Georgia Tech PHYS 6124 Mathematical Methods of Physics I

Size: px
Start display at page:

Download "Georgia Tech PHYS 6124 Mathematical Methods of Physics I"

Transcription

1 Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October == show all your work for maxmum credt == put labels ttle legends on any graphs == acknowledge study group member f collectve effort [All problems n ths set are from Goldbart] Problem ) Moton of a classcal partcle Consder a classcal partcle of unt mass movng along the x-axs. Suppose that the moton s free except that at tme τ = t wth 0 < t < T the partcle receves an mpulse of unt strength. a) Wrte Newton s equaton descrbng the moton of the partcle. Suppose that at tme τ = 0 the partcle s located at poston x and that at tme τ = T t s located at poston x 2. b) Sketch the poston velocty and acceleraton of the partcle as a functon of tme for 0 < τ < T. c) Compute the Green functon for the moton of the partcle.e. solve d 2 G(τ t)/dτ 2 = δ(τ t). d) Consder the appled force f (τ) (wth 0 τ T) to be a sequence of mpulses. Hence establsh the trajectory of the partcle n terms of an ntegral over the appled force. e) Suppose that the force takes the form f (τ) = τ 2 /2. Fnd the moton of the partcle. f) (optonal) Rather than solve for the Green functon drectly as you dd n part (c) construct the Green functon usng the egenfuncton expanson technque. Show by Fourer analyss that the two schemes for computng the Green functon gve equvalent results. Problem 4) Green functons Consder the second order nhomogeneous lnear dfferental equaton: y y = f (x).

2 a) Suppose that the boundary condtons for y are y(± ) = 0. By solvng the assocated homogeneous equaton construct the approprate Green functon for ths equaton. b) Solve the dfferental equaton when the source term s gven by f (x) = 2 e x. Problem 6) Green functons and ther nverses The purpose of ths queston s to show that f G s the Green functon assocated wth some operator L then L s the Green functon assocated wth the operator G. Consder the operator L ( d 2 /dθ 2 + q 2 /2π) n whch the real varable q does not vansh. The functons u(θ) on whch L acts are chosen to be perodc functons on the nterval 0 θ < 2π.e. u(2π) = u(0) and u (2π) = u (0). a) By usng the egenfuncton expanson show that the Green functon assocated wth the operator L s gven by G(θ θ ) = 2πq 2 + cos n(θ θ π ) n= q 2 + n 2. Now show that L s the Green functon for G n the followng sense. Consder the nhomogeneous ntegral equaton 2π 0 dθ G(θ θ ) y(θ ) = f (θ) for the unknown functon y(θ) n whch the source term f (θ) s presumed known. (b) Show that the soluton y(θ) s gven by. Optonal problems y(θ) = (L f )(θ) ( d 2 /dθ 2 + q 2 /2π) f (θ). Problem 2) Varaton of parameters Use the method of varaton of parameters to fnd the most general soluton to the second order lnear nhomogeneous dfferental equaton: x 2 y 4xy + 6y = x 4. Problem 3) General soluton to an nhomogeneous equaton Suppose y = x y = x 2 and y = x 3 each satsfy the second-order nhomogeneous equaton Ly = f (x). Fnd the general soluton. 2

3 Problem 5) More Green functons Fnd the Green functon for these operators: ) Ly (xy ) ; ) Ly (xy ) n 2 y/x; ) Ly x 2 y + xy + (k 2 x 2 )y. Problem 7) Egenfuncton expanson for the bowed strng Green functon In class we studed the Green functon for the response at the drvng frequency ω( ck) of a bowed stretched strng of length l by solvng drectly the equaton ( d 2 dx 2 + k2) G(x x ) = δ(x x ) subject to the boundary condtons G(x x ) x=0l = 0. Use the egenfuncton expanson to produce an alternatve dervaton of ths Green functon and demonstrate that the verson thus obtaned s equvalent to the verson G(x x ) = sn(kx <) sn k(x > l) k sn(kl) obtaned n class n whch x < mn(x x ) and x > max(x x ). Problem 8) Drac delta functon In ths queston we shall explore some of the propertes of the Drac delta functon (the generalsaton to the contnuum of the Kronecker delta symbol). The Drac delta functon δ(x) has as ts argument the real varable x. It has the followng rather strkng propertes:() δ(x) = 0 for x = 0; and dx δ(x) =. From these propertes you can see that δ(x) has an nfntely hgh spke at the orgn s zero elsewhere and has unt area. a) Gve a heurstc (.e. sloppy) proof that f f (x) s a suffcently smooth functon then dx δ(x) f (x) = f (0). b) Show that dx δ(x a) f (x) = f (a). Often we shall neglect to wrte the lmts on ntegrals. c) Usng ntegraton by parts show that dx δ (x) f (x) = f (0) where f (x) d f (x)/dx and δ (x) dδ(x)/dx. d) Consder the famly of gaussan functons ξ (x) { exp x2 } 2πξ 2 2ξ 2 parametrsed by the wdth ξ. Show that the defnng propertes of δ(x) are satsfed by lm ξ 0 ξ (x) and hence that ths lmt represents δ(x). Hnt: recall that dy exp( y2 /2) = 2π. 3

4 e) By consderng the Fourer transform and ts nverse establsh that f (y) = Thus demonstrate that f (q) = f (x) = δ(y) = dx f (x) dx f (x) e qx 2π dq f (q) eqx 2π dq 2π eqy. dq 2π eq(y x). Ths ntegral representaton of the Drac delta functon s known as the Fourer representaton of the Drac delta functon. f) By consderng the Fourer representaton show that δ(ax) = a δ(x). g) Prove ths result startng wth the gaussan representaton of part (d). We can also have hgher dmensonal delta functons by whch we mean delta functons wth vector arguments. For example f r s a three-dmensonal cartesan vector wth cartesan components x y and z then the delta functon δ (3) ( r) s defned to be the product δ (3) ( r) = δ(x) δ(y) δ(z). Often the superscrpt (3) s omtted. h) Wrte down the Fourer representaton of δ (3) ( r) n terms of the vector r and an ntegral over the vector q. ) Wrte down a representaton for δ (3) ( r) n terms of the gaussan functon. j) A real functon of a sngle varable f (x) has zeros at the set of ponts {x }. At these zeros the gradent of f s non-vanshng and takes the values { f () }. Show that δ ( f (x) ) = δ(x x ) f (). k) A real symmetrc 3 matrx A s non-sngular (.e. all ts egenvalues are non-zero real numbers). Prove usng ether the gaussan or the Fourer representaton that δ (3) (A x) = deta δ(3) ( x). 4

5 A thrd representaton of the delta functon s very convenent n the context of complex varables. Consder the real varables ω and. If Im denotes the magnary part use the followng strategy to show that l.) Show that l.) Show that lm lm ω 0 0 lm 0 ω = 0 π Im ω = δ(ω). Im ω = ω lm lm 0 ω 0 ω =. l.) Usng the substtuton ω = tan θ or otherwse show that dω ω = π. l.v) Put together these peces to argue that ndeed lm 0 π Im ω = δ(ω) s a representaton of the delta functon. 5

), it produces a response (output function g (x)

), it produces a response (output function g (x) Lnear Systems Revew Notes adapted from notes by Mchael Braun Typcally n electrcal engneerng, one s concerned wth functons of tme, such as a voltage waveform System descrpton s therefore defned n the domans

More information

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

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

More information

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1

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

More information

APPENDIX A Some Linear Algebra

APPENDIX A Some Linear Algebra APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

More information

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM

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

More information

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

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

More information

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2 Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to

More information

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0 MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector

More information

Mathematical Preparations

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

More information

(c) (cos θ + i sin θ) 5 = cos 5 θ + 5 cos 4 θ (i sin θ) + 10 cos 3 θ(i sin θ) cos 2 θ(i sin θ) 3 + 5cos θ (i sin θ) 4 + (i sin θ) 5 (A1)

(c) (cos θ + i sin θ) 5 = cos 5 θ + 5 cos 4 θ (i sin θ) + 10 cos 3 θ(i sin θ) cos 2 θ(i sin θ) 3 + 5cos θ (i sin θ) 4 + (i sin θ) 5 (A1) . (a) (cos θ + sn θ) = cos θ + cos θ( sn θ) + cos θ(sn θ) + (sn θ) = cos θ cos θ sn θ + ( cos θ sn θ sn θ) (b) from De Movre s theorem (cos θ + sn θ) = cos θ + sn θ cos θ + sn θ = (cos θ cos θ sn θ) +

More information

Lecture 12: Discrete Laplacian

Lecture 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 information

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

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,

More information

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017 U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that

More information

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

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,

More information

For now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results.

For now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results. Neural Networks : Dervaton compled by Alvn Wan from Professor Jtendra Malk s lecture Ths type of computaton s called deep learnng and s the most popular method for many problems, such as computer vson

More information

Robert Eisberg Second edition CH 09 Multielectron atoms ground states and x-ray excitations

Robert Eisberg Second edition CH 09 Multielectron atoms ground states and x-ray excitations Quantum Physcs 量 理 Robert Esberg Second edton CH 09 Multelectron atoms ground states and x-ray exctatons 9-01 By gong through the procedure ndcated n the text, develop the tme-ndependent Schroednger equaton

More information

From Biot-Savart Law to Divergence of B (1)

From Biot-Savart Law to Divergence of B (1) From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to

More information

Errors for Linear Systems

Errors for Linear Systems Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch

More information

Physics 5153 Classical Mechanics. Principle of Virtual Work-1

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

More information

Math 217 Fall 2013 Homework 2 Solutions

Math 217 Fall 2013 Homework 2 Solutions Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has

More information

8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before

8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before .1 Arc Length hat s the length of a curve? How can we approxmate t? e could do t followng the pattern we ve used before Use a sequence of ncreasngly short segments to approxmate the curve: As the segments

More information

Canonical transformations

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,

More information

THEOREMS OF QUANTUM MECHANICS

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

More information

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng

More information

Module 1 : The equation of continuity. Lecture 1: Equation of Continuity

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

More information

Digital Signal Processing

Digital Signal Processing Dgtal Sgnal Processng Dscrete-tme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over

More information

Math1110 (Spring 2009) Prelim 3 - Solutions

Math1110 (Spring 2009) Prelim 3 - Solutions Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.

More information

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

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

More information

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

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

More information

Week 9 Chapter 10 Section 1-5

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,

More information

10/23/2003 PHY Lecture 14R 1

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

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order:

1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order: 68A Solutons to Exercses March 05 (a) Usng a Taylor expanson, and notng that n 0 for all n >, ( + ) ( + ( ) + ) We can t nvert / because there s no Taylor expanson around 0 Lets try to calculate the nverse

More information

ON MECHANICS WITH VARIABLE NONCOMMUTATIVITY

ON MECHANICS WITH VARIABLE NONCOMMUTATIVITY ON MECHANICS WITH VARIABLE NONCOMMUTATIVITY CIPRIAN ACATRINEI Natonal Insttute of Nuclear Physcs and Engneerng P.O. Box MG-6, 07725-Bucharest, Romana E-mal: acatrne@theory.npne.ro. Receved March 6, 2008

More information

The Feynman path integral

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

More information

Classical Field Theory

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

More information

Causal Diamonds. M. Aghili, L. Bombelli, B. Pilgrim

Causal Diamonds. M. Aghili, L. Bombelli, B. Pilgrim Causal Damonds M. Aghl, L. Bombell, B. Plgrm Introducton The correcton to volume of a causal nterval due to curvature of spacetme has been done by Myrhem [] and recently by Gbbons & Solodukhn [] and later

More information

IMGS-261 Solutions to Homework #9

IMGS-261 Solutions to Homework #9 IMGS-6 Solutons to Homework #9. For f [] SINC [] sn[π], use the modulaton theorem to evaluate and sketch π the Fourer transform of f [] f [] f [] (f []) Soluton: We know that F{RECT []} SINC [] so we use

More information

π e ax2 dx = x 2 e ax2 dx or x 3 e ax2 dx = 1 x 4 e ax2 dx = 3 π 8a 5/2 (a) We are considering the Maxwell velocity distribution function: 2πτ/m

π e ax2 dx = x 2 e ax2 dx or x 3 e ax2 dx = 1 x 4 e ax2 dx = 3 π 8a 5/2 (a) We are considering the Maxwell velocity distribution function: 2πτ/m Homework Solutons Problem In solvng ths problem, we wll need to calculate some moments of the Gaussan dstrbuton. The brute-force method s to ntegrate by parts but there s a nce trck. The followng ntegrals

More information

Chapter 4 The Wave Equation

Chapter 4 The Wave Equation Chapter 4 The Wave Equaton Another classcal example of a hyperbolc PDE s a wave equaton. The wave equaton s a second-order lnear hyperbolc PDE that descrbes the propagaton of a varety of waves, such as

More information

ρ some λ THE INVERSE POWER METHOD (or INVERSE ITERATION) , for , or (more usually) to

ρ some λ THE INVERSE POWER METHOD (or INVERSE ITERATION) , for , or (more usually) to THE INVERSE POWER METHOD (or INVERSE ITERATION) -- applcaton of the Power method to A some fxed constant ρ (whch s called a shft), x λ ρ If the egenpars of A are { ( λ, x ) } ( ), or (more usually) to,

More information

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1 C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned

More information

Module 3: Element Properties Lecture 1: Natural Coordinates

Module 3: Element Properties Lecture 1: Natural Coordinates Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers

More information

Error Bars in both X and Y

Error Bars in both X and Y Error Bars n both X and Y Wrong ways to ft a lne : 1. y(x) a x +b (σ x 0). x(y) c y + d (σ y 0) 3. splt dfference between 1 and. Example: Prmordal He abundance: Extrapolate ft lne to [ O / H ] 0. [ He

More information

Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry

Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department

More information

Structure and Drive Paul A. Jensen Copyright July 20, 2003

Structure and Drive Paul A. Jensen Copyright July 20, 2003 Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.

More information

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

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

More information

PHYS 705: Classical Mechanics. Calculus of Variations II

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

More information

Modeling of Dynamic Systems

Modeling 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 information

Physics 181. Particle Systems

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

More information

A Hybrid Variational Iteration Method for Blasius Equation

A 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 information

Yong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )

Yong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 ) Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often

More information

Strong Markov property: Same assertion holds for stopping times τ.

Strong Markov property: Same assertion holds for stopping times τ. Brownan moton Let X ={X t : t R + } be a real-valued stochastc process: a famlty of real random varables all defned on the same probablty space. Defne F t = nformaton avalable by observng the process up

More information

(δr i ) 2. V i. r i 2,

(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

More information

10-801: Advanced Optimization and Randomized Methods Lecture 2: Convex functions (Jan 15, 2014)

10-801: Advanced Optimization and Randomized Methods Lecture 2: Convex functions (Jan 15, 2014) 0-80: Advanced Optmzaton and Randomzed Methods Lecture : Convex functons (Jan 5, 04) Lecturer: Suvrt Sra Addr: Carnege Mellon Unversty, Sprng 04 Scrbes: Avnava Dubey, Ahmed Hefny Dsclamer: These notes

More information

Module 14: THE INTEGRAL Exploring Calculus

Module 14: THE INTEGRAL Exploring Calculus Module 14: THE INTEGRAL Explorng Calculus Part I Approxmatons and the Defnte Integral It was known n the 1600s before the calculus was developed that the area of an rregularly shaped regon could be approxmated

More information

Three views of mechanics

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

More information

PHYS 705: Classical Mechanics. Newtonian Mechanics

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 information

Integrals and Invariants of Euler-Lagrange Equations

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,

More information

Modelli Clamfim Equazioni differenziali 22 settembre 2016

Modelli Clamfim Equazioni differenziali 22 settembre 2016 CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 22 settembre 2016 professor Danele Rtell danele.rtell@unbo.t 1/22? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp

More information

10-701/ Machine Learning, Fall 2005 Homework 3

10-701/ Machine Learning, Fall 2005 Homework 3 10-701/15-781 Machne Learnng, Fall 2005 Homework 3 Out: 10/20/05 Due: begnnng of the class 11/01/05 Instructons Contact questons-10701@autonlaborg for queston Problem 1 Regresson and Cross-valdaton [40

More information

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not

More information

12. The Hamilton-Jacobi Equation Michael Fowler

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

More information

Uniqueness of Weak Solutions to the 3D Ginzburg- Landau Model for Superconductivity

Uniqueness of Weak Solutions to the 3D Ginzburg- Landau Model for Superconductivity Int. Journal of Math. Analyss, Vol. 6, 212, no. 22, 195-114 Unqueness of Weak Solutons to the 3D Gnzburg- Landau Model for Superconductvty Jshan Fan Department of Appled Mathematcs Nanjng Forestry Unversty

More information

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 16

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 16 STAT 39: MATHEMATICAL COMPUTATIONS I FALL 218 LECTURE 16 1 why teratve methods f we have a lnear system Ax = b where A s very, very large but s ether sparse or structured (eg, banded, Toepltz, banded plus

More information

Tensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q

Tensor 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 information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as:

This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as: 1 Problem set #1 1.1. A one-band model on a square lattce Fg. 1 Consder a square lattce wth only nearest-neghbor hoppngs (as shown n the fgure above): H t, j a a j (1.1) where,j stands for nearest neghbors

More information

Modelli Clamfim Equazione del Calore Lezione ottobre 2014

Modelli Clamfim Equazione del Calore Lezione ottobre 2014 CLAMFIM Bologna Modell 1 @ Clamfm Equazone del Calore Lezone 17 15 ottobre 2014 professor Danele Rtell danele.rtell@unbo.t 1/24? Convoluton The convoluton of two functons g(t) and f(t) s the functon (g

More information

THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions

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

More information

1 Matrix representations of canonical matrices

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:

More information

14 The Postulates of Quantum mechanics

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

More information

Week 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product

Week 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 information

Solution 1 for USTC class Physics of Quantum Information

Solution 1 for USTC class Physics of Quantum Information Soluton 1 for 018 019 USTC class Physcs of Quantum Informaton Shua Zhao, Xn-Yu Xu and Ka Chen Natonal Laboratory for Physcal Scences at Mcroscale and Department of Modern Physcs, Unversty of Scence and

More information

How Differential Equations Arise. Newton s Second Law of Motion

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

More information

Lecture 2: Numerical Methods for Differentiations and Integrations

Lecture 2: Numerical Methods for Differentiations and Integrations Numercal Smulaton of Space Plasmas (I [AP-4036] Lecture 2 by Lng-Hsao Lyu March, 2018 Lecture 2: Numercal Methods for Dfferentatons and Integratons As we have dscussed n Lecture 1 that numercal smulaton

More information

Topic 5: Non-Linear Regression

Topic 5: Non-Linear Regression Topc 5: Non-Lnear Regresson The models we ve worked wth so far have been lnear n the parameters. They ve been of the form: y = Xβ + ε Many models based on economc theory are actually non-lnear n the parameters.

More information

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system

Transfer 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 information

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

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

More information

NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS

NON-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 information

Modelli Clamfim Equazioni differenziali 7 ottobre 2013

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

More information

3 Basic boundary value problems for analytic function in the upper half plane

3 Basic boundary value problems for analytic function in the upper half plane 3 Basc boundary value problems for analytc functon n the upper half plane 3. Posson representaton formulas for the half plane Let f be an analytc functon of z throughout the half plane Imz > 0, contnuous

More information

Lecture 21: Numerical methods for pricing American type derivatives

Lecture 21: Numerical methods for pricing American type derivatives Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)

More information

9 Characteristic classes

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

More information

Prof. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model

Prof. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several

More information

Rate of Absorption and Stimulated Emission

Rate of Absorption and Stimulated Emission MIT Department of Chemstry 5.74, Sprng 005: Introductory Quantum Mechancs II Instructor: Professor Andre Tokmakoff p. 81 Rate of Absorpton and Stmulated Emsson The rate of absorpton nduced by the feld

More information

Week 8: Chapter 9. Linear Momentum. Newton Law and Momentum. Linear Momentum, cont. Conservation of Linear Momentum. Conservation of Momentum, 2

Week 8: Chapter 9. Linear Momentum. Newton Law and Momentum. Linear Momentum, cont. Conservation of Linear Momentum. Conservation of Momentum, 2 Lnear omentum Week 8: Chapter 9 Lnear omentum and Collsons The lnear momentum of a partcle, or an object that can be modeled as a partcle, of mass m movng wth a velocty v s defned to be the product of

More information

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.

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

More information

SIO 224. m(r) =(ρ(r),k s (r),µ(r))

SIO 224. m(r) =(ρ(r),k s (r),µ(r)) SIO 224 1. A bref look at resoluton analyss Here s some background for the Masters and Gubbns resoluton paper. Global Earth models are usually found teratvely by assumng a startng model and fndng small

More information

Homework Notes Week 7

Homework Notes Week 7 Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we

More information

Mechanics Physics 151

Mechanics 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 information

Snce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t

Snce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t 8.5: Many-body phenomena n condensed matter and atomc physcs Last moded: September, 003 Lecture. Squeezed States In ths lecture we shall contnue the dscusson of coherent states, focusng on ther propertes

More information

Complex Numbers Alpha, Round 1 Test #123

Complex Numbers Alpha, Round 1 Test #123 Complex Numbers Alpha, Round Test #3. Wrte your 6-dgt ID# n the I.D. NUMBER grd, left-justfed, and bubble. Check that each column has only one number darkened.. In the EXAM NO. grd, wrte the 3-dgt Test

More information

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede Fall 0 Analyss of Expermental easurements B. Esensten/rev. S. Errede We now reformulate the lnear Least Squares ethod n more general terms, sutable for (eventually extendng to the non-lnear case, and also

More information

CHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)

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

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

The equation of motion of a dynamical system is given by a set of differential equations. That is (1)

The equation of motion of a dynamical system is given by a set of differential equations. That is (1) Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence

More information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons

More information

EN40: Dynamics and Vibrations. Homework 4: Work, Energy and Linear Momentum Due Friday March 1 st

EN40: Dynamics and Vibrations. Homework 4: Work, Energy and Linear Momentum Due Friday March 1 st EN40: Dynamcs and bratons Homework 4: Work, Energy and Lnear Momentum Due Frday March 1 st School of Engneerng Brown Unversty 1. The fgure (from ths publcaton) shows the energy per unt area requred to

More information

Erratum: A Generalized Path Integral Control Approach to Reinforcement Learning

Erratum: A Generalized Path Integral Control Approach to Reinforcement Learning Journal of Machne Learnng Research 00-9 Submtted /0; Publshed 7/ Erratum: A Generalzed Path Integral Control Approach to Renforcement Learnng Evangelos ATheodorou Jonas Buchl Stefan Schaal Department of

More information