THE GRAVITATIONAL TWO-BODY PROBLEM
|
|
- Abigayle Welch
- 5 years ago
- Views:
Transcription
1 THE GRAVITATIONAL TWO-BODY PROBLEM Danel J. Scheeres Department of Aerospace Engneerng Scences, The Unversty of Colorado, USA Keywords: Two-body problem, Kepleran soluton, dstrbuted bodes, rotatonal moton, translatonal moton. Contents. Introducton. Body and Mass Dstrbuton Specfcatons 3. Newtonan Gravtatonal Attracton 4. Equatons of Moton 5. Conservaton Prncples and Constrants 6. Constrants on Moton: Escape and Impact 7. Partcular Solutons 8. Soluton of the -Pont Mass or -Sphere Problem 9. Concluson Acknowledgements Glossary Bblography Bographcal Sketch Summary The gravtatonal two-body problem s defned and descrbed. The classcal Kepleran soluton for the moton of two pont masses s just one specalzed verson of ths problem, and n general the only one whch s completely ntegrable. Ths chapter wll provde a general defnton of the two-body problem makng no assumptons on the form of the mass dstrbutons that are mutually gravtatng. Then varous levels of approxmaton wll be ntroduced, descrbng the constrants and general results whch exst for ths problem. The general, or full, two-body problem actually couples rotatonal and translatonal moton n the general case, formng a non-ntegrable problem. Despte ths, there are strong constrants on the Hll and mpact stablty of ths problem. In addton, relatve equlbra and ther stablty can be dscussed n a general settng. The chapter culmnates n a dervaton of the Kepleran soluton for the dynamcs of two pont mass bodes orbtng each other.. Introducton The most basc problem n celestal mechancs s the gravtatonal two-body problem, specfcally the moton of two mutually attractng mass dstrbutons. The understandng of the smplest verson of ths problem occurred wth Coperncus placement of the sun at the center of the solar system and Kepler s subsequent frst descrpton of the ellptcal moton of the planets n the solar system. What Kepler dscovered based on observaton and deducton was a specal soluton to the general gravtatonal two-body problem, where the bodes n moton can be approxmated as spheres or pont masses.
2 Kepler s solutons provded the descrptve geometry of planetary moton. These results were not placed onto a frm mechancal bass untl Newton s celebrated development of the law of gravtaton, hs laws of moton, and the calculus. Taken together these advancements provded a complete development of Kepler s solutons based on physcal prncples. Beyond ths sgnfcant advance, Newton s work also enabled the complete gravtatonal two-body problem to be fully posed and analyzed. Detals of the dynamcs and solutons of the unapproxmated problem stll requred sgnfcant development and nsght, represented by many of the greats of celestal mechancs, physcs and mathematcs. Ths chapter addresses ths most fundamental problem n a general settng, placng an emphass on recent scholarshp and on developng a unfed vew of the two-body problem. Most strkng, t s noted that the full two-body problem s not solvable n most of ts statements, and can only be solved under some very restrctve approxmatons consstent wth Kepler s orgnal soluton. A complete dervaton of Kepler s soluton s gven at the close of the chapter, but frst we develop and focus on the more general statement of ths problem and detal the constrants, solutons and approxmatons that enable these problems to be understood. Ths general approach has been motvated by recent scholarshp on the general, or so-called full, two-body problem that accounts for the coupled rotatonal and translatonal moton of the two mass dstrbutons. Early work that focused on the couplng between translatonal and rotatonal moton traces back to Cassn s Laws on the moton of the moon, although n these cases the lbratonal rotaton s drven by the orbt, whle the nfluence of the rotaton on the orbt s generally neglected. Startng n the 950s, Duboshn studed the dynamcs of coupled rotatonal and translatonal moton and stated the general set of dfferental equatons and ther fundamental ntegrals of moton. In the 970s Knoshta developed perturbaton theores for these problems under the assumpton of relatvely weak couplng between the dfferent modes of moton. In the 990s research nto the generalzed verson of ths problem was explored by a number of researchers. Wang, Krshnaprasad, and Maddocks approached the problem from a geometrcal mechancs approach and explored the exstence and stablty of relatve equlbra, albet under the assumpton that one of the bodes was a sphere. Macejewsk consdered the fully general problem and explored the propertes of relatve equlbra as well as several dfferent ways to pose the problem. In the 000s a seres of artcles by Scheeres explored constrants on the solutons to the full two-body problem and appled ths problem to the dynamcs of bnary asterods and the evoluton of rubble ple asterods. Sgnfcant advances n the study of the averaged full two-body problem was made by Boué and Laskar, generalzng the classcal Cassn states and showng how they ft nto a larger set of ntegrable motons for the averaged problem. The approach taken n ths chapter s focused on sharp results that do not make strong assumptons on the moton, such as are found n averagng theores. Solutons whch yeld general and partcular solutons to the problem are gven, and constrants for the system whch act on the full soluton space are emphaszed. The goal of the current chapter s to develop a consstent and unfed approach to ths problem, relyng on classcal mechancs formulatons. Several theorems are ntroduced and developed to capture key results that hold for the general system and some specal subsystems. There are several results for the general full two-body problem that are not well known and are
3 somewhat surprsng and at odds wth the classcal Kepler soluton. Frst s that the full two-body problem s n general non-ntegrable and can exhbt chaotc moton (although we wll not establsh these results here). Ths occurs ether due to the couplng of rotatonal moton of the two bodes wth ther relatve translatonal moton or due to the non-sphercal mass dstrbutons of ether one of the bodes. Second, for the general evoluton of the two-sphere problem s that a system wth a fxed value of angular momentum can have multple crcular orbts, some of whch can be unstable. Only when the system s lmted to two pont-mass dstrbutons does t become a fully ntegrable problem. In the followng we work through a number of these dfferent results and only arrve at the ntegrable verson of the problem n the last secton.. Body and Mass Dstrbuton Specfcatons The core assumpton we make n ths chapter s that the mass dstrbutons of the two bodes are rgd, meanng that we do not account for any deformaton n ther shape or mass dstrbuton. Fgure provdes a graphcal defnton of the problem, wth the followng secton provdng a mathematcal descrpton. Fgure. The full gravtatonal -Body Problem and ts degrees of freedom.. Mass and Center of Mass Assume that there exst two rgd bodes wth dstrbuted masses, characterzed by havng fnte denstes and well defned lmts. Ther dfferental masses are defned as dm ρ dv () where s the varyng densty of the th body,,, and ρ s the poston of the mass element referenced to some frame. The densty s zero when ρ s taken outsde of the body and s fnte wthn the body, denoted as. Wth ths defnton the total mass of each body equals
4 M dm () and the locaton of each of the body s center of mass s computed as r dm M ρ (3) The jont mass dstrbuton of the system s defned va the dfferental mass element dm dm dm (4) and the jont bodes as,. Ths allows the total mass of the system to be defned as MM dm (5) The barycenter of the system s then found as R dm M M ρ (6) As wll be dscussed later, we can take.. Relatve Orentatons R 0 n general. The orentaton of the bodes s defned by rotaton dyadcs that transform a vector n the body-fxed frame nto nertal space, denoted as A and A. Then the complete specfcaton of each body n an nertal frame s r, r, A, A and the relatve poston and orentaton of the bodes wth respect to each other s ra, where the relatve poston and atttude of body relatve to body s defned as r r r (7) T A A A (8) As each of these terms represent 3 degrees of freedom, to specfy the relatve poston and orentaton of two rgd bodes requres a total of 6 degrees of freedom. These are called the nternal or relatve degrees of freedom. To orent these nternal degrees of freedom wth respect to an nertal frame requres an addtonal 3 degrees of freedom, represented by the rotaton dyadc A. Fgure shows a graphcal representaton of ths full system. We note that each of the rgd bodes has an angular velocty that defnes the nstantaneous rate of rotaton between the body-fxed frame and an nertal frame.
5 .3. Moments of Inerta An addtonal mass dstrbuton quantty of nterest s the nerta dyadcs of each body, defned by I U ρρ dm (9) where U s the dentty dyadc and the product of two vectors, e.g. ρρ, s a dyad. The coeffcents of ths dyadc can be computed n closed form for specal cases such as constant densty spheres, ellpsods, and polyhedra. The nerta dyadc of the entre system can also be specfed as I U ρρ dm (0) and smplfes nto the ndvdual nerta dyadcs and the nerta dyadc of the two masses f the system s computed relatve to the barycenter MM I U rr ˆˆ I I () r M M where the hat desgnates a vector as a unt vector. Of nterest later s the moment of nerta relatve to a fxed drecton, Ĥ, computed as I Hˆ IH ˆ () H Note that I H s a functon of the relatve poston of the two masses and ther orentaton, all relatve to the unt vector Ĥ. Also of nterest s the polar moment of nerta, computed as one-half of the trace of the total nerta IP trace I (3) MM trace r M M I I (4) Note that the polar moment of nerta for a -body system s only a functon of the separaton between the bodes, or I r, and s ndependent of the orentaton of the P two mass dstrbutons. An mportant nequalty can be proven between the polar moment of nerta and the moment of nerta relatve to a fxed drecton
6 IH I (5) P whch holds for any fxed drecton Ĥ..4. Body Shapes and Geometry In the followng we wll assume that both bodes are convex. Ths s not an essental assumpton but makes t smpler to dscuss stuatons when the two bodes can come nto contact. If they are both convex, then at every relatve confguraton of the system d r,a ˆ defned as the there s a well defned mnmum dstance between the bodes radus at whch the two bodes touch for ther relatve confguraton. Ths dstance changes smoothly wth ˆr and A and s constant for all rotatons of A about the unt vector ˆr. If non-convex bodes are assumed then there s the potental for multple dstances between the dstrbutons at a gven relatve confguraton and dscontnutes n the mnmum dstance as a functon of ˆr and A. The maxmum of these mnmum dstances can be specfed as D max d, ra ra. ˆ, ˆ Ths quantty s of fundamental nterest for any two body system as beyond ths dstance the two bodes can never mpact wth each other. Ths lmt can be defned ndependent of whether the bodes are convex or not. 3. Newtonan Gravtatonal Attracton Havng defned the two bodes and ther mass dstrbuton, we next consder the relatve forces that these dstrbutons apply to each other due to Newton s law of gravtatonal attracton. 3.. Relatve Forces Newton s fundamental law of gravtatonal attracton states that two dfferental mass elements wll experence an attracton between them proportonal to the product of the masses, nversely proportonal to the square of the dstance between them and drected along ther relatve poston. Gven our defnton of the relatve poston vector r as gong from body to body the dfferental force that a mass element n body places on a mass element n body s dm dm df r A ρ A ρ 3 r A ρ A ρ (6) where we recall that r r r and ρ s the poston of a mass element of body relatve to ts center of mass. The dfferental force of a mass element n body on a mass element n body s smply df n accordance wth Newton s law of acton and reacton. To compute the total force that body exerts on body, and vce-versa, these dfferentals are ntegrated over both mass dstrbutons:
7 F df (7) and F F. The dfferental force can also be derved from a scalar potental functon, defned as the dfferental potental gravtatonal energy between the bodes dm dm (8) r A ρ A ρ du then df df du r ρ du r ρ (9) (0) Note that du du r ρ r n general. The gravtatonal potental energy between the bodes s then defned by ntegratng ths dfferental potental over both bodes U r, r, A, A du () dm ρ dm ρ r r A ρ A ρ () where the ntegraton varables ρ are expressed n ther respectve body-fxed frames and all terms n the vector magntude are specfed n nertal space. The forces of these bodes on each other are then computed as F U r (3) F U r (4) 3.. Relatve Moments For bodes wth fnte mass dstrbutons t s also necessary to compute the mutual moments of the force (or moments) that are exerted on each other. The moment that
8 body exerts on body s found by ntegratng the dfferental moment over both bodes: dm A ρ df (5) M A ρ df (6) The moment can also be related to the mutual potental, although the detals are more nvolved. Ths s found by takng the partal of the mutual potental wth respect to the nfntesmal rotatons about each axs of body, whch we represent as M U θ. In practce, once the relatve atttude of body s defned explctly usng A the partals wth respect to the angles can be computed from ths relatonshp. The moment actng on body, M, s smlarly computed as U θ. It must be noted that the mutual moments are not equal and opposte, but that ther sum equals the negatve moment of the total gravtatonal force between the bodes M M r r F 0 (7) meanng that we do not need to solve for one of the moments f the force s known TO ACCESS ALL THE 45 PAGES OF THIS CHAPTER, Vst: Bblography A.J. Macejewsk. Reducton, relatve equlbra and potental n the two rgd bodes problem. Celestal Mechancs and Dynamcal Astronomy, 63(): 8, 995. [A comprehensve summary and dscusson of the Full -Body problem] D.J. Scheeres. Stablty n the full two-body problem. Celestal Mechancs and Dynamcal Astronomy, 83():55 69, 00. [Ths paper presents rgorous results on the stablty of moton n the Full -Body problem] D.J. Scheeres. Relatve Equlbra for General Gravty Felds n the Sphere-Restrcted Full -Body Problem. Celestal Mechancs and Dynamcal Astronomy, 94:37 349, March 006. [Ths paper presents an algorthm for computng general relatve equlbra for the sphere-restrcted Full -Body problem] D.J. Scheeres. Stablty of the planar full -body problem. Celestal Mechancs and Dynamcal Astronomy, 04():03 8, 009. [Ths paper presents detaled examples of the Full -Body problem when both bodes are non-sphercal] D.J. Scheeres. Orbtal Moton n Strongly Perturbed Envronments : Applcatons to Asterod, Comet and Planetary Satellte Orbters. Sprnger-Praxs, London (UK), 0. [Ths book contans much nformaton
9 on the computaton of gravtatonal potentals for the modelng of ther dynamcal evoluton] D.J. Scheeres. Mnmum energy confguratons n the n-body problem and the celestal mechancs of granular systems. Celestal Mechancs and Dynamcal Astronomy, 3: 9030, 0. [Treats fnte densty mnmum energy confguratons n the N -body problem] G. Boué and J. Laskar. Spn axs evoluton of two nteractng bodes. Icarus, 0(): , 009. [Ths paper presents recent results on averaged dynamcs n the Full -Body problem] G.N. Duboshn. The dfferental equatons of translatonal-rotatonal moton of mutally attractng rgd bodes. Sovet Astronomy, :39, 958. [A classcal dscusson of the couplng of rotatonal and translatonal moton] H. Knoshta. Frst-order perturbatons of the two fnte body problem. Publcatons of the Astronomcal Socety of Japan, 4:43, 97. [A classcal treatment of the Full -Body problem] H. Pollard. Celestal mechancs. The Carus Mathematcal Monographs, Provdence: Mathematcal Assocaton of Amerca, 976. [A classc text on celestal mechancs] L.S. Wang, P.S. Krshnaprasad, and JH Maddocks. Hamltonan dynamcs of a rgd body n a central gravtatonal feld. Celestal Mechancs and Dynamcal Astronomy, 50(4): , 990. [Ths paper presents a geometrc mechancs approach to the Full -Body problem] P. Pravec, D. Vokrouhlckỳ, D. Polshook, DJ Scheeres, AW Harrs, A. Galád, O. Vaduvescu, F. Pozo, A. Barr, P. Longa, et al. Formaton of asterod pars by rotatonal fsson. Nature, 466(730): , 00. [Ths paper provdes evdence of the fsson of asterods nto rgd bodes whch subsequently escape from each other] S. A. Jacobson and D. J. Scheeres. Dynamcs of rotatonally fssoned asterods: Source of observed small asterod systems. Icarus, 4:6 78, July 0. [Ths paper presents numercal experments regardng the dsrupton of Full -Body problems wth postve energy] S. Smale. Topology and mechancs.. Inventones mathematcae, ():45 64, 970. [Develops the theory behnd the amended potental] V.I. Arnold, V.V. Kozlov, and A.I. Neshtadt. Mathematcal aspects of classcal and celestal mechancs. Sprnger, 006. [Ths book provdes a clear explanaton of the applcaton of the amended potental to the N -body problem] V. Gubout and D.J. Scheeres. Stablty of Surface Moton on a Rotatng Ellpsod. Celestal Mechancs and Dynamcal Astronomy, 87:63 90, November 003. [Ths paper dscusses the stablty of surface restng ponts on a rotatng ellpsod] Bographcal Sketch Danel Jay Scheeres (born n 963 n Royal Oak, Mchgan, USA) has a Bachelor s of Scence n Letters and Engneerng from Calvn College (985), a Bachelor s and Master s of Scence n Aerospace Engneerng from The Unversty of Mchgan (987 and 988, respectvely), and a PhD. n Aerospace Engneerng from The Unversty of Mchgan (99) where he studed wth Nguyen Xuan Vnh. Snce 008 he has been the A. Rchard Seebass Endowed Char Professor n the Department of Aerospace Engneerng Scences at The Unversty of Colorado Boulder. Pror to ths he held academc postons at the Unversty of Mchgan and Iowa State Unversty. Pror to that he was a Senor Member of the Techncal Staff at the Jet Propulson Laboratory / Calforna Insttute of Technology. He s past char of the Amercan Astronomcal Socety s Dvson on Dynamcal Astronomy and the vce-presdent of the Celestal Mechancs Insttute. Hs research nterests nclude celestal mechancs of dstrbuted bodes, the mechancs of comets and asterods, the navgaton and dynamcs of spacecraft, and the long-term evoluton of orbt debrs.
PHYS 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 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 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 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 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 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 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 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 informationMathematical 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 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 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 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 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 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 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 informationIntegrals and Invariants of Euler-Lagrange Equations
Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
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 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 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 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. 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 information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
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 informationConservation of Angular Momentum = "Spin"
Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts
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 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 informationIndeterminate pin-jointed frames (trusses)
Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationIterative General Dynamic Model for Serial-Link Manipulators
EEL6667: Knematcs, Dynamcs and Control of Robot Manpulators 1. Introducton Iteratve General Dynamc Model for Seral-Lnk Manpulators In ths set of notes, we are gong to develop a method for computng a general
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 informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More informationRotational Dynamics. Physics 1425 Lecture 19. Michael Fowler, UVa
Rotatonal Dynamcs Physcs 1425 Lecture 19 Mchael Fowler, UVa Rotatonal Dynamcs Newton s Frst Law: a rotatng body wll contnue to rotate at constant angular velocty as long as there s no torque actng on t.
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 informationTensor Smooth Length for SPH Modelling of High Speed Impact
Tensor Smooth Length for SPH Modellng of Hgh Speed Impact Roman Cherepanov and Alexander Gerasmov Insttute of Appled mathematcs and mechancs, Tomsk State Unversty 634050, Lenna av. 36, Tomsk, Russa RCherepanov82@gmal.com,Ger@npmm.tsu.ru
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 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 informationErrors in Nobel Prize for Physics (7) Improper Schrodinger Equation and Dirac Equation
Errors n Nobel Prze for Physcs (7) Improper Schrodnger Equaton and Drac Equaton u Yuhua (CNOOC Research Insttute, E-mal:fuyh945@sna.com) Abstract: One of the reasons for 933 Nobel Prze for physcs s for
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 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 informationPES 1120 Spring 2014, Spendier Lecture 6/Page 1
PES 110 Sprng 014, Spender Lecture 6/Page 1 Lecture today: Chapter 1) Electrc feld due to charge dstrbutons -> charged rod -> charged rng We ntroduced the electrc feld, E. I defned t as an nvsble aura
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 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 informationPhysics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints
Physcs 106a, Caltech 11 October, 2018 Lecture 4: Constrants, Vrtual Work, etc. Many, f not all, dynamcal problems we want to solve are constraned: not all of the possble 3 coordnates for M partcles (or
More informationPhysics 141. Lecture 14. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 14, Page 1
Physcs 141. Lecture 14. Frank L. H. Wolfs Department of Physcs and Astronomy, Unversty of Rochester, Lecture 14, Page 1 Physcs 141. Lecture 14. Course Informaton: Lab report # 3. Exam # 2. Mult-Partcle
More informationAppendix B. The Finite Difference Scheme
140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton
More informationPhysics 207 Lecture 6
Physcs 207 Lecture 6 Agenda: Physcs 207, Lecture 6, Sept. 25 Chapter 4 Frames of reference Chapter 5 ewton s Law Mass Inerta s (contact and non-contact) Frcton (a external force that opposes moton) Free
More informationIntegrals and Invariants of
Lecture 16 Integrals and Invarants of Euler Lagrange Equatons NPTEL Course Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng, Indan Insttute of Scence, Banagalore
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 informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationProbabilistic method to determine electron correlation energy
Probablstc method to determne electron elaton energy T.R.S. Prasanna Department of Metallurgcal Engneerng and Materals Scence Indan Insttute of Technology, Bombay Mumba 400076 Inda A new method to determne
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationLagrangian Field Theory
Lagrangan Feld Theory Adam Lott PHY 391 Aprl 6, 017 1 Introducton Ths paper s a summary of Chapter of Mandl and Shaw s Quantum Feld Theory [1]. The frst thng to do s to fx the notaton. For the most part,
More informationThe non-negativity of probabilities and the collapse of state
The non-negatvty of probabltes and the collapse of state Slobodan Prvanovć Insttute of Physcs, P.O. Box 57, 11080 Belgrade, Serba Abstract The dynamcal equaton, beng the combnaton of Schrödnger and Louvlle
More informationThermodynamics General
Thermodynamcs General Lecture 1 Lecture 1 s devoted to establshng buldng blocks for dscussng thermodynamcs. In addton, the equaton of state wll be establshed. I. Buldng blocks for thermodynamcs A. Dmensons,
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 informationA PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY.
Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng -- ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng -- ABCM, Curtba, Brazl,- Dec. 5-8, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR
More informationSecond Order Analysis
Second Order Analyss In the prevous classes we looked at a method that determnes the load correspondng to a state of bfurcaton equlbrum of a perfect frame by egenvalye analyss The system was assumed to
More informationNON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS
IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc
More informationRemarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence
Remarks on the Propertes of a Quas-Fbonacc-lke Polynomal Sequence Brce Merwne LIU Brooklyn Ilan Wenschelbaum Wesleyan Unversty Abstract Consder the Quas-Fbonacc-lke Polynomal Sequence gven by F 0 = 1,
More information829. An adaptive method for inertia force identification in cantilever under moving mass
89. An adaptve method for nerta force dentfcaton n cantlever under movng mass Qang Chen 1, Mnzhuo Wang, Hao Yan 3, Haonan Ye 4, Guola Yang 5 1,, 3, 4 Department of Control and System Engneerng, Nanng Unversty,
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 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 informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
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 informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
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 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 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 informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
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 informationNumerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
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 informationComparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method
Appled Mathematcal Scences, Vol. 7, 0, no. 47, 07-0 HIARI Ltd, www.m-hkar.com Comparson of the Populaton Varance Estmators of -Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method
More informationTranslational Equations of Motion for A Body Translational equations of motion (centroidal) for a body are m r = f.
Lesson 12: Equatons o Moton Newton s Laws Frst Law: A artcle remans at rest or contnues to move n a straght lne wth constant seed there s no orce actng on t Second Law: The acceleraton o a artcle s roortonal
More informationEPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski
EPR Paradox and the Physcal Meanng of an Experment n Quantum Mechancs Vesseln C Nonnsk vesselnnonnsk@verzonnet Abstract It s shown that there s one purely determnstc outcome when measurement s made on
More informationPower law and dimension of the maximum value for belief distribution with the max Deng entropy
Power law and dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy Bngy Kang a, a College of Informaton Engneerng, Northwest A&F Unversty, Yanglng, Shaanx, 712100, Chna. Abstract Deng
More informationDETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM
Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for S111 DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM by Davood Domr GANJI
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 informationAssessment of Site Amplification Effect from Input Energy Spectra of Strong Ground Motion
Assessment of Ste Amplfcaton Effect from Input Energy Spectra of Strong Ground Moton M.S. Gong & L.L Xe Key Laboratory of Earthquake Engneerng and Engneerng Vbraton,Insttute of Engneerng Mechancs, CEA,
More informationLecture 4. Macrostates and Microstates (Ch. 2 )
Lecture 4. Macrostates and Mcrostates (Ch. ) The past three lectures: we have learned about thermal energy, how t s stored at the mcroscopc level, and how t can be transferred from one system to another.
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 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 informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
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 informationr i r j 3. (2) Gm j m i r i (r i r j ) r i r j 3. (3)
N-body problem There s a lot of nterestng physcs related to the nteractons of a few bodes, but there are also many systems that can be well-approxmated by a large number of bodes that nteract exclusvely
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 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 informationCIE4801 Transportation and spatial modelling Trip distribution
CIE4801 ransportaton and spatal modellng rp dstrbuton Rob van Nes, ransport & Plannng 17/4/13 Delft Unversty of echnology Challenge the future Content What s t about hree methods Wth specal attenton for
More information9 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 - Chapter 9R -Davd Klenfeld - Fall 2005 9 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys a set
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and
More informationExercises of Chapter 2
Exercses of Chapter Chuang-Cheh Ln Department of Computer Scence and Informaton Engneerng, Natonal Chung Cheng Unversty, Mng-Hsung, Chay 61, Tawan. Exercse.6. Suppose that we ndependently roll two standard
More informationFUZZY FINITE ELEMENT METHOD
FUZZY FINITE ELEMENT METHOD RELIABILITY TRUCTURE ANALYI UING PROBABILITY 3.. Maxmum Normal tress Internal force s the shear force, V has a magntude equal to the load P and bendng moment, M. Bendng moments
More information1 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 -Davd Klenfeld - Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys
More informationTensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q
For orthogonal curvlnear coordnates, eˆ grad a a= ( aˆ ˆ e). h q (98) Expandng the dervatve, we have, eˆ aˆ ˆ e a= ˆ ˆ a h e + q q 1 aˆ ˆ ˆ a e = ee ˆˆ ˆ + e. h q h q Now expandng eˆ / q (some of the detals
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 information