Mathematical Preparations
|
|
- Bartholomew Griffin Dean
- 5 years ago
- Views:
Transcription
1 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 velocty of EM radaton was found to be un-affected by the moton of source of radaton wth respect to the coordnate system,.e. the space through whch the EM radaton travels. On the other hand, Newton postulated that n the absence of forces, the spatal coordnates of a movng pont are lnear n tme. Thus the poston of a partcle measured n a rest frame of reference, X, related to ts poston n a movng frame, X, s gven by X = X + V 0 t. Usng ths coordnate transformaton, the veloctes n two systems movng wth constant velocty, V 0, relatve to each other s; dx dt = dx dt + V 0 V = V + V 0 The above equatons represent a spatal transformaton between coordnate frames movng wth constant velocty, V 0, wth respect to each other. It s presumed that there s a fxed, unversal coordnate frame (nertal frame), and all frames movng wth constant velocty wth respect to ths frame have the same acceleraton; d 2 X dt 2 = a = a The nvarance of the laws of physcs n dfferent coordante frames s a symmetry called the Prncple of Relatvty. In the above case, Newton s laws of moton are the same n all nertal frames, as the force (acceleraton) F = M a s ndependent of the nertal frame. However, the ndependence of the velocty of EM waves between dfferent coordnate frames s not consstent wth Neutonan physcs. 2 Gallean transformaton To be consstent, the mathematcal form of all physcs laws cannot be changed by a coordnate transformaton. In the case of Newtonan physcs, a transformaton between nertal frames preserves Newton s laws of mechancs, and s called a Gallean transformaton. The Gallean transformaton transformaton s defned below. 1
2 Suppose 2 reference frames related to each other by a constant velocty along the Z axs, (X, Y, Z) and (X, Y, Z ). A system not subject to a force experences no force n any nertal system. Thus f the force n one frame s gven by F = M a then the force n the other frame s; F = F = M a wth a = a ; d 2 X dt 2 = d2 X dt 2 = a The Gallean transformaton between nertal systems must take the form; X Y Z t = X = Y = Z + V 0 t = t Although Newton s laws of mechancs are nvarent under a Gallean transformaton, Maxwell s equatons whch descrbe electrodynamcs are not, and ths was recognzed long before the theory of relatvty. Thus the descrpton of electromagnetc radaton was nconsstent wth a Galelan transformaton. It was orgnally thought that Maxwell s equatons were ncomplete, and theores were proposed to correct EM under the assumpton that a Galelan transformaton correctly descrbed the coordante transformaton between movng bodes. We now know of course, that EM was correct and Newtonan mechancs requred modfcaton. 2.1 Generalzed coordnates Because 4 rather that 3 dmenson (3 spatal and one tme coordnate) are necessary to descrbe the relatve moton of systems, t s mportant to frst dscuss geometry and transformatons n a generalzed set of coordnates. Most students have been mnmaly exposed to ths mathematcs. However, only the parts of tensor analyss requred for specal relatvty are developed here. General relatvty requres more n-depth development whch s not necessary for the study of classcal electrodynamcs. All coordnate systems are defned relatve to a Cartesan set of axes. For 3-D wrte (x 1, x 2, x 2 ), although extenson to more spatal dmensons s trval. Thus there s a 3- D functon of the coordnates whch locates some pont n space. Ths pont can also be located n a dfferent coordnate frame, ζ ( = 1, 2, 3); ζ (x 1, x 2, x 3 ) = 1, 2, 3 2
3 There also exsts a unque nverse of the transformaton functon between the coordnates. Mathematcally, ths s descrbed by a one-to-one mappng of each pont n one frane to one pont n the other. Ths mappng must have a unque nverse so that each pont has only one locaton n all frames of reference. x (ζ 1, ζ 2, ζ 3 ) = 1, 2, 3 Now at the ntersecton of the planes; ζ = constant = 1, 2, 3 defne a set of unt vectors, â, perpendcular to each surface. If these vectors are mutually orthorgonal, an orthorgonal coordnate system s defned. A reference frame wth orthogonal coordnates s not necessary n general, but orthogonal coordnates greately smplfes the mathematcs. The drecton cosnes of the coordnates wth respect to the set of Cartesan unt vectors are; â 1 ˆx = α = γ 1 â 2 ˆx = β = γ 2 â 3 ˆx = γ = γ 3 For an orthorgonal system, the 3 non-trval drecton cosnes are related, as may be shown by calculatng â â j for, j = 1, 2, 3. Then; 3 s=1 γ ms γ ns = 3 s=1 γ sm γ sn = δ mn â n = j γ nj ˆx j ˆx j = n γ nj â n Now consder the dfferental element of length, ds. In the Cartesan system, the square of ths element s; ds ds = 3 dx 2 =1 Suppose a general curvlnear set of coordnates s ntroduced as defned above. 3
4 dx = 3 j=1 dζ j The square of the length elements s then ds 2 = 3 3 j,k=1 =1 Ths s rewrtten as ; g jk = 3 =1 ζ k ζ k dζ j dζ k where g jk are the metrc elements whch defne the space. Therefore; ds 2 = jk g jk dζ j dζ k In the case of an orthorgonal system g jk = 0 f j k, so defne a scale factor, h 2 j = Note that h j dζ j s the length element for the j th coordnate. ( ) 2. ds 2 = (h dζ ) 2 Usng ths, one can obtan the dfferental volume and area elements; dτ = (h 1 dζ 1 )(h 2 dζ 2 )(h 3 dζ 3 ) dσ k = (h dζ )(h j dζ j ) To obtan the varous surface areas n the above, apply cyclc permentatons of, j, k. In general h vares at each pont n the coordnate space. The drecton cosnes along the new coordnate axes (ONLY for an orthorgonal system) are; γ n = (1/h n ) ζ n = h n ζ n (no sum) Not only do the scale factors change wth poston, but also the unt vectors change drectons, Fg. 1. For example, ˆx h j ζj â j = â j = ζ k ζ k Whch can be reduced to; ˆx h j 4
5 a^ 2 α a^ 1 d ζ 1 dh 2 dζ 1 d ζ 2 h 2 dζ 2 a^ 1 h 1 dζ 1 a^2 a^ 1 Fgure 1: A cross secton of an area element n a generalzed coordnate system â j ζ = â h h j It s then nterestng to apply these equatons to a famlar coordnate system. Use sphercal coordnates for ths example. x = rcos(φ)sn(θ) â 1 = sn(θ) cos(φ) ˆx + sn(θ) sn(φ) ŷ + cos(θ) ẑ y = rsn(φ)sn(θ) â 2 = cos(θ) cos(φ) ˆx + cos(θ) sn(φ) ŷ sn(θ) ẑ x = rcos(θ) â 3 = sn(φ) ˆx + cos(φ) ŷ Take the partal dervatves to show that an orthorgonal system s produced ( 0). The square of the metrc length s; ζ k = ds 2 = dr 2 + r 2 dθ 2 + r 2 sn 2 (θ)dφ 2 as expected. Unt vectors, volume/area elements, and the vector operatons gradent, dv, and curl can be obtaned from the physcal defnton of these operators. 2.2 Tensors Tensors are defned by consderng the transformaton propertes of functons under a coordnate rotaton and reflecton. Thus a scalar functon does not change value under rotaton or reflecton. As an example the functon f = 3 (x x 0, ) 2 remans constant and for ths example, s the magntude of a vector. On the other hand f we consder; =1 5
6 f = 3 =1 f ˆx then f transforms as a vector whch preserves magntude but changes drecton. It also changes sgn upon reflecton x x. All these propertes are preserved when a coordnate transformaton s appled so that the representaton of a vector s ndependent of the coordnate frame. A true scalar remans the same under all coordnate transformatons, ncludng reflectons. However, f a scalar functon s constant under rotaton but changes sgn under reflecton t s a pseudo-scalar. Smlarly f a vector does not change sgn under reflecton t s a pseudo-vector. As an example, a pseudo-vector s the result of the cross product of 2 true vectors as wll be observed below. Now generalze ths descrpton of functons by defnng a scalar functon as a tensor of rank 0, and a vector functon as a tensor of rank 1. Ths can be generalzed by extendng the transformaton propertes to hgher rank. To help wth notaton, the summaton conventon s employed unless t leads to ambgutes. The summaton conventon suppresses the symbol and s represented by a repeated ndex on the varables. Thus the defnton; x k = x k Suppose an n-dnemsonal space, wth N ndependent varables x = 1,, n. The set of x defne a pont n ths space. Now defne a set of n lnearly ndependent functons ζ (x 1,, x n ) = 1,, n. The Jacoban of a set of lneraly ndependent functons does not vansh. J = ζ 1 x 1 ζ n x 1 0 ζ 1 ζ n x n x n The functons, ζ, defne a new coordnate system. Make the substtuton x = ζ, and evaluate x k for future use. x k In addton; = δ j = x k x dx = x dx j 6
7 3 Tensor contracton and drect product In the followng, use the results of the dfferental operatons between the prmed and unprmed frame whch were obtaned n the last secton. The dfferental quanttes dx and dx j are related by a lnear transformaton, x. A tensor functon s defned by the lnear transformaton of ts dfferental form between two coordnate frames. Thus a tensor, A, of rank 1 (a vector) has the transformaton propertes; A = j A j For the record, ths s a contravarent tensor ndcated by the super-scrpted ndex. A subscrpted ndex ndcates a covarent tensor, and hgher order tensors wth both super- and sub-scrpts are called a mxed tensor. A = j x A j For Cartesan coordnates covarent and contravarent tensors are dentcal snce; x k = δ jk The contracton of any tensor by a vector for example (for 2 vectors ths s the dot product) reduces the order of the tensor by one (the rank of the tensor less the rank of the vector). In the case of contractng 2 vectors a scalar s produced. A B = jk x A k = k j A j B j On the other hand, the drect product of 2 tensors multples each element of a tensor by the elements of the other tensor. Ths ncreases the rank of the tensor by the sum of the ranks of each tensor. Thus the drect product of a tensor of rank 1 (a vector) wth another tensor of rank 1, produces a tensor of rank 2 (a matrx). A B l = j,m x x l A B m m Note the above form transforms lke a tensor of rank 2. 4 The metrc tensor As prevously, the square of the length element s; 7
8 ds 2 = dx dx = g jk dx j dx k The g j form a tensor of 2 nd rank called the metrc tensor of the space. The determnant s g = g j 0. It s possble n general to have ds 2 < 0, however, ths would not be consstent wth length, so the measure of the space s taken as the absolute value of ds 2. Note that ds 2 s a tensor of rank 0, e a scalar quantty. 5 Lev-Cvta tensor It s useful to defne the followng tensor of rank 3 or hgher. ǫ jk = d The constant d takes on the followng values. ǫ jk = 1 when, j, k = 1, 2, 3 f j k and wth an even permutaton of 1, 2, 3. The tensor equals -1 f the ndces are an odd permutaton of 1, 2, 3 and the tensor s 0 f any of the ndces have the same value. Ths tensor s a pseudo-tensor, e a tensor wth nverted symmetry upon nterchange of ndces. A conjugate tensor wth the same propertes can also be defned. The contracton of a pseudo-scalar tensor tensor wth another tensor produces another pseudo-tensor, perhaps a pseudo-scalar or a pseudo-vector. Ths leads to the defnton of dual tensors to be defned below. The vector cross product (vector product) s a tensor of rank 2 but t has a dual representaton as a pseudo-vector. 6 Contracton wth the Lev-Cvta tensor Suppose an ant-symmetrc tensor of rank 2, A j = A j. We contract ths tensor wth the Lev-Cvta tensor of rank 3, ǫ jk 0 A 12 A 13 [A j ] = A 21 0 A 23 A 31 A 32 0 Thus; A j ǫ jk = [A j A j ] k j It s obvous that ths results n a form whch transforms lke a tensor of rank 1, but does not change sgn under a coordnate nverson. It s then a pseudo-vector and a dual of the tensor of second rank. It s also clear that ths represents the cross product of two vectors. 8
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 informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More 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 informationTensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q
For orthogonal curvlnear coordnates, eˆ grad a a= ( aˆ ˆ e). h q (98) Expandng the dervatve, we have, eˆ aˆ ˆ e a= ˆ ˆ a h e + q q 1 aˆ ˆ ˆ a e = ee ˆˆ ˆ + e. h q h q Now expandng eˆ / q (some of the detals
More informationFrom 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 informationKinematics 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
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More 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 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 informationare called the contravariant components of the vector a and the a i are called the covariant components of the vector a.
Non-Cartesan Coordnates The poston of an arbtrary pont P n space may be expressed n terms of the three curvlnear coordnates u 1,u,u 3. If r(u 1,u,u 3 ) s the poston vector of the pont P, at every such
More informationSalmon: 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 informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More 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 informationWeek 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product
The Vector Product Week 11: Chapter 11 Angular Momentum There are nstances where the product of two vectors s another vector Earler we saw where the product of two vectors was a scalar Ths was called the
More informationClassical Mechanics ( Particles and Biparticles )
Classcal Mechancs ( Partcles and Bpartcles ) Alejandro A. Torassa Creatve Commons Attrbuton 3.0 Lcense (0) Buenos Ares, Argentna atorassa@gmal.com Abstract Ths paper consders the exstence of bpartcles
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 informationTHEOREMS 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 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 informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
More informationSo far: simple (planar) geometries
Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector
More informationLecture 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(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 informationPY2101 Classical Mechanics Dr. Síle Nic Chormaic, Room 215 D Kane Bldg
PY2101 Classcal Mechancs Dr. Síle Nc Chormac, Room 215 D Kane Bldg s.ncchormac@ucc.e Lectures stll some ssues to resolve. Slots shared between PY2101 and PY2104. Hope to have t fnalsed by tomorrow. Mondays
More informationLAGRANGIAN 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,
More informationPhysics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.
Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays
More informationOn the symmetric character of the thermal conductivity tensor
On the symmetrc character of the thermal conductvty tensor Al R. Hadjesfandar Department of Mechancal and Aerospace Engneerng Unversty at Buffalo, State Unversty of New York Buffalo, NY 146 USA ah@buffalo.edu
More informationGeorgia 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
More informationn 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 information11. Dynamics in Rotating Frames of Reference
Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons
More 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 informationSpin-rotation coupling of the angularly accelerated rigid body
Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationIterative General Dynamic Model for Serial-Link Manipulators
EEL6667: Knematcs, Dynamcs and Control of Robot Manpulators 1. Introducton Iteratve General Dynamc Model for Seral-Lnk Manpulators In ths set of notes, we are gong to develop a method for computng a general
More informationNEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system).
EWTO S LAWS Consder two partcles. 1 1. If 1 0 then 0 wth p 1 m1v. 1 1 2. 1.. 3. 11 These laws only apply when vewed from an nertal coordnate system (unaccelerated system). consder a collecton of partcles
More informationWeek 9 Chapter 10 Section 1-5
Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informationError 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 informationMEV442 Introduction to Robotics Module 2. Dr. Santhakumar Mohan Assistant Professor Mechanical Engineering National Institute of Technology Calicut
MEV442 Introducton to Robotcs Module 2 Dr. Santhakumar Mohan Assstant Professor Mechancal Engneerng Natonal Insttute of Technology Calcut Jacobans: Veloctes and statc forces Introducton Notaton for tme-varyng
More informationModeling of Dynamic Systems
Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how
More informationModule 1 : The equation of continuity. Lecture 1: Equation of Continuity
1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationStructure 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 informationLinear 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
More informationChapter 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 information1. Review of Mechanics Newton s Laws
. Revew of Mechancs.. Newton s Laws Moton of partcles. Let the poston of the partcle be gven by r. We can always express ths n Cartesan coordnates: r = xˆx + yŷ + zẑ, () where we wll always use ˆ (crcumflex)
More informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
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 information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationErrors 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 informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O
More information10/23/2003 PHY Lecture 14R 1
Announcements. Remember -- Tuesday, Oct. 8 th, 9:30 AM Second exam (coverng Chapters 9-4 of HRW) Brng the followng: a) equaton sheet b) Calculator c) Pencl d) Clear head e) Note: If you have kept up wth
More informationPhysics 111: Mechanics Lecture 11
Physcs 111: Mechancs Lecture 11 Bn Chen NJIT Physcs Department Textbook Chapter 10: Dynamcs of Rotatonal Moton q 10.1 Torque q 10. Torque and Angular Acceleraton for a Rgd Body q 10.3 Rgd-Body Rotaton
More informationMEM 255 Introduction to Control Systems Review: Basics of Linear Algebra
MEM 255 Introducton to Control Systems Revew: Bascs of Lnear Algebra Harry G. Kwatny Department of Mechancal Engneerng & Mechancs Drexel Unversty Outlne Vectors Matrces MATLAB Advanced Topcs Vectors A
More informationConservation of Angular Momentum = "Spin"
Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts
More informationRigid body simulation
Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum
More informationSCALARS AND VECTORS All physical quantities in engineering mechanics are measured using either scalars or vectors.
SCALARS AND ECTORS All phscal uanttes n engneerng mechancs are measured usng ether scalars or vectors. Scalar. A scalar s an postve or negatve phscal uantt that can be completel specfed b ts magntude.
More informationAP Physics 1 & 2 Summer Assignment
AP Physcs 1 & 2 Summer Assgnment AP Physcs 1 requres an exceptonal profcency n algebra, trgonometry, and geometry. It was desgned by a select group of college professors and hgh school scence teachers
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More information5.76 Lecture #21 2/28/94 Page 1. Lecture #21: Rotation of Polyatomic Molecules I
5.76 Lecture # /8/94 Page Lecture #: Rotaton of Polatomc Molecules I A datomc molecule s ver lmted n how t can rotate and vbrate. * R s to nternuclear as * onl one knd of vbraton A polatomc molecule can
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationThe classical spin-rotation coupling
LOUAI H. ELZEIN 2018 All Rghts Reserved The classcal spn-rotaton couplng Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 louaelzen@gmal.com Abstract Ths paper s prepared to show that a rgd
More informationCelestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestial-mechanics - J. Hedberg
PHY 454 - celestal-mechancs - J. Hedberg - 207 Celestal Mechancs. Basc Orbts. Why crcles? 2. Tycho Brahe 3. Kepler 4. 3 laws of orbtng bodes 2. Newtonan Mechancs 3. Newton's Laws. Law of Gravtaton 2. The
More informationIn this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
More 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 informationcoordinates. Then, the position vectors are described by
Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,
More informationPhysics 207 Lecture 6
Physcs 207 Lecture 6 Agenda: Physcs 207, Lecture 6, Sept. 25 Chapter 4 Frames of reference Chapter 5 ewton s Law Mass Inerta s (contact and non-contact) Frcton (a external force that opposes moton) Free
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationLectures - 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 informationRIGID BODY MOTION. Next, we rotate counterclockwise about ξ by angle. Next we rotate counterclockwise about γ by angle to get the final set (x,y z ).
RGD BODY MOTON We now consder the moton of rgd bodes. The frst queston s what coordnates are needed to specf the locaton and orentaton of such an object. Clearl 6 are needed 3 to locate a partcular pont
More informationAdvanced Quantum Mechanics
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
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 informationGeometry and Screw Theory for Robotics
Tutoral (T9) Geometry and Screw Theory for Robotcs Stefano Stramgol and Herman Bruynnckx March 15, 2001 2 Contents 1 Moton of a Rgd Body 5 1.1 Introducton... 5 1.2 The Eucldean Space... 6 1.2.1 Ponts versus
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 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 informationNotes on Frequency Estimation in Data Streams
Notes on Frequency Estmaton n Data Streams In (one of) the data streamng model(s), the data s a sequence of arrvals a 1, a 2,..., a m of the form a j = (, v) where s the dentty of the tem and belongs to
More informationLAB # 4 - Torque. d (1)
LAB # 4 - Torque. Introducton Through the use of Newton's three laws of moton, t s possble (n prncple, f not n fact) to predct the moton of any set of partcles. That s, n order to descrbe the moton of
More informationAn Algorithm to Solve the Inverse Kinematics Problem of a Robotic Manipulator Based on Rotation Vectors
An Algorthm to Solve the Inverse Knematcs Problem of a Robotc Manpulator Based on Rotaton Vectors Mohamad Z. Al-az*, Mazn Z. Othman**, and Baker B. Al-Bahr* *AL-Nahran Unversty, Computer Eng. Dep., Baghdad,
More informationChapter 11 Angular Momentum
Chapter 11 Angular Momentum Analyss Model: Nonsolated System (Angular Momentum) Angular Momentum of a Rotatng Rgd Object Analyss Model: Isolated System (Angular Momentum) Angular Momentum of a Partcle
More informationPerfect Fluid Cosmological Model in the Frame Work Lyra s Manifold
Prespacetme Journal December 06 Volume 7 Issue 6 pp. 095-099 Pund, A. M. & Avachar, G.., Perfect Flud Cosmologcal Model n the Frame Work Lyra s Manfold Perfect Flud Cosmologcal Model n the Frame Work Lyra
More informationSrednicki Chapter 34
Srednck Chapter 3 QFT Problems & Solutons A. George January 0, 203 Srednck 3.. Verfy that equaton 3.6 follows from equaton 3.. We take Λ = + δω: U + δω ψu + δω = + δωψ[ + δω] x Next we use equaton 3.3,
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationQuantum Mechanics I - Session 4
Quantum Mechancs I - Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationThe equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More 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 information9 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 informationIntroduction and Fundamental Concepts
1 Introducton and Fundamental Concepts The numercal expresson of a scentfc statement has tradtonally been the manner by whch scentsts have verfed a theoretcal descrpton of the physcal world. Durng ths
More information2. Differentiable Manifolds and Tensors
. Dfferentable Manfolds and Tensors.1. Defnton of a Manfold.. The Sphere as a Manfold.3. Other Examples of Manfolds.4. Global Consderatons.5. Curves.6. Functons on M.7. Vectors and Vector Felds.8. Bass
More information1 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 informationThe 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
More informationThermodynamics General
Thermodynamcs General Lecture 1 Lecture 1 s devoted to establshng buldng blocks for dscussng thermodynamcs. In addton, the equaton of state wll be establshed. I. Buldng blocks for thermodynamcs A. Dmensons,
More informationFall 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 informationAPPENDIX 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 informationComparative Studies of Law of Conservation of Energy. and Law Clusters of Conservation of Generalized Energy
Comparatve Studes of Law of Conservaton of Energy and Law Clusters of Conservaton of Generalzed Energy No.3 of Comparatve Physcs Seres Papers Fu Yuhua (CNOOC Research Insttute, E-mal:fuyh1945@sna.com)
More 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 information. 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
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More 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 informationFirst Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.
Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act
More information