Elshaboury SM et al.; Sch. J. Phys. Math. Stat., 2015; Vol-2; Issue-2B (Mar-May); pp
|
|
- Isaac Harmon
- 5 years ago
- Views:
Transcription
1 Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp Scholars Journal of Phscs Mathematcs Statstcs Sch. J. Phs. Math. Stat. 5; (B):69-75 Scholars Academc Scentfc Publshers (SAS Publshers) (An Internatonal Publsher for Academc Scentfc Resources) ISSN (Prnt) ISSN (Onlne) The Stablt of the Trangular Ponts of the Restrcted three Bod Problem when both the Prmares are Traal Rgd Bodes S. M. Elshabour M.R. Amn Department of Math. Facult of Scence An Shams Unverst Caro Egpt Departement of theoretcal phscs Natonal Research Centre Caro Egpt. *Correspondng Author: M.R. Amn Emal: Abstract: The locaton the stablt of the trangular ponts of the planar restrcted three bod problem have been dscussed when both the prmares are traal rgd bodes consderng the case of statonar rotatonal moton of the bgger prmar of the smaller prmar are respectvel. Kewords: restrcted three bod problem; trangular ponts; traal rgd bodes. INTRODUCTION The problem of stablt condtons of trangular lbraton ponts was assumed b Gascheau [] then b Routh[5]. In recent tmes man perturbng forces.e. oblateness radaton forces of the prmares Corols centrfugal forces etc. have been ncluded n the stud of the restrcted three bod problem. Bhatnagar Gupta[] show the estence of 6 statonar motons each correspondng to the constant values of the non-cclc generalzed coordnates thus dependng on the Euleran angles of both the bodes. Khanna Bhatnagar [] have studed the problem when the smaller prmar s a traal rgd bod. Also Sharma et.al.[5] have studed the problem when both the prmares are traal rgd bodes n the case of statonar rotatonal moton are small quanttes. In ths paper we consder the restrcted three bod problem when both the prmares are traal rgd bodes wth the statonar rotatonal moton of the bgger prmar of the smaller prmar. EQUATIONS OF MOTION We shall adopt the notaton termnolog of Szebehl [7]. As a consequence the dstance between the prmares does not change s taken equal to one; the sum of masses of the prmares s also taken one. The unt of tme s chosen so as to make the gravtatonal constant unt. Besdes ths the prncple aes of the prmares are orented to the snodc aes b Euler's angels. Snce the aes are supposed to rotate wth the same angular veloct as that of the rgd bodes the bodes are movng around ther center of mass wthout rotaton the Euler's angles reman constant throughout the moton. Usng dmensonless varables Avalable Onlne: 69
2 Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp Fg-: Left: the crcular restrcted three bod problem n the snodcal reference sstem wth a dmensonal unts. Rght: the fve equlbrum ponts assocated wth the problem. the equatons of moton of the nfntesmal mass m n a snodc coordnate sstem are... n... n n r r I I I I r r m r I ' I ' I ' I ' mr () () r r () Here s the rato of mass of the smaller prmar to the total mass of prmares.e. m wth m m beng the masses of the prmares. m m I I I are the prncpal moments of nerta of the traal rgd bod of mass m at ts center of mass wth a b c as ts aes. I s the moment of nerta about a lne jonng the center of the rgd bod of mass m the nfntesmal bod of mass m s gven b I I l I m I n ' ' ' l m n are the drectonal cosnes of the lne respect to ts prncpal aes. ' ' ' Avalable Onlne: 7
3 Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp I ' I ' I ' are the prncpal moments of nerta of the traal rgd bod of mass m at ts center of mass wth a ' b ' c ' as ts aes. I' s the moment of nerta about a lne jonng the center of the rgd bod of mass m the nfntesmal bod of mass m s gven b I I l I m I n ' ' ' ' ' ' ' l ' m ' n ' are the drectonal cosnes of the lne respect to ts prncpal aes. We denote the unt vectors along the prncple aes at por p b jk the unt vectors parallel to the snodc aes b IJK wth the help of Euler's angles. The are connected b Snge Grffth (7) (959) I a b jc k J a b jc k K a b jc k a Sn Sn Cos Cos Cos a Cos Sn Cos Sn Cos a Sn Cos b Sn Cos Cos Cos Sn b Cos Cos Cos Sn Sn b Sn Sn c Sn Cos c Sn Sn c Cos. The aes O z have been defned b Szebehel [7]. Now n equaton () can be wrtten as n r n r r r ' ' A' A' A' A' A' b r r ' ' Avalable Onlne: 7 A A a a A A A A A b b r r A A c c A a A b A c 5R 5R 5R A A a a b A A c c (4)
4 Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp a ' b ' c ' A ' A ' A ' (5) 5R 5R 5R R s the dstance between the prmares. The mean moton n s gven b n A A A a A A b A A c A A (6) A ' ' ' A A a A ' ' A b A ' ' A c A ' ' A. Equaton () permt an ntegral analogous to Jacob ntegral & & C. The lberaton ponts are the sngulartes of the manfold && & & f C. Therefore these ponts are the solutons of the equatons. are establshed b Sharma [4]. Let the other elements are equal to zero; We have a b c a b c the other elements are equal to zero n r n r r r r r of the bgger prmar n ths case of the smaller prmar A A A A A A A r r A' 4 A' A' A' A' A' A' r r n r n r r r r r 5 5 A 4A 4A A A A A r r A' 4 A' A' A' A' A' A' r r n A A A A' A' A'. (7) TRIANGULAR LIBERATION POINTS The trangular lberaton ponts are the solutons of the equaton (7). A A' are equal to zero we smpl get r r If the values of. When ' A A aren t equal to zero we suppose that r r =. (8) Puttng the values of r r from equaton (8) n equaton () we get Avalable Onlne: 7
5 Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp (9) Puttng the values of r r from equaton (8) from equaton (7) rejectng hgher order terms we get A A 7 A 4 A A A A 4 A A 6 A A () STABILITY ANALYSIS Assumng denote small dsplacement of the nfntesmal partcle from the equlbrum ponts. X X Y Y () Now o o Epng b talor s epanson consderng onl frst orders we have Where s the value of at the pont ( ) smlarl the other values the respectve values at the ponts ( ). At the equlbrum ponts we have () are Hence the equaton of moton of the nfntesmal partcle s d d n dt dt () d d n. dt dt In order to solve equaton (6) substtute t t Ae Be (4) Where AB λ are parameters. Ths gves that t t A e Bn e (5) t t A n e B e. The set of equaton (5) has nontrval soluton f Where n 4 4 (6) s defned at L 4 when the prmares are traal rgd bodes as o o o A 4A 7A o n 4A 4A 7A (7) Avalable Onlne: 7
6 Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp o 5 5 n 7A 68A 4A (8) 68AA 4AA 7AA o A 8A 7A (9) 5 8A 7A A 6 We can rewrte equaton (6) as P Q () Where P 4 n Q = The stablt of the trangular ponts requres that = must be negatve to obtan pure magnar roots.e. the dscrnnant of equaton (7) s P 4Qthat s the condton for stablt mples that: ( 7 7 ) ( ) A ( ) A 8 4 ( ) A ( 5 ) A () 8 ( ) A ( ) A 6 6 If A A ( = ) are equal to zero then the stablt condton s Szebehl [7]. And f A A ( = ) are not equal to zero we suppose that crt. p A p A p A p4 A p5 A p6 A p p p p4 p5 p 6 are to be determned therefore we have p p 7( ) 6( ) p p p4 7( ) 8( ) p6 44( ) 44( ) And the stablt condton for the trangular ponts s crt A A A A A. 446 A. () CONCLUSION Avalable Onlne: 74
7 Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp We construct the locaton the stablt condton of the trangular ponts of the restrcted three bod problem wth traal prmares consderng a statonar rotatonal moton of the bgger prmar of the smaller prmar we conclude that the stablt condton s depend on that orentatons. Also we found the stablt condton n our partcular case. REFERENCES. Bhatnagar KB Gupta U; The estence stablt of the equlbrum ponts of a traal rgd bod movng around another traal rgd bod. Celestal mechancs 986; 9(): Gascheau M; Eamen d'une class d'equatons dfférentelles et applcaton a un cas partculer du probleme des tros corps. Comptes Rendus 84; 6:9 94. Khanna MONA Bhatnagar KB; Estence stablt of lbraton ponts n the restrcted three bod problem when the smaller prmar s a traal rgd bod the bgger one an oblate spherod. Indan Journal of Pure Appled Mathematcs 999; (): Sharma RK Taqv ZA Bhatnagar KB; Estence of lbraton ponts n the restrcted three bod problem when both the prmares are traal rgd bodes. Indan journal of pure appled mathematcs ;(): Routh EJ; Proc. Lond. Math. Soc 875; 6: Snge Grffth; Prncples of Mechancs Thrd Edton Mc Graw-Hll Szbehel V; Theor of Orbts Academc Press New York 967. Avalable Onlne: 75
Mathematical Applications in Modern Science
Effect of Perturbatons n the Corols Centrfugal Forces on the Locaton Stablty of the Equlbrum Solutons n Robe s Restrcted Problem of + Bodes BHAVNEET KAUR Lady Shr Ram College for Women Department of Mathematcs
More informationSCALARS AND VECTORS All physical quantities in engineering mechanics are measured using either scalars or vectors.
SCALARS AND ECTORS All phscal uanttes n engneerng mechancs are measured usng ether scalars or vectors. Scalar. A scalar s an postve or negatve phscal uantt that can be completel specfed b ts magntude.
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationSo far: simple (planar) geometries
Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector
More informationWeek 9 Chapter 10 Section 1-5
Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informationOne Dimensional Axial Deformations
One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the
More informationChapter 11: Angular Momentum
Chapter 11: ngular Momentum Statc Equlbrum In Chap. 4 we studed the equlbrum of pontobjects (mass m) wth the applcaton of Newton s aws F 0 F x y, 0 Therefore, no lnear (translatonal) acceleraton, a0 For
More informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O
More informationExponential Type Product Estimator for Finite Population Mean with Information on Auxiliary Attribute
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 193-9466 Vol. 10, Issue 1 (June 015), pp. 106-113 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) Exponental Tpe Product Estmator
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationPlate Theories for Classical and Laminated plates Weak Formulation and Element Calculations
Plate heores for Classcal and Lamnated plates Weak Formulaton and Element Calculatons PM Mohte Department of Aerospace Engneerng Indan Insttute of echnolog Kanpur EQIP School on Computatonal Methods n
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationCelestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestial-mechanics - J. Hedberg
PHY 454 - celestal-mechancs - J. Hedberg - 207 Celestal Mechancs. Basc Orbts. Why crcles? 2. Tycho Brahe 3. Kepler 4. 3 laws of orbtng bodes 2. Newtonan Mechancs 3. Newton's Laws. Law of Gravtaton 2. The
More information10/23/2003 PHY Lecture 14R 1
Announcements. Remember -- Tuesday, Oct. 8 th, 9:30 AM Second exam (coverng Chapters 9-4 of HRW) Brng the followng: a) equaton sheet b) Calculator c) Pencl d) Clear head e) Note: If you have kept up wth
More informationModule 1 : The equation of continuity. Lecture 1: Equation of Continuity
1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationChapter 8. Potential Energy and Conservation of Energy
Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and non-conservatve forces Mechancal Energy Conservaton of Mechancal
More informationPhysics 2A Chapter 3 HW Solutions
Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C
More informationRigid body simulation
Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationThe Photogravitational Circular Restricted Four-body Problem with Variable Masses
The Photogravtatonal Crcular Restrcted Four-body Problem wth Varable Masses Abdullah A. Ansar Department of Mathematcs College of Scence Majmaah Unversty Al-Zulf Kngdom of Saud Araba Abstract a.ansar@mu.edu.sa
More informationPhysics 111: Mechanics Lecture 11
Physcs 111: Mechancs Lecture 11 Bn Chen NJIT Physcs Department Textbook Chapter 10: Dynamcs of Rotatonal Moton q 10.1 Torque q 10. Torque and Angular Acceleraton for a Rgd Body q 10.3 Rgd-Body Rotaton
More informationWeek 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product
The Vector Product Week 11: Chapter 11 Angular Momentum There are nstances where the product of two vectors s another vector Earler we saw where the product of two vectors was a scalar Ths was called the
More informationChapter Eight. Review and Summary. Two methods in solid mechanics ---- vectorial methods and energy methods or variational methods
Chapter Eght Energy Method 8. Introducton 8. Stran energy expressons 8.3 Prncpal of statonary potental energy; several degrees of freedom ------ Castglano s frst theorem ---- Examples 8.4 Prncpal of statonary
More informationPart C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis
Part C Dynamcs and Statcs of Rgd Body Chapter 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More information5.76 Lecture #21 2/28/94 Page 1. Lecture #21: Rotation of Polyatomic Molecules I
5.76 Lecture # /8/94 Page Lecture #: Rotaton of Polatomc Molecules I A datomc molecule s ver lmted n how t can rotate and vbrate. * R s to nternuclear as * onl one knd of vbraton A polatomc molecule can
More informationSpin-rotation coupling of the angularly accelerated rigid body
Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationPhysics 207: Lecture 20. Today s Agenda Homework for Monday
Physcs 207: Lecture 20 Today s Agenda Homework for Monday Recap: Systems of Partcles Center of mass Velocty and acceleraton of the center of mass Dynamcs of the center of mass Lnear Momentum Example problems
More informationPhysics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.
Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays
More informationSpring 2002 Lecture #13
44-50 Sprng 00 ecture # Dr. Jaehoon Yu. Rotatonal Energy. Computaton of oments of nerta. Parallel-as Theorem 4. Torque & Angular Acceleraton 5. Work, Power, & Energy of Rotatonal otons Remember the md-term
More informationComparative Studies of Law of Conservation of Energy. and Law Clusters of Conservation of Generalized Energy
Comparatve Studes of Law of Conservaton of Energy and Law Clusters of Conservaton of Generalzed Energy No.3 of Comparatve Physcs Seres Papers Fu Yuhua (CNOOC Research Insttute, E-mal:fuyh1945@sna.com)
More informationModeling of Dynamic Systems
Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how
More informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
More informationInfinitesimal Orbits Near Any Point on Fawzy Equilibrium Circle in the Three Dimensional Restricted Three Body Problem
Earth Scence Inda, eissn: 097 850 Vol. 6 (I, January, 0, pp. - http://www.earthscencenda.nfo/ Infntesmal Orbts Near ny Pont on Fawzy Equlbrum Crcle n the hree Dmensonal Restrcted hree Body Problem F..
More informationKinematics in 2-Dimensions. Projectile Motion
Knematcs n -Dmensons Projectle Moton A medeval trebuchet b Kolderer, c1507 http://members.net.net.au/~rmne/ht/ht0.html#5 Readng Assgnment: Chapter 4, Sectons -6 Introducton: In medeval das, people had
More informationStereo 3D Simulation of Rigid Body Inertia Ellipsoid for The Purpose of Unmanned Helicopter Autopilot Tuning
nternatonal Journal of Engneerng Scence nventon SSN (Onlne): 319 6734, SSN (Prnt): 319 676 www.jes.org Volume 3 ssue 8 ǁ August 14 ǁ PP.8-35 Stereo 3D Smulaton of Rgd Bod nerta Ellpsod for The Purpose
More informationA NUMERICAL COMPARISON OF LANGRANGE AND KANE S METHODS OF AN ARM SEGMENT
Internatonal Conference Mathematcal and Computatonal ology 0 Internatonal Journal of Modern Physcs: Conference Seres Vol. 9 0 68 75 World Scentfc Publshng Company DOI: 0.4/S009450059 A NUMERICAL COMPARISON
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More information11. Dynamics in Rotating Frames of Reference
Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons
More informationStudy Guide For Exam Two
Study Gude For Exam Two Physcs 2210 Albretsen Updated: 08/02/2018 All Other Prevous Study Gudes Modules 01-06 Module 07 Work Work done by a constant force F over a dstance s : Work done by varyng force
More informationQuantum Mechanics I Problem set No.1
Quantum Mechancs I Problem set No.1 Septembe0, 2017 1 The Least Acton Prncple The acton reads S = d t L(q, q) (1) accordng to the least (extremal) acton prncple, the varaton of acton s zero 0 = δs = t
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationMEASUREMENT OF MOMENT OF INERTIA
1. measurement MESUREMENT OF MOMENT OF INERTI The am of ths measurement s to determne the moment of nerta of the rotor of an electrc motor. 1. General relatons Rotatng moton and moment of nerta Let us
More informationChapter 11 Angular Momentum
Chapter 11 Angular Momentum Analyss Model: Nonsolated System (Angular Momentum) Angular Momentum of a Rotatng Rgd Object Analyss Model: Isolated System (Angular Momentum) Angular Momentum of a Partcle
More informationPHYS 1443 Section 003 Lecture #17
PHYS 144 Secton 00 ecture #17 Wednesda, Oct. 9, 00 1. Rollng oton of a Rgd od. Torque. oment of Inerta 4. Rotatonal Knetc Energ 5. Torque and Vector Products Remember the nd term eam (ch 6 11), onda, Nov.!
More informationPoisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
More informationcoordinates. Then, the position vectors are described by
Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More information9. Complex Numbers. 1. Numbers revisited. 2. Imaginary number i: General form of complex numbers. 3. Manipulation of complex numbers
9. Comple Numbers. Numbers revsted. Imagnar number : General form of comple numbers 3. Manpulaton of comple numbers 4. The Argand dagram 5. The polar form for comple numbers 9.. Numbers revsted We saw
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationArmy Ants Tunneling for Classical Simulations
Electronc Supplementary Materal (ESI) for Chemcal Scence. Ths journal s The Royal Socety of Chemstry 2014 electronc supplementary nformaton (ESI) for Chemcal Scence Army Ants Tunnelng for Classcal Smulatons
More informationPY2101 Classical Mechanics Dr. Síle Nic Chormaic, Room 215 D Kane Bldg
PY2101 Classcal Mechancs Dr. Síle Nc Chormac, Room 215 D Kane Bldg s.ncchormac@ucc.e Lectures stll some ssues to resolve. Slots shared between PY2101 and PY2104. Hope to have t fnalsed by tomorrow. Mondays
More information6.3.7 Example with Runga Kutta 4 th order method
6.3.7 Example wth Runga Kutta 4 th order method Agan, as an example, 3 machne, 9 bus system shown n Fg. 6.4 s agan consdered. Intally, the dampng of the generators are neglected (.e. d = 0 for = 1, 2,
More informationVEKTORANALYS. GAUSS s THEOREM and STOKES s THEOREM. Kursvecka 3. Kapitel 6-7 Sidor 51-82
VEKTORANAY Kursvecka 3 GAU s THEOREM and TOKE s THEOREM Kaptel 6-7 dor 51-82 TARGET PROBEM EECTRIC FIED MAGNETIC FIED N + Magnetc monopoles do not est n nature. How can we epress ths nformaton for E and
More informationT f. Geometry. R f. R i. Homogeneous transformation. y x. P f. f 000. Homogeneous transformation matrix. R (A): Orientation P : Position
Homogeneous transformaton Geometr T f R f R T f Homogeneous transformaton matr Unverst of Genova T f Phlppe Martnet = R f 000 P f 1 R (A): Orentaton P : Poston 123 Modelng and Control of Manpulator robots
More informationRIGID BODY MOTION. Next, we rotate counterclockwise about ξ by angle. Next we rotate counterclockwise about γ by angle to get the final set (x,y z ).
RGD BODY MOTON We now consder the moton of rgd bodes. The frst queston s what coordnates are needed to specf the locaton and orentaton of such an object. Clearl 6 are needed 3 to locate a partcular pont
More informationCOMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
More informationPHYS 705: Classical Mechanics. Hamilton-Jacobi Equation
1 PHYS 705: Classcal Mechancs Hamlton-Jacob Equaton Hamlton-Jacob Equaton There s also a very elegant relaton between the Hamltonan Formulaton of Mechancs and Quantum Mechancs. To do that, we need to derve
More information12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationThe classical spin-rotation coupling
LOUAI H. ELZEIN 2018 All Rghts Reserved The classcal spn-rotaton couplng Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 louaelzen@gmal.com Abstract Ths paper s prepared to show that a rgd
More informationFinslerian Nonholonomic Frame For Matsumoto (α,β)-metric
Internatonal Journal of Mathematcs and Statstcs Inventon (IJMSI) E-ISSN: 2321 4767 P-ISSN: 2321-4759 ǁ Volume 2 ǁ Issue 3 ǁ March 2014 ǁ PP-73-77 Fnsleran Nonholonomc Frame For Matsumoto (α,)-metrc Mallkarjuna
More informationGrid Generation around a Cylinder by Complex Potential Functions
Research Journal of Appled Scences, Engneerng and Technolog 4(): 53-535, 0 ISSN: 040-7467 Mawell Scentfc Organzaton, 0 Submtted: December 0, 0 Accepted: Januar, 0 Publshed: June 0, 0 Grd Generaton around
More informationClassical Field Theory
Classcal Feld Theory Before we embark on quantzng an nteractng theory, we wll take a dverson nto classcal feld theory and classcal perturbaton theory and see how far we can get. The reader s expected to
More informationClassical Mechanics Virtual Work & d Alembert s Principle
Classcal Mechancs Vrtual Work & d Alembert s Prncple Dpan Kumar Ghosh UM-DAE Centre for Excellence n Basc Scences Kalna, Mumba 400098 August 15, 2016 1 Constrants Moton of a system of partcles s often
More informationCHAPTER 4. Vector Spaces
man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear
More informationSolution of Linear System of Equations and Matrix Inversion Gauss Seidel Iteration Method
Soluton of Lnear System of Equatons and Matr Inverson Gauss Sedel Iteraton Method It s another well-known teratve method for solvng a system of lnear equatons of the form a + a22 + + ann = b a2 + a222
More informationThe equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More informationIrregular vibrations in multi-mass discrete-continuous systems torsionally deformed
(2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationSINGLE EVENTS, TIME SERIES ANALYSIS, AND PLANETARY MOTION
SINGLE EVENTS, TIME SERIES ANALYSIS, AND PLANETARY MOTION John N. Harrs INTRODUCTION The advent of modern computng devces and ther applcaton to tme-seres analyses permts the nvestgaton of mathematcal and
More informationEN40: Dynamics and Vibrations. Homework 7: Rigid Body Kinematics
N40: ynamcs and Vbratons Homewor 7: Rgd Body Knematcs School of ngneerng Brown Unversty 1. In the fgure below, bar AB rotates counterclocwse at 4 rad/s. What are the angular veloctes of bars BC and C?.
More informationAngular Momentum and Fixed Axis Rotation. 8.01t Nov 10, 2004
Angular Momentum and Fxed Axs Rotaton 8.01t Nov 10, 2004 Dynamcs: Translatonal and Rotatonal Moton Translatonal Dynamcs Total Force Torque Angular Momentum about Dynamcs of Rotaton F ext Momentum of a
More informationPhysics 443, Solutions to PS 7
Physcs 443, Solutons to PS 7. Grffths 4.50 The snglet confguraton state s χ ) χ + χ χ χ + ) where that second equaton defnes the abbrevated notaton χ + and χ. S a ) S ) b χ â S )ˆb S ) χ In sphercal coordnates
More informationFirst Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.
Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act
More informationNew Method for Solving Poisson Equation. on Irregular Domains
Appled Mathematcal Scences Vol. 6 01 no. 8 369 380 New Method for Solvng Posson Equaton on Irregular Domans J. Izadan and N. Karamooz Department of Mathematcs Facult of Scences Mashhad BranchIslamc Azad
More informationAn Algorithm to Solve the Inverse Kinematics Problem of a Robotic Manipulator Based on Rotation Vectors
An Algorthm to Solve the Inverse Knematcs Problem of a Robotc Manpulator Based on Rotaton Vectors Mohamad Z. Al-az*, Mazn Z. Othman**, and Baker B. Al-Bahr* *AL-Nahran Unversty, Computer Eng. Dep., Baghdad,
More informationVEKTORANALYS GAUSS THEOREM STOKES THEOREM. and. Kursvecka 3. Kapitel 6 7 Sidor 51 82
VEKTORANAY Kursvecka 3 GAU THEOREM and TOKE THEOREM Kaptel 6 7 dor 51 82 TARGET PROBEM Do magnetc monopoles est? EECTRIC FIED MAGNETIC FIED N +? 1 TARGET PROBEM et s consder some EECTRIC CHARGE 2 - + +
More informationAP Physics 1 & 2 Summer Assignment
AP Physcs 1 & 2 Summer Assgnment AP Physcs 1 requres an exceptonal profcency n algebra, trgonometry, and geometry. It was desgned by a select group of college professors and hgh school scence teachers
More information2-π STRUCTURES ASSOCIATED TO THE LAGRANGIAN MECHANICAL SYSTEMS UDC 531.3: (045)=111. Victor Blãnuţã, Manuela Gîrţu
FACTA UNIVERSITATIS Seres: Mechancs Automatc Control and Robotcs Vol. 6 N o 1 007 pp. 89-95 -π STRUCTURES ASSOCIATED TO THE LAGRANGIAN MECHANICAL SYSTEMS UDC 531.3:53.511(045)=111 Vctor Blãnuţã Manuela
More informationA Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph
A Tale o Frcton Basc Rollercoaster Physcs Fahrenhet Rollercoaster, Hershey, PA max heght = 11 t max speed = 58 mph PLAY PLAY PLAY PLAY Rotatonal Movement Knematcs Smlar to how lnear velocty s dened, angular
More informationEuler-Lagrange Elasticity: Differential Equations for Elasticity without Stress or Strain
Journal of Appled Mathematcs and Physcs,,, 6- Publshed Onlne December (http://wwwscrporg/ournal/amp) http://dxdoorg/46/amp74 Euler-Lagrange Elastcty: Dfferental Equatons for Elastcty wthout Stress or Stran
More information3 d Rotations Rotating dumbbells in lab frame Moment of Inertial Tensor
d Rotatons Rotatng dumells n la frame Moment of Inertal Tensor Revew of BCS and FCS sstems Component notaton for I β Moments and Products of Inerta I x, I, I z P x, P xz, P z Moment of Inerta for a cue
More informationSCHOOL OF COMPUTING, ENGINEERING AND MATHEMATICS SEMESTER 2 EXAMINATIONS 2011/2012 DYNAMICS ME247 DR. N.D.D. MICHÉ
s SCHOOL OF COMPUTING, ENGINEERING ND MTHEMTICS SEMESTER EXMINTIONS 011/01 DYNMICS ME47 DR. N.D.D. MICHÉ Tme allowed: THREE hours nswer: ny FOUR from SIX questons Each queston carres 5 marks Ths s a CLOSED-BOOK
More informationCS 468 Lecture 16: Isometry Invariance and Spectral Techniques
CS 468 Lecture 16: Isometry Invarance and Spectral Technques Justn Solomon Scrbe: Evan Gawlk Introducton. In geometry processng, t s often desrable to characterze the shape of an object n a manner that
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationChapter 20 Rigid Body: Translation and Rotational Motion Kinematics for Fixed Axis Rotation
Chapter 20 Rgd Body: Translaton and Rotatonal Moton Knematcs for Fxed Axs Rotaton 20.1 Introducton... 1 20.2 Constraned Moton: Translaton and Rotaton... 1 20.2.1 Rollng wthout slppng... 5 Example 20.1
More informationPhysics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2
Physcs 607 Exam 1 Please be well-organzed, and show all sgnfcant steps clearly n all problems. You are graded on your wor, so please do not just wrte down answers wth no explanaton! Do all your wor on
More informationIn this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More information1. Review of Mechanics Newton s Laws
. Revew of Mechancs.. Newton s Laws Moton of partcles. Let the poston of the partcle be gven by r. We can always express ths n Cartesan coordnates: r = xˆx + yŷ + zẑ, () where we wll always use ˆ (crcumflex)
More information