# Mechanics Physics 151

Size: px
Start display at page:

Transcription

1 Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2)

2 What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and reacton! Introduced constrants! Generalzed coordnates! Introduced Lagrange s Equatons!... and ddn t do the dervaton " Let s pck t up and start from there

3 Today s Goals! Derve Lagrange s Eqn from Newton s Eqn! Use D Alembert s prncple! There wll be a few assumptons! Wll make them clear as we go! Introduce Hamlton s Prncple! Equvalent to Lagrange s Equatons! Whch n turn s equvalent to Newton s Equatons! Does not depend on coordnates by constructon! Dervaton n the next lecture

4 Lagrange s Equatons Recpe d L L = dt q! q 0 Knetc energy Lqqt (,!, ) T V Potental energy Lagrangan! Express L = T V n terms of generalzed coordnates { q }, ther tme-dervatves { q! }, and tme t! The potental V = V(q, t) must exst!.e. all forces must be conservatve

5 Vrtual Dsplacement! Consder a system wth constrants! Ordnary coordnates r ( = 1...N)! Generalzed coordnates q ( = 1...n)! Imagne movng all the partcles slghtly r r +δ r r1 = r1( q1, q2,..., qn, t) r2 = r2( q1, q2,..., qn, t) " rn = rn( q1, q2,..., qn, t) Vrtual dsplacement! Note that δr must satsfy the constrants 3N coordnates not ndependent δr q q +δ q = r q δq n coordnates ndependent

6 D Alembert s Prncple! From Newton s Equaton of Moton F = p! F p! = 0! Part of the force F must be due to constrants F = F + f ( )! Appled force s known! Constrant force f (usually) does no work! Movement s perpendcular to the force! Excepton: frcton appled force constrant force ( a)! Now multply F + f p! = 0 by δr and sum over F F ( r, r,..., r,..., r, ) ( a) = ( a) t 1 2 N fδ r = 0

7 D Alembert s Prncple F p! r ( a) ( ) δ = 0! Force of constrants dropped out because! Called D Alembert s Prncple (1743) constrant force s out of the game. You can forget (a) fδ r = 0! Now we swtch from r to q 1st term r = F q Q q q δ = δ Q F r q! Unt of Q not always [force]! Q q s always [work] Generalzed force

8 D Alembert s Prncple 2nd term δ r δ r = p! r = p! q = m!! r δ q q, q! A bt of work can show! D Alembert s Prncple becomes r!! d T T = dt q! q 2 2 r d v v q dt q! 2 q 2 δ q d T T Q δ q = 0 dt q! q mv T 2 2

9 Lagrange s Equatons d T T Q δ q = dt q! q! Generalzed coordnates q are ndependent d T T = Q dt q! q! Assume forces are conservatve Q r V = V r F = q q q 0 F These are free Almost there! = V Throw ths back n

10 Lagrange s Equatons ( T V) d T = 0 dt q! q V q! = 0 q! d L L Fnally = 0 L= T( q dt q, q!, t) V( q, t)! q! Assume that V does not depend on Done!

11 Assumptons We Made! Constrants are holonomc! We always assume ths! Constrant forces do no work! Forget frctons F = V! Lagrange s Eqn. tself s OK f V depends explctly on t V q! = 0 q!! Appled forces are conservatve! Potental V does not depend on Wll revew the last assumpton later r = r( q, q,..., q, t) 1 2 n fδ r = 0

12 Example: Tme-Dependent! Transformaton functons may depend on t! Generalzed coordnate system may move! E.g. coordnate system fxed to the Earth! An example r = r( q, t) sprng constant K natural length l mass m on a ral l + r angular velocty α

13 Example: Tme-Dependent x= ( l+ r)cosαt! Transformaton functons: y = ( l + r)snαt m 2 2 m 2 2 2! Knetc energy T = x! + y! = r! + ( l+ r) α 2 2! Potental energy V = K 2 { } { } r 2 m { ( ) } K L= r! + l+ r α r Lagrange s Equaton d L L mr m α 2 ( l r ) Kr 0 dt = + + = r!!! r

14 Example: Tme-Dependent d L L mr m α 2 ( l r ) Kr 0 dt = + + = r!!! r 2 2 mα l mr!! + ( K mα ) r 0 2 = K mα! If K > mα 2, a harmonc oscllator wth! Center of oscllaton s shfted by! If K < mα 2, moves away exponentally! If K = mα 2, velocty s constant ω =! Centrpetal force balances wth the sprng force K 2 mα m

15 Note on Arbtrarty! Lagrangan s not unque for a gven system! If a Lagrangan L descrbes a system df( q, t) L = L+ works as well for any functon F dt! One can prove d df df dt q dt! q dt = 0 usng df F F = q! + dt q t

16 Assumptons We Made! Constrants are holonomc! We always assume ths! Constrant forces do no work! Forget frctons F = V! Lagrange s Eqn. tself s OK f V depends explctly on t V q! = 0 q!! Appled forces are conservatve! Potental V does not depend on Let s revew the last assumpton r = r( q, q,..., q, t) 1 2 n fδ r = 0

17 Velocty-Dependent Potental! We assumed Q and = 0 so that d T T = Q dt q! q! We could do the same f we had Q U d U = + q dt q! V = q L= T( q, q!, t) U( q, q!, t) V q! Ths had to be 0 d ( T V) ( T V) = 0 dt q! q U = U( q, q!, t) Generalzed, or veloctydependent potental

18 EM Force on Partcle! Lorentz force on a charged partcle F = q[ E+ ( v B)]! E and B felds are gven by A E = φ t B= A! Force s v-dependent " Need a v-dependent potental U = qφ qa v works check Velocty-dependent. Can t fnd a usual potental V Physcs 15b! Lagrangan s 1 = + A v 2 2 L mv qφ q

19 Monogenc System! If all forces n a system are derved from a generalzed potental, U d U ts called a monogenc system Q! U s a functon of! Lorentz force s monogenc! A monogenc system s conservatve only f! Or U q! U = = t 0 qqt,!,! Lagrange s Equaton works on a monogenc system = + q dt q! U = U( q)

20 Hamlton s Prncple! We derved Lagrange s Eqn from Newton s Eqn usng a dfferental prncple! D Alembert s prncple uses nfntesmal dsplacements! It s possble to do t wth an ntegral prncple Hamlton s Prncple

21 Confguraton Space! Generalzed coordnates q 1,...,q n fully descrbe the system s confguraton at any moment! Imagne an n-dmensonal space! Each pont n ths space (q 1,...,q n ) corresponds to one confguraton of the system! Tme evoluton of the system " A curve n the confguraton space real space confguraton space confguraton space

22 Acton Integral! A system s movng as! Lagrangan s q = q () t = 1... n Lqqt (,!,) = Lqt ((), qt!(),) t ntegrate I t 2 = Ldt Acton, or acton ntegral t1! Acton I depends on the entre path from t 1 to t 2! Choce of coordnates q does not matter! Acton s nvarant under coordnate transformaton

23 Hamlton s Prncple The acton ntegral of a physcal system s statonary for the actual path! Ths s equvalent to Lagrange s Equatons! We wll prove ths! Three equvalent formulatons! Newton s Eqn depends explctly on x-y-z coordnates! Lagrange s Eqn s same for any generalzed coordnates! Hamlton s Prncple refers to no coordnates! Everythng s n the acton ntegral We wll also defne statonary Hamlton s Prncple s more fundamental probably...

24 Statonary! Consder two paths that are close to each other! Dfference s nfntesmal! Statonary means that the dfference of the acton ntegrals s zero to the 1st order of δq(t)! Smlar to frst dervatve = 0 t δ ( δ,! δ!,) (,!,) 0 I 2 Lq qq qtdt 2 Lqqtdt t1 t1 = + + =! Almost same as sayng mnmum! It could as well be maxmum t confguraton space t 1 qt () t 2 qt () +δ qt () δqt ( ) = δqt ( ) = 0 1 2

25 Infntesmal Path Dfference! What s δq(t)?! It s arbtrary sort of! It has to be zero at t 1 and t 2! It s well-behavng Contnuous, non-sngular, contnuous 1 st and 2 nd dervatves Don t worry too much confguraton space qt () t 2 qt () +δ qt ()! Have to shrnk t to zero! Trck: wrte t as δqt () = αη() t! α s a parameter, whch we ll make " 0! η(t) s an arbtrary well-behavng functon t 1 η( t ) = η( t ) = 0 1 2

26 Hamlton " Lagrange! To derve Lagrange s Eqns from Hamlton s Prncple t δ ( δ,! δ!,) (,!,) 0 I 2 Lq qq qtdt 2 Lqqtdt t1 t1 = + + = t! Defne t I Lqt t qt!! t tdt ( 2 α ) ( ( ) + αη ( ), ( ) + αη ( ), ) t1! δi s then lm ( ) (0) α [ I α I ] 0 I α α = 0 dα I! We must show that = 0 leads to Lagrange s Eqns α α = 0 A bt of work. Wll do t on Thursday

27 Summary! Derved Lagrange s Eqn from Newton s Eqn! Usng D Alembert s Prncple Dfferental approach! Assumptons we made:! Constrants are holonomc " Generalzed coordnates! Forces of constrants do no work " No frctons! Other forces are monogenc " Generalzed potental! Introduced Hamlton s Prncple! Integral approach! Defned the acton ntegral and statonary! Dervaton n the next lecture Q U d U = + q dt q!

### Physics 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

### Mechanics 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 ( )

### Mechanics 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

### PHYS 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

### Lagrange 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

### coordinates. 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,

### Physics 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

### CHAPTER 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

### Quantum 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

### 11. 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

### Notes 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

### Physics 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

### 12. 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

### Poisson 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

### 10. 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

### Lecture 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

### Canonical 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,

### The 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

### Classical 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

### Week3, 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

### LAGRANGIAN MECHANICS

LAGRANGIAN MECHANICS Generalzed Coordnates State of system of N partcles (Newtonan vew): PE, KE, Momentum, L calculated from m, r, ṙ Subscrpt covers: 1) partcles N 2) dmensons 2, 3, etc. PE U r = U x 1,

### A 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,

### Lagrangian 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,

### Physics 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

### Mechanics 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

### Classical 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

### Mechanics 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

### 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)]

### 4. Laws of Dynamics: Hamilton s Principle and Noether's Theorem

4. Laws of Dynamcs: Hamlton s Prncple and Noether's Theorem Mchael Fowler Introducton: Galleo and Newton In the dscusson of calculus of varatons, we antcpated some basc dynamcs, usng the potental energy

### PHYS 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

### Study 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

### Integrals 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,

### Moving coordinate system

Chapter Movng coordnate system Introducton Movng coordnate systems are mportant because, no materal body s at absolute rest As we know, even galaxes are not statonary Therefore, a coordnate frame at absolute

### Lecture 10: Euler s Equations for Multivariable

Lecture 0: Euler s Equatons for Multvarable Problems Let s say we re tryng to mnmze an ntegral of the form: {,,,,,, ; } J f y y y y y y d We can start by wrtng each of the y s as we dd before: y (, ) (

### 6. Hamilton s Equations

6. Hamlton s Equatons Mchael Fowler A Dynamcal System s Path n Confguraton Sace and n State Sace The story so far: For a mechancal system wth n degrees of freedom, the satal confguraton at some nstant

### Moments 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

### The Second Anti-Mathima on Game Theory

The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player

### Mechanics Physics 151

Mechancs Physcs 5 Lecture 3 Contnuous Systems an Fels (Chapter 3) Where Are We Now? We ve fnshe all the essentals Fnal wll cover Lectures through Last two lectures: Classcal Fel Theory Start wth wave equatons

### Week 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,

### Rigid 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

### PHYS 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

### Spring Force and Power

Lecture 13 Chapter 9 Sprng Force and Power Yeah, energy s better than orces. What s net? Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi IN THIS CHAPTER, you wll learn how to solve problems

### Analytical classical dynamics

Analytcal classcal ynamcs by Youun Hu Insttute of plasma physcs, Chnese Acaemy of Scences Emal: yhu@pp.cas.cn Abstract These notes were ntally wrtten when I rea tzpatrck s book[] an were later revse to

### Symmetric Lie Groups and Conservation Laws in Physics

Symmetrc Le Groups and Conservaton Laws n Physcs Audrey Kvam May 1, 1 Abstract Ths paper eamnes how conservaton laws n physcs can be found from analyzng the symmetrc Le groups of certan physcal systems.

### Physics 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

### CHAPTER 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

### Lecture 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.

### A Review of Analytical Mechanics

Chapter 1 A Revew of Analytcal Mechancs 1.1 Introducton These lecture notes cover the thrd course n Classcal Mechancs, taught at MIT snce the Fall of 01 by Professor Stewart to advanced undergraduates

### 1 Matrix representations of canonical matrices

1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:

### 9 Characteristic classes

THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct

### Chapter 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

### NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582

NMT EE 589 & UNM ME 48/58 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 48/58 7. Robot Dynamcs 7.5 The Equatons of Moton Gven that we wsh to fnd the path q(t (n jont space) whch mnmzes the energy

### Physics 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

### Three 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

### Kinematics of Fluids. Lecture 16. (Refer the text book CONTINUUM MECHANICS by GEORGE E. MASE, Schaum s Outlines) 17/02/2017

17/0/017 Lecture 16 (Refer the text boo CONTINUUM MECHANICS by GEORGE E. MASE, Schaum s Outlnes) Knematcs of Fluds Last class, we started dscussng about the nematcs of fluds. Recall the Lagrangan and Euleran

### Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

### Lecture 10. Reading: Notes and Brennan Chapter 5

Lecture tatstcal Mechancs and Densty of tates Concepts Readng: otes and Brennan Chapter 5 Georga Tech C 645 - Dr. Alan Doolttle C 645 - Dr. Alan Doolttle Georga Tech How do electrons and holes populate

### Mathematical 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

### 10/23/2003 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

### . The kinetic energy of this system is T = T i. m i. Now let s consider how the kinetic energy of the system changes in time. Assuming each.

Chapter 2 Systems of Partcles 2. Work-Energy Theorem Consder a system of many partcles, wth postons r and veloctes ṙ. The knetc energy of ths system s T = T = 2 mṙ2. 2. Now let s consder how the knetc

### The Principles of Dynamics

The Prncples of Dynamcs Dr C. T. Whelan. 1 Lent 000. 1 L A TEXed by Tom Bentley Techncaltes Copyrght & Dsclamer c 000, T. J. Bentley, Pembroke Collge, Cambrdge. Permsson s granted to copy and dstrbute

### 2D Motion of Rigid Bodies: Falling Stick Example, Work-Energy Principle

Example: Fallng Stck 1.003J/1.053J Dynamcs and Control I, Sprng 007 Professor Thomas Peacock 3/1/007 ecture 10 D Moton of Rgd Bodes: Fallng Stck Example, Work-Energy Prncple Example: Fallng Stck Fgure

### Homework & Solution. Contributors. Prof. Lee, Hyun Min. Particle Physics Winter School. Park, Ye

Homework & Soluton Prof. Lee, Hyun Mn Contrbutors Park, Ye J(yej.park@yonse.ac.kr) Lee, Sung Mook(smlngsm0919@gmal.com) Cheong, Dhong Yeon(dhongyeoncheong@gmal.com) Ban, Ka Young(ban94gy@yonse.ac.kr) Ro,

### Inner 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,

### Module 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

### THEOREMS OF QUANTUM MECHANICS

THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat many-electron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn

### First 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

### Group Analysis of Ordinary Differential Equations of the Order n>2

Symmetry n Nonlnear Mathematcal Physcs 997, V., 64 7. Group Analyss of Ordnary Dfferental Equatons of the Order n> L.M. BERKOVICH and S.Y. POPOV Samara State Unversty, 4430, Samara, Russa E-mal: berk@nfo.ssu.samara.ru

Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton

### Linear Momentum. Center of Mass.

Lecture 6 Chapter 9 Physcs I 03.3.04 Lnear omentum. Center of ass. Course webste: http://faculty.uml.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcssprng.html

### (δr i ) 2. V i. r i 2,

Cartesan coordnates r, = 1, 2,... D for Eucldean space. Dstance by Pythagoras: (δs 2 = (δr 2. Unt vectors ê, dsplacement r = r ê Felds are functons of poston, or of r or of {r }. Scalar felds Φ( r, Vector

### PY2101 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

### Chapter 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

### Physics 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

### 1 What is a conservation law?

MATHEMATICS 7302 (Analytcal Dynamcs) YEAR 2016 2017, TERM 2 HANDOUT #6: MOMENTUM, ANGULAR MOMENTUM, AND ENERGY; CONSERVATION LAWS In ths handout we wll develop the concepts of momentum, angular momentum,

### UNIT-I 1.0 INTRODUCTION

BLOCK-I Ths Block-I of Mechancs s devoted to the man topcs of Classcal Mechancs. Tll the tme of standardzaton of Basc Unts of measurement--m L T Kelvn Ampere and Mole -- the laws about the behavour of

### Chapter 3. r r. Position, Velocity, and Acceleration Revisited

Chapter 3 Poston, Velocty, and Acceleraton Revsted The poston vector of a partcle s a vector drawn from the orgn to the locaton of the partcle. In two dmensons: r = x ˆ+ yj ˆ (1) The dsplacement vector

### Iterative 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

### χ 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

### Physics 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

### Integrals 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

### High-Order Hamilton s Principle and the Hamilton s Principle of High-Order Lagrangian Function

Commun. Theor. Phys. Bejng, Chna 49 008 pp. 97 30 c Chnese Physcal Socety Vol. 49, No., February 15, 008 Hgh-Orer Hamlton s Prncple an the Hamlton s Prncple of Hgh-Orer Lagrangan Functon ZHAO Hong-Xa an

### Modelli Clamfim Equazioni differenziali 7 ottobre 2013

CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 7 ottobre 2013 professor Danele Rtell danele.rtell@unbo.t 1/18? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp

### Advanced Circuits Topics - Part 1 by Dr. Colton (Fall 2017)

Advanced rcuts Topcs - Part by Dr. olton (Fall 07) Part : Some thngs you should already know from Physcs 0 and 45 These are all thngs that you should have learned n Physcs 0 and/or 45. Ths secton s organzed

### Lecture 16. Chapter 11. Energy Dissipation Linear Momentum. Physics I. Department of Physics and Applied Physics

Lecture 16 Chapter 11 Physcs I Energy Dsspaton Lnear Momentum Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi Department o Physcs and Appled Physcs IN IN THIS CHAPTER, you wll learn

### Lecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2

P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The non-relatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons

### Conservation 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

### THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions

THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George

### 14 The Postulates of Quantum mechanics

14 The Postulates of Quantum mechancs Postulate 1: The state of a system s descrbed completely n terms of a state vector Ψ(r, t), whch s quadratcally ntegrable. Postulate 2: To every physcally observable

### How Differential Equations Arise. Newton s Second Law of Motion

page 1 CHAPTER 1 Frst-Order Dfferental Equatons Among all of the mathematcal dscplnes the theory of dfferental equatons s the most mportant. It furnshes the explanaton of all those elementary manfestatons

### 1. 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)

### Physics 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

### ELASTIC 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

### Spring 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

### ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION

Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION

### 1.2 Single Particle Kinematics

1.. SINGLE PARTICLE KINEMATICS 3 multpole moments of the sun forces due to other planets effects of correctons to Newtonan gravty due to general relatvty frcton due to the solar wnd and gas n the solar

### Physics 114 Exam 2 Fall 2014 Solutions. Name:

Physcs 114 Exam Fall 014 Name: For gradng purposes (do not wrte here): Queston 1. 1... 3. 3. Problem Answer each of the followng questons. Ponts for each queston are ndcated n red. Unless otherwse ndcated,

### Georgia Tech PHYS 6124 Mathematical Methods of Physics I

Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends