Mechanics Physics 151
|
|
- Naomi Wilson
- 5 years ago
- Views:
Transcription
1 Mechancs Physcs 151 Lectue 18 Hamltonan Equatons of Moton (Chapte 8) What s Ahead We ae statng Hamltonan fomalsm Hamltonan equaton Today and 11/6 Canoncal tansfomaton 1/3, 1/5, 1/10 Close lnk to non-elatvstc QM May not cove Hamlton-Jacob theoy Cute but not vey elevant What shall we do n the last lectues? Classcal chaos? Petubaton theoy? Classcal feld theoy? Send me e-mal f you have pefeence! Hamltonan Fomalsm Newtonan Lagangan Hamltonan Descbe same physcs and poduce same esults Dffeence s n the vewponts Symmetes and nvaance moe appaent Flexblty of coodnate tansfomaton Hamltonan fomalsm lnked to the development of Hamlton-Jacob theoy Classcal petubaton theoy Quantum mechancs Statstcal mechancs 1
2 Lagange Hamlton Lagange s equatons fo n coodnates d L L = 0 = 1,, n nd -ode dffeental dt q q equaton of n vaables n equatons n ntal condtons q ( t = 0) q ( t = 0) Can we do wth 1 st -ode dffeental equatons? Yes, but you ll need n equatons We keep q and eplace q wth somethng smla Lq ( j, q j, t) We take the conjugate momenta p q Confguaton Space We consdeed ( q1,, q n ) as a pont n an n-dm. space Called confguaton space Moton of the system q = q() t A cuve n the confg space When we take vaatons, we consde q and q as ndependent vaables.e., we have n ndependent vaables n n-dm. space Isn t t moe natual to consde the moton n n-dm space? Phase Space Consde coodnates and momenta as ndependent State of the system s gven by ( q1,, qn, p1,, pn) Consde t a pont n the n-dmensonal phase space We ae swtchng the ndependent vaables ( q, q, t) ( q, p, t) A bt of mathematcal tck s needed to do ths q = q() t p = p () t
3 Legende Tansfomaton Stat fom a functon of two vaables Total devatve s f f df = dx + dy udx + vdy x y Defne g f ux f ( xy, ) and consde ts total devatve dg = df d( ux) = udx + vdy udx xdu = vdy xdu.e. g s a functon of u and y g g If f = L and ( x, y) = ( q, q) = v = x y u Lqq (, ) g( pq, ) = L pq Ths s what we need Hamltonan Opposte sgn fom Legende tansf. Defne Hamltonan: Hqpt (,, ) = qp Lqqt (,, ) Total devatve s L L L dh = pdq + q dp dq dq dt q q t L Lagange s equatons say = p q L dh = q dp p dq dt t Ths must be equvalent to Puttng them togethe gves dh = dp + dq + dt p q t Hamlton s Equatons L We fnd = q = p and = p q t t n equatons eplacng the n Lagange s equatons 1 st -ode dffeental nstead of nd -ode Symmety between q and p s appaent Thee s nothng new We just eaanged equatons Fst equaton lnks momentum to velocty Ths elaton s gven n Newtonan fomalsm Second equaton s equvalent to Newton s/lagange s equatons of moton 3
4 Quck Example Patcle unde Hooke s law foce F = kx m k L L= x x p = = mx x m k H = xp L= x + x p Replace x wth p k x m = + m Hamlton s equatons p x = = p = = kx p m x Usual hamonc oscllato Enegy Functon Defnton of Hamltonan s dentcal to the enegy L functon hqqt (,, ) = q Lqqt (,, ) q Dffeence s subtle: H s a functon of (q, p, t) Ths equals to the total enegy f Lagangan s L= L0( qt, ) + L1( qtq, ) + L( qtqq, ) j k Constants ae tme-ndependent Ths makes T = L ( q, t ) q j q k Foces ae consevatve See Lectue 4, o Ths makes V = L ( q) Goldsten Secton.7 0 Hamltonan and Total Enegy If the condtons make h to be total enegy, we can skp calculatng L and go dectly to H Fo the patcle unde Hooke s law foce p k H = E = T + V = + x m Ths woks often, but not always when the coodnate system s tme-dependent e.g., otatng (non-netal) coodnate system when the potental s velocty-dependent e.g., patcle n an EM feld Let s look at ths 4
5 Patcle n EM Feld Fo a patcle n an EM feld m L= x qφ + qax p = mx + qa We can t jump on H = E because of the last tem, but mx H = ( mx + qa) x L= + qφ Ths s n fact E We d be done f we wee calculatng h Fo H, we must ewte t usng p = mx + qa ( p qa) H( x, p) = + qφ m Patcle n EM Feld ( p qa) H( x, p) = + qφ m Hamlton s equatons ae p qa pj qaj Aj φ x = = p = = q q p m x m x x Ae they equvalent to the usual Loentz foce? Check ths by elmnatng p d Aj φ ( mx + qa ) = qx q dt x x d ( mv ) = qe + q ( v B) dt A bt of wok Consevaton of Hamltonan Consde tme-devatve of Hamltonan dh ( q, p, t) = q + p + dt q p t Hamltonan s = pq + qp + conseved f t does not t depend explctly on t H may o may not be total enegy If t s, ths means enegy consevaton Even f t sn t, H s stll a constant of moton 5
6 Cyclc Coodnates A cyclc coodnate does not appea n L By constucton, t does not appea n H ethe H ( q, pt, ) = qp L( q, qt, ) Hamlton s equaton says p = = 0 q Conjugate momentum of a cyclc coodnate s conseved Exactly the same as n the Lagangan fomalsm Cyclc Example Cental foce poblem n dmensons m L= ( + θ ) V( ) = m p = m θ p θ 1 p θ H = p + () + V θ s cyclc p m = const = θ l Hamlton s equatons 1 l = p + + V() p l V() m = p = 3 m m Cyclc vaable dops off by tself Gong Relatvstc Pactcal appoach Fnd a Hamltonan that woks Does t epesent the total enegy? Pust appoach Constuct covaant Hamltonan fomalsm Fo one patcle n an EM feld Don t expect macles Fundamental dffcultes eman the same 6
7 Pactcal Appoach Stat fom the elatvstc Lagangan that woks L mc β V = 1 ( x) L mv p = = v 1 β 4 H = h= p c + m c + V( x) It does equal to the total enegy Hamlton s equatons pc p x = = = p 4 pc + mc mγ Dd ths last tme V p = = = F x x Pactcal Appoach w/ EM Feld Consde a patcle n an EM feld L= mc 1 β qφ( x) + q( v A) Hamltonan s stll total enegy H = mγc + q φ 4 = m γ v c + m c + qφ Can be easly checked Dffeence s n the momentum p = mγ v + qa 4 H = ( p qa) c + m c + qφ Not the usual lnea momentum! Pactcal Appoach w/ EM Feld 4 H = ( p qa) c + m c + qφ Consde H qφ 4 ( H qφ ) ( p qa) c = m c constant It means that ( H qφ, pc qac) s a 4-vecto, and so s ( H, pc) Smla to 4-momentum (E/c, p) of a elatvstc patcle Remembe p hee s not the lnea momentum! Ths patcula Hamltonan + canoncal momentum tansfoms as a 4-vecto Tue only fo well-defned 4-potental such as EM feld 7
8 Pust Appoach Covaant Lagangan fo a fee patcle Λ= Λ p p p = = mu H = u m We know that p 0 s E/c We also know that x 0 s ct Geneally tue fo any covaant Lagangan You know the coespondng elatonshp n QM 1 mu uν ν Enegy s the conjugate momentum of tme Pust Appoach Value of Hamltonan s p p mc H = = m What s mpotant s H s dependence on p Hamlton s equatons dx p = = p m dp = = 0 x Tme components ae dct ( ) E dec ( ) = = γ c = 0 mc Ths s constant! 4-momentum consevaton Enegy defnton and consevaton Pust Appoach w/ EM Feld Wth EM feld, Lagangan becomes Λ ( x, u ) = mu u + qu A p = mu + qa 1 mu u ( p qa )( p qa ) H = = m Hamlton s equatons ae dx p qa dp ( pν qaν) A = = = = p m x m x A bt of wok can tun them nto du ν A A m = q uν = K x x ν ν 4-foce 8
9 EM Feld and Hamltonan In Hamltonan fomalsm, EM feld always modfy the canoncal momentum as pa = p0 + qa Wth EM feld Wthout EM feld A handy ule: Hamltonan wth EM feld s gven by eplacng p n the feld-fee Hamltonan wth p qa Often used n elatvstc QM to ntoduce EM nteacton Summay Constucted Hamltonan fomalsm Equvalent to Lagangan fomalsm Smple, but twce as many, equatons Hamltonan s conseved (unless explctly t-dependent) Equals to total enegy (unless t sn t) (duh) Cyclc coodnates dops out qute easly A few new nsghts fom elatvstc Hamltonans Conjugate of tme = enegy p qa ule fo ntoducng EM nteacton 9
PHYS 705: Classical Mechanics. Derivation of Lagrange Equations from D Alembert s Principle
1 PHYS 705: Classcal Mechancs Devaton of Lagange Equatons fom D Alembet s Pncple 2 D Alembet s Pncple Followng a smla agument fo the vtual dsplacement to be consstent wth constants,.e, (no vtual wok fo
More informationMechanics Physics 151
Mechancs Physcs 5 Lecture 7 Specal Relatvty (Chapter 7) What We Dd Last Tme Worked on relatvstc knematcs Essental tool for epermental physcs Basc technques are easy: Defne all 4 vectors Calculate c-o-m
More informationIntegral Vector Operations and Related Theorems Applications in Mechanics and E&M
Dola Bagayoko (0) Integal Vecto Opeatons and elated Theoems Applcatons n Mechancs and E&M Ι Basc Defnton Please efe to you calculus evewed below. Ι, ΙΙ, andιιι notes and textbooks fo detals on the concepts
More informationEnergy in Closed Systems
Enegy n Closed Systems Anamta Palt palt.anamta@gmal.com Abstact The wtng ndcates a beakdown of the classcal laws. We consde consevaton of enegy wth a many body system n elaton to the nvese squae law and
More informationPhysics 11b Lecture #2. Electric Field Electric Flux Gauss s Law
Physcs 11b Lectue # Electc Feld Electc Flux Gauss s Law What We Dd Last Tme Electc chage = How object esponds to electc foce Comes n postve and negatve flavos Conseved Electc foce Coulomb s Law F Same
More informationMechanics Physics 151
Mechancs Physcs 5 Lecture 0 Canoncal Transformatons (Chapter 9) What We Dd Last Tme Hamlton s Prncple n the Hamltonan formalsm Dervaton was smple δi δ p H(, p, t) = 0 Adonal end-pont constrants δ t ( )
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 informationMechanics Physics 151
Mechanics Physics 151 Lectue 5 Cental Foce Poblem (Chapte 3) What We Did Last Time Intoduced Hamilton s Pinciple Action integal is stationay fo the actual path Deived Lagange s Equations Used calculus
More informationChapter 23: Electric Potential
Chapte 23: Electc Potental Electc Potental Enegy It tuns out (won t show ths) that the tostatc foce, qq 1 2 F ˆ = k, s consevatve. 2 Recall, fo any consevatve foce, t s always possble to wte the wok done
More informationPhysics 2A Chapter 11 - Universal Gravitation Fall 2017
Physcs A Chapte - Unvesal Gavtaton Fall 07 hese notes ae ve pages. A quck summay: he text boxes n the notes contan the esults that wll compse the toolbox o Chapte. hee ae thee sectons: the law o gavtaton,
More informationSet of square-integrable function 2 L : function space F
Set of squae-ntegable functon L : functon space F Motvaton: In ou pevous dscussons we have seen that fo fee patcles wave equatons (Helmholt o Schödnge) can be expessed n tems of egenvalue equatons. H E,
More informationPart V: Velocity and Acceleration Analysis of Mechanisms
Pat V: Velocty an Acceleaton Analyss of Mechansms Ths secton wll evew the most common an cuently pactce methos fo completng the knematcs analyss of mechansms; escbng moton though velocty an acceleaton.
More information10. Canonical Transformations Michael Fowler
10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst
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 information24-2: Electric Potential Energy. 24-1: What is physics
D. Iyad SAADEDDIN Chapte 4: Electc Potental Electc potental Enegy and Electc potental Calculatng the E-potental fom E-feld fo dffeent chage dstbutons Calculatng the E-feld fom E-potental Potental of a
More informationV. Principles of Irreversible Thermodynamics. s = S - S 0 (7.3) s = = - g i, k. "Flux": = da i. "Force": = -Â g a ik k = X i. Â J i X i (7.
Themodynamcs and Knetcs of Solds 71 V. Pncples of Ievesble Themodynamcs 5. Onsage s Teatment s = S - S 0 = s( a 1, a 2,...) a n = A g - A n (7.6) Equlbum themodynamcs detemnes the paametes of an equlbum
More informationRigid Bodies: Equivalent Systems of Forces
Engneeng Statcs, ENGR 2301 Chapte 3 Rgd Bodes: Equvalent Sstems of oces Intoducton Teatment of a bod as a sngle patcle s not alwas possble. In geneal, the se of the bod and the specfc ponts of applcaton
More informationChapter Fifiteen. Surfaces Revisited
Chapte Ffteen ufaces Revsted 15.1 Vecto Descpton of ufaces We look now at the vey specal case of functons : D R 3, whee D R s a nce subset of the plane. We suppose s a nce functon. As the pont ( s, t)
More informationPhysics 202, Lecture 2. Announcements
Physcs 202, Lectue 2 Today s Topcs Announcements Electc Felds Moe on the Electc Foce (Coulomb s Law The Electc Feld Moton of Chaged Patcles n an Electc Feld Announcements Homewok Assgnment #1: WebAssgn
More informationPHY126 Summer Session I, 2008
PHY6 Summe Sesson I, 8 Most of nfomaton s avalable at: http://nngoup.phscs.sunsb.edu/~chak/phy6-8 ncludng the sllabus and lectue sldes. Read sllabus and watch fo mpotant announcements. Homewok assgnment
More informationPhysics 207 Lecture 16
Physcs 07 Lectue 6 Goals: Lectue 6 Chapte Extend the patcle odel to gd-bodes Undestand the equlbu of an extended object. Analyze ollng oton Undestand otaton about a fxed axs. Eploy consevaton of angula
More informationScalars and Vectors Scalar
Scalas and ectos Scala A phscal quantt that s completel chaacteed b a eal numbe (o b ts numecal value) s called a scala. In othe wods a scala possesses onl a magntude. Mass denst volume tempeatue tme eneg
More informationRemember: When an object falls due to gravity its potential energy decreases.
Chapte 5: lectc Potental As mentoned seveal tmes dung the uate Newton s law o gavty and Coulomb s law ae dentcal n the mathematcal om. So, most thngs that ae tue o gavty ae also tue o electostatcs! Hee
More informationDYNAMICS VECTOR MECHANICS FOR ENGINEERS: Kinematics of Rigid Bodies in Three Dimensions. Seventh Edition CHAPTER
Edton CAPTER 8 VECTOR MECANCS FOR ENGNEERS: DYNAMCS Fednand P. Bee E. Russell Johnston, J. Lectue Notes: J. Walt Ole Teas Tech Unvest Knematcs of Rgd Bodes n Thee Dmensons 003 The McGaw-ll Companes, nc.
More informationChapter 8. Linear Momentum, Impulse, and Collisions
Chapte 8 Lnea oentu, Ipulse, and Collsons 8. Lnea oentu and Ipulse The lnea oentu p of a patcle of ass ovng wth velocty v s defned as: p " v ote that p s a vecto that ponts n the sae decton as the velocty
More informationChapter I Matrices, Vectors, & Vector Calculus 1-1, 1-9, 1-10, 1-11, 1-17, 1-18, 1-25, 1-27, 1-36, 1-37, 1-41.
Chapte I Matces, Vectos, & Vecto Calculus -, -9, -0, -, -7, -8, -5, -7, -36, -37, -4. . Concept of a Scala Consde the aa of patcles shown n the fgue. he mass of the patcle at (,) can be epessed as. M (,
More informationPHYS Week 5. Reading Journals today from tables. WebAssign due Wed nite
PHYS 015 -- Week 5 Readng Jounals today fom tables WebAssgn due Wed nte Fo exclusve use n PHYS 015. Not fo e-dstbuton. Some mateals Copyght Unvesty of Coloado, Cengage,, Peason J. Maps. Fundamental Tools
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 informationPhysics Exam II Chapters 25-29
Physcs 114 1 Exam II Chaptes 5-9 Answe 8 of the followng 9 questons o poblems. Each one s weghted equally. Clealy mak on you blue book whch numbe you do not want gaded. If you ae not sue whch one you do
More information4.4 Continuum Thermomechanics
4.4 Contnuum Themomechancs The classcal themodynamcs s now extended to the themomechancs of a contnuum. The state aables ae allowed to ay thoughout a mateal and pocesses ae allowed to be eesble and moe
More information3. A Review of Some Existing AW (BT, CT) Algorithms
3. A Revew of Some Exstng AW (BT, CT) Algothms In ths secton, some typcal ant-wndp algothms wll be descbed. As the soltons fo bmpless and condtoned tansfe ae smla to those fo ant-wndp, the pesented algothms
More informationCSU ATS601 Fall Other reading: Vallis 2.1, 2.2; Marshall and Plumb Ch. 6; Holton Ch. 2; Schubert Ch r or v i = v r + r (3.
3 Eath s Rotaton 3.1 Rotatng Famewok Othe eadng: Valls 2.1, 2.2; Mashall and Plumb Ch. 6; Holton Ch. 2; Schubet Ch. 3 Consde the poston vecto (the same as C n the fgue above) otatng at angula velocty.
More informationCapítulo. Three Dimensions
Capítulo Knematcs of Rgd Bodes n Thee Dmensons Mecánca Contents ntoducton Rgd Bod Angula Momentum n Thee Dmensons Pncple of mpulse and Momentum Knetc Eneg Sample Poblem 8. Sample Poblem 8. Moton of a Rgd
More informationThermodynamics of solids 4. Statistical thermodynamics and the 3 rd law. Kwangheon Park Kyung Hee University Department of Nuclear Engineering
Themodynamcs of solds 4. Statstcal themodynamcs and the 3 d law Kwangheon Pak Kyung Hee Unvesty Depatment of Nuclea Engneeng 4.1. Intoducton to statstcal themodynamcs Classcal themodynamcs Statstcal themodynamcs
More informationCOMPLEMENTARY ENERGY METHOD FOR CURVED COMPOSITE BEAMS
ultscence - XXX. mcocd Intenatonal ultdscplnay Scentfc Confeence Unvesty of skolc Hungay - pl 06 ISBN 978-963-358-3- COPLEENTRY ENERGY ETHOD FOR CURVED COPOSITE BES Ákos József Lengyel István Ecsed ssstant
More informationLarge scale magnetic field generation by accelerated particles in galactic medium
Lage scale magnetc feld geneaton by acceleated patcles n galactc medum I.N.Toptygn Sant Petesbug State Polytechncal Unvesty, depatment of Theoetcal Physcs, Sant Petesbug, Russa 2.Reason explonatons The
More information8 Baire Category Theorem and Uniform Boundedness
8 Bae Categoy Theoem and Unfom Boundedness Pncple 8.1 Bae s Categoy Theoem Valdty of many esults n analyss depends on the completeness popety. Ths popety addesses the nadequacy of the system of atonal
More informationReview of Vector Algebra and Vector Calculus Operations
Revew of Vecto Algeba and Vecto Calculus Opeatons Tpes of vaables n Flud Mechancs Repesentaton of vectos Dffeent coodnate sstems Base vecto elatons Scala and vecto poducts Stess Newton s law of vscost
More informationStellar Astrophysics. dt dr. GM r. The current model for treating convection in stellar interiors is called mixing length theory:
Stella Astophyscs Ovevew of last lectue: We connected the mean molecula weght to the mass factons X, Y and Z: 1 1 1 = X + Y + μ 1 4 n 1 (1 + 1) = X μ 1 1 A n Z (1 + ) + Y + 4 1+ z A Z We ntoduced the pessue
More informationDynamics of Rigid Bodies
Dynamcs of Rgd Bodes A gd body s one n whch the dstances between consttuent patcles s constant thoughout the moton of the body,.e. t keeps ts shape. Thee ae two knds of gd body moton: 1. Tanslatonal Rectlnea
More information2/24/2014. The point mass. Impulse for a single collision The impulse of a force is a vector. The Center of Mass. System of particles
/4/04 Chapte 7 Lnea oentu Lnea oentu of a Sngle Patcle Lnea oentu: p υ It s a easue of the patcle s oton It s a vecto, sla to the veloct p υ p υ p υ z z p It also depends on the ass of the object, sla
More informationIf there are k binding constraints at x then re-label these constraints so that they are the first k constraints.
Mathematcal Foundatons -1- Constaned Optmzaton Constaned Optmzaton Ma{ f ( ) X} whee X {, h ( ), 1,, m} Necessay condtons fo to be a soluton to ths mamzaton poblem Mathematcally, f ag Ma{ f ( ) X}, then
More informationApplied Statistical Mechanics Lecture Note - 13 Molecular Dynamics Simulation
Appled Statstcal Mechancs Lectue Note - 3 Molecula Dynamcs Smulaton 고려대학교화공생명공학과강정원 Contents I. Basc Molecula Dynamcs Smulaton Method II. Popetes Calculatons n MD III. MD n Othe Ensembles I. Basc MD Smulaton
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 informationOne-dimensional kinematics
Phscs 45 Fomula Sheet Eam 3 One-dmensonal knematcs Vectos dsplacement: Δ total dstance taveled aveage speed total tme Δ aveage veloct: vav t t Δ nstantaneous veloct: v lm Δ t v aveage acceleaton: aav t
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 informationLINEAR MOMENTUM. product of the mass m and the velocity v r of an object r r
LINEAR MOMENTUM Imagne beng on a skateboad, at est that can move wthout cton on a smooth suace You catch a heavy, slow-movng ball that has been thown to you you begn to move Altenatvely you catch a lght,
More informationContact, information, consultations
ontact, nfomaton, consultatons hemsty A Bldg; oom 07 phone: 058-347-769 cellula: 664 66 97 E-mal: wojtek_c@pg.gda.pl Offce hous: Fday, 9-0 a.m. A quote of the week (o camel of the week): hee s no expedence
More information19 The Born-Oppenheimer Approximation
9 The Bon-Oppenheme Appoxmaton The full nonelatvstc Hamltonan fo a molecule s gven by (n a.u.) Ĥ = A M A A A, Z A + A + >j j (883) Lets ewte the Hamltonan to emphasze the goal as Ĥ = + A A A, >j j M A
More informationLecture 20: Noether s Theorem
Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external
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 informationNotes on Analytical Dynamics
Notes on Analytcal Dynamcs Jan Peters & Mchael Mstry October 7, 004 Newtonan Mechancs Basc Asssumptons and Newtons Laws Lonely pontmasses wth postve mass Newtons st: Constant velocty v n an nertal frame
More informationReview. Physics 231 fall 2007
Reew Physcs 3 all 7 Man ssues Knematcs - moton wth constant acceleaton D moton, D pojectle moton, otatonal moton Dynamcs (oces) Enegy (knetc and potental) (tanslatonal o otatonal moton when detals ae not
More informationThree views of mechanics
Three vews of mechancs John Hubbard, n L. Gross s course February 1, 211 1 Introducton A mechancal system s manfold wth a Remannan metrc K : T M R called knetc energy and a functon V : M R called potental
More informationTest 1 phy What mass of a material with density ρ is required to make a hollow spherical shell having inner radius r i and outer radius r o?
Test 1 phy 0 1. a) What s the pupose of measuement? b) Wte all fou condtons, whch must be satsfed by a scala poduct. (Use dffeent symbols to dstngush opeatons on ectos fom opeatons on numbes.) c) What
More informationGenerating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences
Geneatng Functons, Weghted and Non-Weghted Sums fo Powes of Second-Ode Recuence Sequences Pantelmon Stăncă Aubun Unvesty Montgomey, Depatment of Mathematcs Montgomey, AL 3614-403, USA e-mal: stanca@studel.aum.edu
More informationA. Thicknesses and Densities
10 Lab0 The Eath s Shells A. Thcknesses and Denstes Any theoy of the nteo of the Eath must be consstent wth the fact that ts aggegate densty s 5.5 g/cm (ecall we calculated ths densty last tme). In othe
More informationCOORDINATE SYSTEMS, COORDINATE TRANSFORMS, AND APPLICATIONS
Dola Bagaoo 0 COORDINTE SYSTEMS COORDINTE TRNSFORMS ND PPLICTIONS I. INTRODUCTION Smmet coce of coodnate sstem. In solvng Pscs poblems one cooses a coodnate sstem tat fts te poblem at and.e. a coodnate
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 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 informationAppendix B The Relativistic Transformation of Forces
Appendix B The Relativistic Tansfomation of oces B. The ou-foce We intoduced the idea of foces in Chapte 3 whee we saw that the change in the fou-momentum pe unit time is given by the expession d d w x
More informationS7: Classical mechanics problem set 2
J. Magoian MT 9, boowing fom J. J. Binney s 6 couse S7: Classical mechanics poblem set. Show that if the Hamiltonian is indepdent of a genealized co-odinate q, then the conjugate momentum p is a constant
More informationComplex atoms and the Periodic System of the elements
Complex atoms and the Peodc System of the elements Non-cental foces due to electon epulson Cental feld appoxmaton electonc obtals lft degeneacy of l E n l = R( hc) ( n δ ) l Aufbau pncple Lectue Notes
More informationALL QUESTIONS ARE WORTH 20 POINTS. WORK OUT FIVE PROBLEMS.
GNRAL PHYSICS PH -3A (D. S. Mov) Test (/3/) key STUDNT NAM: STUDNT d #: -------------------------------------------------------------------------------------------------------------------------------------------
More informationSummer Workshop on the Reaction Theory Exercise sheet 8. Classwork
Joned Physcs Analyss Cente Summe Wokshop on the Reacton Theoy Execse sheet 8 Vncent Matheu Contact: http://www.ndana.edu/~sst/ndex.html June June To be dscussed on Tuesday of Week-II. Classwok. Deve all
More informationEvent Shape Update. T. Doyle S. Hanlon I. Skillicorn. A. Everett A. Savin. Event Shapes, A. Everett, U. Wisconsin ZEUS Meeting, October 15,
Event Shape Update A. Eveett A. Savn T. Doyle S. Hanlon I. Skllcon Event Shapes, A. Eveett, U. Wsconsn ZEUS Meetng, Octobe 15, 2003-1 Outlne Pogess of Event Shapes n DIS Smla to publshed pape: Powe Coecton
More informationAPPLICATIONS OF SEMIGENERALIZED -CLOSED SETS
Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : 22776982 Volume Issue 4 (Apl 202) http://www.mes.com/ https://stes.google.com/ste/mesounal/ APPLICATIONS OF SEMIGENERALIZED CLOSED SETS G.SHANMUGAM,
More informationMachine Learning 4771
Machne Leanng 4771 Instucto: Tony Jebaa Topc 6 Revew: Suppot Vecto Machnes Pmal & Dual Soluton Non-sepaable SVMs Kenels SVM Demo Revew: SVM Suppot vecto machnes ae (n the smplest case) lnea classfes that
More informationChapter 3 Waves in an Elastic Whole Space. Equation of Motion of a Solid
Chapte 3 Waves n an Elastc Whole Space Equaton of Moton of a Sold Hopefully, many of the topcs n ths chapte ae evew. Howeve, I fnd t useful to dscuss some of the key chaactestcs of elastc contnuous meda.
More informationEngineering Mechanics. Force resultants, Torques, Scalar Products, Equivalent Force systems
Engneeng echancs oce esultants, Toques, Scala oducts, Equvalent oce sstems Tata cgaw-hll Companes, 008 Resultant of Two oces foce: acton of one bod on anothe; chaacteed b ts pont of applcaton, magntude,
More informationSlide 1. Quantum Mechanics: the Practice
Slde Quantum Mecancs: te Pactce Slde Remnde: Electons As Waves Wavelengt momentum = Planck? λ p = = 6.6 x 0-34 J s Te wave s an exctaton a vbaton: We need to know te ampltude of te exctaton at evey pont
More informationElectron density: Properties of electron density (non-negative): => exchange-correlation functionals should respect these conditions.
lecton densty: ρ ( =... Ψ(,,..., ds d... d Pobablty of fndng one electon of abtay spn wthn a volume element d (othe electons may be anywhee. s Popetes of electon densty (non-negatve:.. 3. ρ ( d = ρ( =
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 informationIntroduction. Chapter Introduction
Chapte Intoducton. Intoducton Mechancs s the theoy of moton of mateal bodes. In ode to dstngush ths theoy fom othe theoes, patculaly quantum mechancs, one also efes to t as Classcal Mechancs. Ths latte
More informationgravity r2,1 r2 r1 by m 2,1
Gavtaton Many of the foundatons of classcal echancs wee fst dscoveed when phlosophes (ealy scentsts and atheatcans) ted to explan the oton of planets and stas. Newton s ost faous fo unfyng the oton of
More informationThe Poisson bracket and magnetic monopoles
FYST420 Advanced electodynamics Olli Aleksante Koskivaaa Final poject ollikoskivaaa@gmail.com The Poisson backet and magnetic monopoles Abstact: In this wok magnetic monopoles ae studied using the Poisson
More informationPHYS 2421 Fields and Waves
PHYS 242 Felds nd Wves Instucto: Joge A. López Offce: PSCI 29 A, Phone: 747-7528 Textook: Unvesty Physcs e, Young nd Feedmn 23. Electc potentl enegy 23.2 Electc potentl 23.3 Clcultng electc potentl 23.4
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 informationA Brief Guide to Recognizing and Coping With Failures of the Classical Regression Assumptions
A Bef Gude to Recognzng and Copng Wth Falues of the Classcal Regesson Assumptons Model: Y 1 k X 1 X fxed n epeated samples IID 0, I. Specfcaton Poblems A. Unnecessay explanatoy vaables 1. OLS s no longe
More informationMULTIPOLE FIELDS. Multipoles, 2 l poles. Monopoles, dipoles, quadrupoles, octupoles... Electric Dipole R 1 R 2. P(r,θ,φ) e r
MULTIPOLE FIELDS Mutpoes poes. Monopoes dpoes quadupoes octupoes... 4 8 6 Eectc Dpoe +q O θ e R R P(θφ) -q e The potenta at the fed pont P(θφ) s ( θϕ )= q R R Bo E. Seneus : Now R = ( e) = + cosθ R = (
More informationMechanics Physics 151
Mechanics Physics 5 Lectue 5 Centa Foce Pobem (Chapte 3) What We Did Last Time Intoduced Hamiton s Pincipe Action intega is stationay fo the actua path Deived Lagange s Equations Used cacuus of vaiation
More informationMechanics Physics 151
Mechancs hyscs 151 Lecture Canoncal Transformatons (Chater 9) What We Dd Last Tme Drect Condtons Q j Q j = = j, Q, j, Q, Necessary and suffcent j j for Canoncal Transf. = = j Q, Q, j Q, Q, Infntesmal CT
More informationChapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 29,
hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe 9, 08 47 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe 9, 08 48 I. ETOR AND TENOR ANALYI I... Tenso functon th Let A
More informationE For K > 0. s s s s Fall Physical Chemistry (II) by M. Lim. singlet. triplet
Eneges of He electonc ψ E Fo K > 0 ψ = snglet ( )( ) s s+ ss αβ E βα snglet = ε + ε + J s + Ks Etplet = ε + ε + J s Ks αα ψ tplet = ( s s ss ) ββ ( αβ + βα ) s s s s s s s s ψ G = ss( αβ βα ) E = ε + ε
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 22 Canoncal Transformatons (Chater 9) What We Dd Last Tme Drect Condtons Q j Q j = = j P, Q, P j, P Q, P Necessary and suffcent P j P j for Canoncal Transf. = = j Q, Q, P j
More informationAnalytical and Numerical Solutions for a Rotating Annular Disk of Variable Thickness
Appled Mathematcs 00 43-438 do:0.436/am.00.5057 Publshed Onlne Novembe 00 (http://www.scrp.og/jounal/am) Analytcal and Numecal Solutons fo a Rotatng Annula Ds of Vaable Thcness Abstact Ashaf M. Zenou Daoud
More informationPHYS 1443 Section 003 Lecture #21
PHYS 443 Secton 003 Lectue # Wednesday, Nov. 7, 00 D. Jaehoon Yu. Gavtatonal eld. negy n Planetay and Satellte Motons 3. scape Speed 4. lud and Pessue 5. Vaaton of Pessue and Depth 6. Absolute and Relatve
More informationTHE MANY-BODY PROBLEM
3.30: Lectue 5 Feb 5 005 THE MANY-BODY PROBLEM Feb 5 005 3.30 Atomstc Modelng of Mateals -- Geband Cede and Ncola Maza When s a patcle lke a wave? Wavelength momentum = Planck λ p = h h = 6.6 x 0-34 J
More informationMechanics Physics 151
Mechanics Physics 5 Lectue 5 Centa Foce Pobem (Chapte 3) What We Did Last Time Intoduced Hamiton s Pincipe Action intega is stationay fo the actua path Deived Lagange s Equations Used cacuus of vaiation
More informationLagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013
Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned
More informationPhysics 1501 Lecture 19
Physcs 1501 ectue 19 Physcs 1501: ectue 19 Today s Agenda Announceents HW#7: due Oct. 1 Mdte 1: aveage 45 % Topcs otatonal Kneatcs otatonal Enegy Moents of Ineta Physcs 1501: ectue 19, Pg 1 Suay (wth copason
More informationSupersymmetry in Disorder and Chaos (Random matrices, physics of compound nuclei, mathematics of random processes)
Supesymmety n Dsoe an Chaos Ranom matces physcs of compoun nucle mathematcs of anom pocesses Lteatue: K.B. Efetov Supesymmety n Dsoe an Chaos Cambge Unvesty Pess 997999 Supesymmety an Tace Fomulae I.V.
More informationAsymptotic Solutions of the Kinetic Boltzmann Equation and Multicomponent Non-Equilibrium Gas Dynamics
Jounal of Appled Mathematcs and Physcs 6 4 687-697 Publshed Onlne August 6 n ScRes http://wwwscpog/jounal/jamp http://dxdoog/436/jamp64877 Asymptotc Solutons of the Knetc Boltzmann Equaton and Multcomponent
More informationMultipole Radiation. March 17, 2014
Multpole Radaton Mach 7, 04 Zones We wll see that the poblem of hamonc adaton dvdes nto thee appoxmate egons, dependng on the elatve magntudes of the dstance of the obsevaton pont,, and the wavelength,
More informationPhysics 201 Lecture 4
Phscs 1 Lectue 4 ltoda: hapte 3 Lectue 4 v Intoduce scalas and vectos v Peom basc vecto aleba (addton and subtacton) v Inteconvet between atesan & Pola coodnates Stat n nteestn 1D moton poblem: ace 9.8
More informationHamiltonian multivector fields and Poisson forms in multisymplectic field theory
JOURNAL OF MATHEMATICAL PHYSICS 46, 12005 Hamltonan multvecto felds and Posson foms n multsymplectc feld theoy Mchael Foge a Depatamento de Matemátca Aplcada, Insttuto de Matemátca e Estatístca, Unvesdade
More informationUNIT10 PLANE OF REGRESSION
UIT0 PLAE OF REGRESSIO Plane of Regesson Stuctue 0. Intoducton Ojectves 0. Yule s otaton 0. Plane of Regesson fo thee Vaales 0.4 Popetes of Resduals 0.5 Vaance of the Resduals 0.6 Summay 0.7 Solutons /
More informationPhysics 1: Mechanics
Physcs : Mechancs Đào Ngọc Hạnh Tâm Offce: A.503, Emal: dnhtam@hcmu.edu.vn HCMIU, Vetnam Natonal Unvesty Acknowledgment: Sldes ae suppoted by Pof. Phan Bao Ngoc Contents of Physcs Pat A: Dynamcs of Mass
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 informationRotational Kinematics. Rigid Object about a Fixed Axis Western HS AP Physics 1
Rotatonal Knematcs Rgd Object about a Fxed Axs Westen HS AP Physcs 1 Leanng Objectes What we know Unfom Ccula Moton q s Centpetal Acceleaton : Centpetal Foce: Non-unfom a F c c m F F F t m ma t What we
More information