD-BODY PERTURBATION USING A SINGLE AVERAGED MODEL
INPE-1183-PRE/67 THIRD-BODY PERTURBATION USING A SINGLE AVERAGED MODEL Carlos Renato Huaura Solórzano Antonio Fernando Bertachini de Almeida Prado ADVANCES IN SPACE DYNAMICS : CELESTIAL MECHANICS AND ASTRONAUTICS,
More informationQuantum Theory and Group Representations
Quantum Theory and Group Representations Peter Woit Columbia University LaGuardia Community College, November 1, 2017 Queensborough Community College, November 15, 2017 Peter Woit (Columbia University)
More informationDynamical properties of the Solar System. Second Kepler s Law. Dynamics of planetary orbits. ν: true anomaly
First Kepler s Law The secondary body moves in an elliptical orbit, with the primary body at the focus Valid for bound orbits with E < 0 The conservation of the total energy E yields a constant semi-major
More informationPC 1141 : AY 2012 /13
NUS Physics Society Past Year Paper Solutions PC 1141 : AY 2012 /13 Compiled by: NUS Physics Society Past Year Solution Team Yeo Zhen Yuan Ryan Goh Published on: November 17, 2015 1. An egg of mass 0.050
More informationTwo models for the parametric forcing of a nonlinear oscillator
Nonlinear Dyn (007) 50:147 160 DOI 10.1007/s11071-006-9148-3 ORIGINAL ARTICLE Two models for the parametric forcing of a nonlinear oscillator Nazha Abouhazim Mohamed Belhaq Richard H. Rand Received: 3
More informationStudy of the Restricted Three Body Problem When One Primary Is a Uniform Circular Disk
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Applications and Applied Mathematics: An International Journal (AAM) Vol. 3, Issue (June 08), pp. 60 7 Study of the Restricted Three Body
More informationIntroduction to Applied Nonlinear Dynamical Systems and Chaos
Stephen Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos Second Edition With 250 Figures 4jj Springer I Series Preface v L I Preface to the Second Edition vii Introduction 1 1 Equilibrium
More informationPreliminary Examination - Day 2 May 16, 2014
UNL - Department of Physics and Astronomy Preliminary Examination - Day May 6, 04 This test covers the topics of Thermodynamics and Statistical Mechanics (Topic ) and Mechanics (Topic ) Each topic has
More informationPADEU. Pulsating zero velocity surfaces and capture in the elliptic restricted three-body problem. 1 Introduction
PADEU PADEU 15, 221 (2005) ISBN 963 463 557 c Published by the Astron. Dept. of the Eötvös Univ. Pulsating zero velocity surfaces and capture in the elliptic restricted three-body problem F. Szenkovits
More informationPenning Traps. Contents. Plasma Physics Penning Traps AJW August 16, Introduction. Clasical picture. Radiation Damping.
Penning Traps Contents Introduction Clasical picture Radiation Damping Number density B and E fields used to increase time that an electron remains within a discharge: Penning, 936. Can now trap a particle
More informationHamiltonian aspects of fluid dynamics
Hamiltonian aspects of fluid dynamics CDS 140b Joris Vankerschaver jv@caltech.edu CDS 01/29/08, 01/31/08 Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 1 / 34 Outline
More informationAdvanced Newtonian gravity
Foundations of Newtonian gravity Solutions Motion of extended bodies, University of Guelph h treatment of Newtonian gravity, the book develops approximation methods to obtain weak-field solutions es the
More informationHamiltonian formulation of reduced Vlasov-Maxwell equations
Hamiltonian formulation of reduced Vlasov-Maxell equations Cristel CHANDRE Centre de Physique Théorique CNRS, Marseille, France Contact: chandre@cpt.univ-mrs.fr importance of stability vs instability in
More information= 0. = q i., q i = E
Summary of the Above Newton s second law: d 2 r dt 2 = Φ( r) Complicated vector arithmetic & coordinate system dependence Lagrangian Formalism: L q i d dt ( L q i ) = 0 n second-order differential equations
More informationChaotic motion. Phys 420/580 Lecture 10
Chaotic motion Phys 420/580 Lecture 10 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t
More informationψ s a ˆn a s b ˆn b ψ Hint: Because the state is spherically symmetric the answer can depend only on the angle between the two directions.
1. Quantum Mechanics (Fall 2004) Two spin-half particles are in a state with total spin zero. Let ˆn a and ˆn b be unit vectors in two arbitrary directions. Calculate the expectation value of the product
More informationA path integral approach to the Langevin equation
A path integral approach to the Langevin equation - Ashok Das Reference: A path integral approach to the Langevin equation, A. Das, S. Panda and J. R. L. Santos, arxiv:1411.0256 (to be published in Int.
More informationPhysics 106b: Lecture 7 25 January, 2018
Physics 106b: Lecture 7 25 January, 2018 Hamiltonian Chaos: Introduction Integrable Systems We start with systems that do not exhibit chaos, but instead have simple periodic motion (like the SHO) with
More informationPhase Synchronization
Phase Synchronization Lecture by: Zhibin Guo Notes by: Xiang Fan May 10, 2016 1 Introduction For any mode or fluctuation, we always have where S(x, t) is phase. If a mode amplitude satisfies ϕ k = ϕ k
More informationUse conserved quantities to reduce number of variables and the equation of motion (EOM)
Physics 106a, Caltech 5 October, 018 Lecture 8: Central Forces Bound States Today we discuss the Kepler problem of the orbital motion of planets and other objects in the gravitational field of the sun.
More informationHamiltonian Dynamics
Hamiltonian Dynamics CDS 140b Joris Vankerschaver jv@caltech.edu CDS Feb. 10, 2009 Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 1 / 31 Outline 1. Introductory concepts; 2. Poisson brackets;
More informationLecture 41: Highlights
Lecture 41: Highlights The goal of this lecture is to remind you of some of the key points that we ve covered this semester Note that this is not the complete set of topics that may appear on the final
More informationAnalysis of frozen orbits for solar sails
Trabalho apresentado no XXXV CNMAC, Natal-RN, 2014. Analysis of frozen orbits for solar sails J. P. S. Carvalho, R. Vilhena de Moraes, Instituto de Ciência e Tecnologia, UNIFESP, São José dos Campos -
More informationUnder evolution for a small time δt the area A(t) = q p evolves into an area
Physics 106a, Caltech 6 November, 2018 Lecture 11: Hamiltonian Mechanics II Towards statistical mechanics Phase space volumes are conserved by Hamiltonian dynamics We can use many nearby initial conditions
More informationQuestion 1: Spherical Pendulum
Question 1: Spherical Pendulum Consider a two-dimensional pendulum of length l with mass M at its end. It is easiest to use spherical coordinates centered at the pivot since the magnitude of the position
More informationTangent and Normal Vectors
Tangent and Normal Vectors MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Navigation When an observer is traveling along with a moving point, for example the passengers in
More informationSymbolic Solution of Kepler s Generalized Equation
Symbolic Solution of Kepler s Generalized Equation Juan Félix San-Juan 1 and Alberto Abad 1 Universidad de La Rioja, 6004 Logroño, Spain juanfelix.sanjuan@dmc.unirioja.es, Grupo de Mecánica Espacial, Universidad
More informationThe two body problem involves a pair of particles with masses m 1 and m 2 described by a Lagrangian of the form:
Physics 3550, Fall 2011 Two Body, Central-Force Problem Relevant Sections in Text: 8.1 8.7 Two Body, Central-Force Problem Introduction. I have already mentioned the two body central force problem several
More informationTowards stability results for planetary problems with more than three bodies
Towards stability results for planetary problems with more than three bodies Ugo Locatelli [a] and Marco Sansottera [b] [a] Math. Dep. of Università degli Studi di Roma Tor Vergata [b] Math. Dep. of Università
More information16. GAUGE THEORY AND THE CREATION OF PHOTONS
6. GAUGE THEORY AD THE CREATIO OF PHOTOS In the previous chapter the existence of a gauge theory allowed the electromagnetic field to be described in an invariant manner. Although the existence of this
More informationAnalytical Estimation of Time Dilation of a Satellite in Elliptical Orbit
Analytical Estimation of Time Dilation of a Satellite in Elliptical Orbit G. PRASAD*,1, P. MANIGANDAN *Corresponding author *,1 Department of Aeronautical Engineering, Bannari amman Institute of Technology,
More informationIllustrating Dynamical Symmetries in Classical Mechanics: The Laplace-Runge-Lenz Vector Revisited
Illustrating Dynamical Symmetries in Classical Mechanics: The Laplace-Runge-Lenz Vector Revisited Ross C. O Connell and Kannan Jagannathan Physics Department, Amherst College Amherst, MA 01002-5000 Abstract
More informationA SEMI-ANALYTICAL ORBIT PROPAGATOR PROGRAM FOR HIGHLY ELLIPTICAL ORBITS
A SEMI-ANALYTICAL ORBIT PROPAGATOR PROGRAM FOR HIGHLY ELLIPTICAL ORBITS M. Lara, J. F. San Juan and D. Hautesserres Scientific Computing Group and Centre National d Études Spatiales 6th International Conference
More information( r) = 1 Z. e Zr/a 0. + n +1δ n', n+1 ). dt ' e i ( ε n ε i )t'/! a n ( t) = n ψ t = 1 i! e iε n t/! n' x n = Physics 624, Quantum II -- Exam 1
Physics 624, Quantum II -- Exam 1 Please show all your work on the separate sheets provided (and be sure to include your name) You are graded on your work on those pages, with partial credit where it is
More informationPrevious Lecture. The Von Zeipel Method. Application 1: The Brouwer model. Application 2: The Cid-Lahulla model. Simplified Brouwer transformation.
2 / 36 Previous Lecture The Von Zeipel Method. Application 1: The Brouwer model. Application 2: The Cid-Lahulla model. Simplified Brouwer transformation. Review of Analytic Models 3 / 36 4 / 36 Review:
More informationIntroduction. Chapter Plasma: definitions
Chapter 1 Introduction 1.1 Plasma: definitions A plasma is a quasi-neutral gas of charged and neutral particles which exhibits collective behaviour. An equivalent, alternative definition: A plasma is a
More informationEva Miranda. UPC-Barcelona and BGSMath. XXV International Fall Workshop on Geometry and Physics Madrid
b-symplectic manifolds: going to infinity and coming back Eva Miranda UPC-Barcelona and BGSMath XXV International Fall Workshop on Geometry and Physics Madrid Eva Miranda (UPC) b-symplectic manifolds Semptember,
More informationThe quantum effects on all Lagrangian points and prospects to measure them in the Earth-Moon system.
The quantum effects on all Lagrangian points and prospects to measure them in the Earth-Moon system. Emmanuele Battista Simone Dell Agnello Giampiero Esposito Jules Simo Outline Restricted three-body problem.
More informationStability of circular orbits
Stability of circular orbits Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2011 September 12, 2011 Angular motion Since the radial Hamiltonian is H = p 2 /2m+U(r), the radial equation of motion
More information1 Summary of Chapter 2
General Astronomy (9:61) Fall 01 Lecture 7 Notes, September 10, 01 1 Summary of Chapter There are a number of items from Chapter that you should be sure to understand. 1.1 Terminology A number of technical
More informationThe Restricted Three-Body Problem : Earth,Jupitar,Sun
The Restricted Three-Body Problem : Earth,Jupitar,Sun Karla Boucher Phys349 Hamiltonian Dynamics March 2004 Abstract How Jupitar s gravitational pull affects the Earth s motion around the sun is known
More informationMotion under the Influence of a Central Force
Copyright 004 5 Motion under the Influence of a Central Force The fundamental forces of nature depend only on the distance from the source. All the complex interactions that occur in the real world arise
More informationGEOMETRIC QUANTIZATION
GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical
More informationPhysics of Leap Second. Takehisa Fujita
Physics of Leap Second Takehisa Fujita 1 Contents [1] Newcomb Time and Atomic Watch [2] Origin of Leap Second [3] Earth s Rotation and Tidal Force [4] Work of Non-conservative Force [5] Leap Second: Prediction
More informationOn the Estimated Precession of Mercury s Orbit
1 On the Estimated Precession of Mercury s Orbit R. Wayte. 9 Audley Way, Ascot, Berkshire, SL5 8EE, England, UK e-mail: rwayte@googlemail.com Research Article, Submitted to PMC Physics A 4 Nov 009 Abstract.
More information06. Lagrangian Mechanics II
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 06. Lagrangian Mechanics II Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationCanonical transformations (Lecture 4)
Canonical transformations (Lecture 4) January 26, 2016 61/441 Lecture outline We will introduce and discuss canonical transformations that conserve the Hamiltonian structure of equations of motion. Poisson
More informationTheory of Adiabatic Invariants A SOCRATES Lecture Course at the Physics Department, University of Marburg, Germany, February 2004
Preprint CAMTP/03-8 August 2003 Theory of Adiabatic Invariants A SOCRATES Lecture Course at the Physics Department, University of Marburg, Germany, February 2004 Marko Robnik CAMTP - Center for Applied
More informationINTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
IC/90/124.-- INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS CENTRIFUGAL FORCE IN ERNST SPACETIME A.R.Prasanna INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION
More informationLecture Tutorial: Angular Momentum and Kepler s Second Law
2017 Eclipse: Research-Based Teaching Resources Lecture Tutorial: Angular Momentum and Kepler s Second Law Description: This guided inquiry paper-and-pencil activity helps students to describe angular
More informationPeriodic Orbits in the Photogravitational Elliptic Restricted Three-Body Problem
54 Advances in Astrophysics, Vol., No., August 8 https://dx.doi.org/.66/adap.8.4 Periodic Orbits in the Photogravitational Elliptic Restricted Three-Body Problem Y. SHARON RUTH, RAM KRISHAN SHARMA Department
More informationAnalysis of Relative Motion of Collocated Geostationary Satellites with Geometric Constraints
www.dlr.de Chart 1 Analysis of Relative Motion of Collocated Geostationary Satellites with Geometric Constraints SFFMT2013, München F. de Bruijn & E. Gill 31 May 2013 www.dlr.de Chart 2 Presentation Outline
More informationTranscendental cases in stability problem. Hamiltonian systems
of Hamiltonian systems Boris S. Bardin Moscow Aviation Institute (Technical University) Faculty of Applied Mathematics and Physics Department of Theoretical Mechanics Hamiltonian Dynamics and Celestial
More informationAnalytical Method for Space Debris propagation under perturbations in the geostationary ring
Analytical Method for Space Debris propagation under perturbations in the geostationary ring July 21-23, 2016 Berlin, Germany 2nd International Conference and Exhibition on Satellite & Space Missions Daniel
More informationResearch Article Stability Analysis of Journal Bearing: Dynamic Characteristics
Research Journal of Applied Sciences, Engineering and Technology 9(1): 47-52, 2015 DOI:10.19026/rjaset.9.1375 ISSN: 2040-7459; e-issn: 2040-7467 2015 Maxwell Scientific Publication Corp. Submitted: July
More informationarxiv:hep-th/ v1 8 Mar 1995
GALILEAN INVARIANCE IN 2+1 DIMENSIONS arxiv:hep-th/9503046v1 8 Mar 1995 Yves Brihaye Dept. of Math. Phys. University of Mons Av. Maistriau, 7000 Mons, Belgium Cezary Gonera Dept. of Physics U.J.A Antwerpen,
More informationRemarks on Quadratic Hamiltonians in Spaceflight Mechanics
Remarks on Quadratic Hamiltonians in Spaceflight Mechanics Bernard Bonnard 1, Jean-Baptiste Caillau 2, and Romain Dujol 2 1 Institut de mathématiques de Bourgogne, CNRS, Dijon, France, bernard.bonnard@u-bourgogne.fr
More informationClassical Mechanics Ph.D. Qualifying Examination. 8 January, :00 to 12:00
UNIVERSITY OF ILLINOIS AT CHICAGO DEPARTMENT OF PHYSICS Classical Mechanics Ph.D. Qualifying Examination 8 January, 2013 9:00 to 12:00 Full credit can be achieved from completely correct answers to 4 questions.
More informationHAMILTONIAN STABILITY OF SPIN ORBIT RESONANCES IN CELESTIAL MECHANICS. 1. Introduction
HAMILTONIAN STABILITY OF SPIN ORBIT RESONANCES IN CELESTIAL MECHANICS ALESSANDRA CELLETTI 1 and LUIGI CHIERCHIA 2 1 Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica,
More informationTopic 6 The Killers LEARNING OBJECTIVES. Topic 6. Circular Motion and Gravitation
Topic 6 Circular Motion and Gravitation LEARNING OBJECTIVES Topic 6 The Killers 1. Centripetal Force 2. Newton s Law of Gravitation 3. Gravitational Field Strength ROOKIE MISTAKE! Always remember. the
More informationGravitation. Kepler s Law. BSc I SEM II (UNIT I)
Gravitation Kepler s Law BSc I SEM II (UNIT I) P a g e 2 Contents 1) Newton s Law of Gravitation 3 Vector representation of Newton s Law of Gravitation 3 Characteristics of Newton s Law of Gravitation
More informationSome Collision solutions of the rectilinear periodically forced Kepler problem
Advanced Nonlinear Studies 1 (2001), xxx xxx Some Collision solutions of the rectilinear periodically forced Kepler problem Lei Zhao Johann Bernoulli Institute for Mathematics and Computer Science University
More informationPage 684. Lecture 40: Coordinate Transformations: Time Transformations Date Revised: 2009/02/02 Date Given: 2009/02/02
Page 684 Lecture 40: Coordinate Transformations: Time Transformations Date Revised: 2009/02/02 Date Given: 2009/02/02 Time Transformations Section 12.5 Symmetries: Time Transformations Page 685 Time Translation
More informationSection B. Electromagnetism
Prelims EM Spring 2014 1 Section B. Electromagnetism Problem 0, Page 1. An infinite cylinder of radius R oriented parallel to the z-axis has uniform magnetization parallel to the x-axis, M = m 0ˆx. Calculate
More informationPhysics 576 Stellar Astrophysics Prof. James Buckley. Lecture 13 Thermodynamics of QM particles
Physics 576 Stellar Astrophysics Prof. James Buckley Lecture 13 Thermodynamics of QM particles Reading/Homework Assignment Read chapter 3 in Rose. Midterm Exam, April 5 (take home) Final Project, May 4
More informationPentahedral Volume, Chaos, and Quantum Gravity
Pentahedral Volume, Chaos, and Quantum Gravity Hal Haggard May 30, 2012 Volume Polyhedral Volume (Bianchi, Doná and Speziale): ˆV Pol = The volume of a quantum polyhedron Outline 1 Pentahedral Volume 2
More informationA = 6561 times greater. B. 81 times greater. C. equally strong. D. 1/81 as great. E. (1/81) 2 = 1/6561 as great Pearson Education, Inc.
Q13.1 The mass of the Moon is 1/81 of the mass of the Earth. Compared to the gravitational force that the Earth exerts on the Moon, the gravitational force that the Moon exerts on the Earth is A. 81 2
More informationFundamentals of Satellite technology
Fundamentals of Satellite technology Prepared by A.Kaviyarasu Assistant Professor Department of Aerospace Engineering Madras Institute Of Technology Chromepet, Chennai Orbital Plane All of the planets,
More informationLectures on A coherent dual vector field theory for gravitation.
Lectures on coherent dual vector field theory for gravitation. The purpose of these lectures is to get more familiarized with gyrotation concepts and with its applications. Lecture : a word on the Maxwell
More informationCELESTIAL MECHANICS. Part I. Mathematical Preambles
Chapter 1. Numerical Methods CELESTIAL MECHANICS Part I. Mathematical Preambles 1.1 Introduction 1.2 Numerical Integration 1.3 Quadratic Equations 1.4 The Solution of f(x) = 0 1.5 The Solution of Polynomial
More information2015 Lloyd G. Elliott University Prize Exam Compiled by the Department of Physics & Astronomy, University of Manitoba
Canadian Association of Physicists SUPPORTING PHYSICS RESEARCH AND EDUCATION IN CANADA 2015 Lloyd G. Elliott University Prize Exam Compiled by the Department of Physics & Astronomy, University of Manitoba
More informationCelestial Mechanics II. Orbital energy and angular momentum Elliptic, parabolic and hyperbolic orbits Position in the orbit versus time
Celestial Mechanics II Orbital energy and angular momentum Elliptic, parabolic and hyperbolic orbits Position in the orbit versus time Orbital Energy KINETIC per unit mass POTENTIAL The orbital energy
More informationA plane autonomous system is a pair of simultaneous first-order differential equations,
Chapter 11 Phase-Plane Techniques 11.1 Plane Autonomous Systems A plane autonomous system is a pair of simultaneous first-order differential equations, ẋ = f(x, y), ẏ = g(x, y). This system has an equilibrium
More informationM02M.1 Particle in a Cone
Part I Mechanics M02M.1 Particle in a Cone M02M.1 Particle in a Cone A small particle of mass m is constrained to slide, without friction, on the inside of a circular cone whose vertex is at the origin
More information1. Electricity and Magnetism (Fall 1995, Part 1) A metal sphere has a radius R and a charge Q.
1. Electricity and Magnetism (Fall 1995, Part 1) A metal sphere has a radius R and a charge Q. (a) Compute the electric part of the Maxwell stress tensor T ij (r) = 1 {E i E j 12 } 4π E2 δ ij both inside
More informationA study upon Eris. I. Describing and characterizing the orbit of Eris around the Sun. I. Breda 1
Astronomy & Astrophysics manuscript no. Eris c ESO 2013 March 27, 2013 A study upon Eris I. Describing and characterizing the orbit of Eris around the Sun I. Breda 1 Faculty of Sciences (FCUP), University
More information( ) /, so that we can ignore all
Physics 531: Atomic Physics Problem Set #5 Due Wednesday, November 2, 2011 Problem 1: The ac-stark effect Suppose an atom is perturbed by a monochromatic electric field oscillating at frequency ω L E(t)
More informationHamilton-Jacobi theory on Lie algebroids: Applications to nonholonomic mechanics. Manuel de León Institute of Mathematical Sciences CSIC, Spain
Hamilton-Jacobi theory on Lie algebroids: Applications to nonholonomic mechanics Manuel de León Institute of Mathematical Sciences CSIC, Spain joint work with J.C. Marrero (University of La Laguna) D.
More informationLecture 1: Oscillatory motions in the restricted three body problem
Lecture 1: Oscillatory motions in the restricted three body problem Marcel Guardia Universitat Politècnica de Catalunya February 6, 2017 M. Guardia (UPC) Lecture 1 February 6, 2017 1 / 31 Outline of the
More information