Notes on Analytical Dynamics

Size: px
Start display at page:

Download "Notes on Analytical Dynamics"

Transcription

1 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 (e, no acceleraton) Newtons nd: F = d(mv)=dt Newtons 3rd: Acton has reacton solated partcles: m v + m v = p =) F + F = 0 Practcal Pendulum Equatons We have a base acceleraton a 0 = x 0 + y 0 j and angular acceleraton k = _! () _k =! Ths mples that a B = a 0 + a Bj0 ; wth a Bj0 = _! R +! (! R) : 3 Energy We de ne T = Z xf x 0 F (x)dx = Z xf x 0 rv (x)dx = mv f mv 0; where we use T = mv = 0 as the knetc energy, and have F (x) = r x V (x) for the potental energy for conservatve systems From ths we also realze that V (x 0 ) + T (x 0 ) = V (x f ) + T (x f ) = const Furthermore, we have mx = F (x)

2 4 Langrangans from Energy From Knetc Energy, we can derve the generalzed equatons of moton Assume we have T = 0:5 P n = m _x as knetc energy for a system of partcle, and the postons as functons of generalzed coordnates, e, x = x(q) In ths case, we also = = _q k m _x = X m _q k = X _x ; ; where the later part s only true for holomonc constrants (e, constrants do only depend on the generalzed postons and tme or, equvalently, x = x (q ; : : : ; q n ; t)) for whch we have dot-cancellaton" When d erentatng the later of the two wth respect to tme, we obtan d = _q k + X _x : When addng up these equatons, we realze = _p = X _q k = k ; where s generalzed force We can repeat the same excercse for any V (q), where we obtan k k _q k = 0 When de nng the Lagrangan ths mples that d dt L = _q = 0, whch s equvalent to sayng that the force derved from V s equal to the one derved from T 5 Duraton of Moton From dx m + V (x) = h; dt we can nfer the duraton of movement for two ponts, e, Z tf r Z m xf dx T = dt = p t 0 h V (x) If T = g(x 0 ; x f ; h) s nvertable, we have x = g (x 0 ; t t 0 ; h) Its usually ntractable x 0

3 An addtonal representaton s gven by V (q) = V 0 (q)+"v (q), whch mples that T (h) = T 0 (h) + "T (h) also has 6 Small Oscllatons Z q0 T 0 (h) p m d p h V0 (q)dq; dh q 0 T (h) p m d Z q0 p dh h V0 (q) dq: q 0 When lnearzng 0:5m _x + V (x) = h 0 around an equlbrum pont x 0 wth a small devaton x, we obtan 0:5m _x +V (x 0 +x) = h 0 +h, and subtractng the two yelds m _x x = h: x0 When comparng ths equaton to a lnear oscllator, t yelds the frequency and ampltude! V ; x0 s h A = V 00 (x 0 ) : 7 Phase planes We can draw phase plane motons usng the potental functon We have have the followng rules, llustrated n Fgure Endponts on the x-axs have V (x) = h These are the ampltudes of the oscllaton Endponts on the _x-axs have T ( _x; x) = h A hll s a seperatrx on the axs, a valley an attractor Hamltonan Dynamcs Basc Idea The basc dea of Hamltonan Dynamcs s to replace the varables x by the generalzed coordnate q and _x by the conjugate momentum p, where x = q, p = m _x: 3

4 Fgure : Ths gure demonstrates how to obtan a phaseplane moton from a potental functon n a conservatve system For example, for the Hamltonan H (q; p; t) = p =(m) + V (q; t), where we see that _q = (q; p; t) = (q; p; t) _p = F (q; t) = When d erentatng a tme-nvarant H (q; p) wth respect to tme, we see that dh (q; p) (q; p) (q; p) dp = dt (q; (q; p; @H (q; p; = 0; e, that all trajectores have constant Hamltonans or are on the soclnes Understandng Hamltonans A quantty s consdered conserved" f t s constant over tme whle the moton proceed, e, t s ndependent of the generalzed veloctes _q k For example, the 4

5 energy E = V + T s conserved We can express ths n general as the d erence of the part of the Langrangan whch depends explctly on _q k and the whole langrangan, e, H _q k _q k k By d erentatng, we obtan dh ; whch mples that H s only conserved f tme does not appear explctly n the Lagrangan Addtonally, we have dh : 3 Area Preservaton When lnearzng the resultng equatons from the Hamltonan, we obtan q(t) = q(t 0 + t) = q(t 0 ) + dq dt t + O(t ) = q(t 0 ) t + O(t ); {z } f(q;p;t) p(t) = p(t 0 + t) = p(t 0 ) + dp dt t + ) = p(t 0 ) t + O(t ): {z } g(q;p;t) Now, we can also lnearze dp and dq, and obtan dp(t) = df df dq + dq dp dp = H dq(t) = dg dg dq + dq dp dp dq + dq H dp: When denoted as dp(t) = dq(t) " H # {z } A dp(t0 ) ; dq(t 0 ) we realze that det A = + O(t ), e, that the area of any pont n state space s constant and we have area preservng mappng Ths area conservaton s equvalent to Energy conservaton, see Fgure 5

6 Fgure : Ths gure llustrates the conservaton of the area n (q; p) space whch s equvalent to the conservaton of energy 4 Hamltonans and Lagrangans The connecton between fx; _xg and fq; pg s not always a smple scalng operaton but are n fact de ned by some operatons We de ne the _q = u, and p k _q k From H X k _q k L = X k u k p k L: Ths yelds for a one partcle system the equaton L (q; u; t) = pu H(q; p; t): We realze that u = (L + H)=p Ths transformaton s llustrated n Fgure 3 Furthermore, we realze _q = : The Lagrangan s de ned as an functon L(q; u; t) n terms of poston and the slope of the path wth respect to q The Hamltonan s de ned as a functon H(q; p; t) n terms of the poston and the momentum Ths s llustrated n Fgure 4 If you are gven a Hamltonan such as H(p) = p =(m), you can always compute the Lagrangan whch would be L(u) = pu H(p) = mu = = m _q = as u = = p=m () p = mu n ths example 6

7 Fgure 3: Ths Fgure llustrates the transformaton of a Lagrangan nto a Hamltonan 3 Systems of Partcles 3 Momentum The moton of a system or partcles can be descrbed n terms of the moton X c of the center of mass (e, the translaton) and the rotaton around the center of mass If we de ne y = x X c, and M = P m, we can obtan L 0 = MX c _X c + X m y _y = X m x _x ; where P m x = MX c 3 Torques The torques around the the center of mass s gven by nx c = y F = d dt =! X m y _y = _L c ; 7

8 Fgure 4: Ths gure llustrates the transformaton of a Hamltonan nto a Lagrangan and by z0 = r M X c + L _ c ; where M X c = P n = F, and r = z 0 X c Ths also yelds z0 = Mr z 0 + _L z0 ; where L z0 represents the angular moment around z 0 Mr z 0 s zero f z 0 = 0, or f r = 0, or f r k z 0 33 Knetc Energy The knetc energy s gven by T = X m _x _x = M _X c _X c + X m _y _y Note that 0:5 P m _y _y = 0:5 P m r! = 0:5I!, and therefore T = M _X c + I! : The rotatonal part s gven by T rot = 0:5!L 0 = 0:5! P m r v 8

9 34 General Representaton The general representaton of a system of partcle s gven by Mx(t) = F(x; _x; t) + F c ; where F represents the external forces whch create work, and F c represents the constrant forces Ths gves us 3n equatons but we have equally many new unknowns as we do not know the constrant forces Let us assume that we are gven some (even non-holomonc) m constrants ' (x; _x; t) = 0; then we can d erentate these and obtan where A s a 3n by m matrx 35 General Soluton A(x; _x; t)x = b(x; _x; t); In general, we the unconstraned acceleraton Ma = F, and the constrant forces F c = M += AM = + (b Aa) combned, ths yelds the fundamental equaton of moton x(t) = M F + M += AM = + b AM F : We can nd M = by ful llng the followng three condtons () M = M = = I, () M = M = = M, and () M = MM = = I For M = dag(m ; : : : ; m n ), we also have M = = dag(= p m ; : : : ; = p m ) For a system wth M = mi, we can smplfy the fundamental equaton to x(t) = m F + b m A m AF : Note that e = (b Aa) s lke an error n a control equaton 36 Interpretaton of the General Soluton We have two nterpretatons for ths result Least-Squares Soluton The general soluton can be nterpreted as the soluton to the mnmzaton problem under the constrant Ax = b mn x (x a) T M (x a) ; 9

10 Nature as a controller We realze that nature s bascally a controller x = x a = M = AM = + (b Aa) = K e; wth K = M = AM = + The motor command s F c = M += AM = + (b Aa) = K e; wth K = M += AM = + 37 Langrange s Vew of Lfe Determne the unconstraned moton from Mx u = F, where M s a n by n matrx and x, F are n vectors Determne the constrants n the form Ax = b where A s a n by m Matrx wth rank A = r 3 Several acceleratons x 0,, x n r are possble acceleratons as they ful ll Ax = b where exactly n r of the n r + are lnear ndependent 4 When subtractng the x 0 we obtan n r vrtual dsplacements x = x x 0 ; for whch Ax = 0 These dsplacements do not create work, e, we have F T c x = 0 (D Alemberts prncple) These can also we obtaned by rewrtng Ax = 0 as creatng vectors lke 3 v (v r+ ; : : : ; v n ) v = v r (v r+ ; : : : ; v n ) v r+ ; v n 0

11 and then obtanng n r vectors as 3 v (0; : : : ; 0; v r+ ; 0; : : : ; 0) v r (0; : : : ; 0; v r+ ; 0; : : : ; 0) 0 x = : Ths gves us Mx = F + F c ; Ax = b; F T c x = 0; whch allows us to determne the n unknowns n x, and the n n F c 38 Gauss vew of Lfe (Le Chatelers Prncple) Determne the unconstraned moton from Mx u = F, where M s a n by n matrx and x, F are n vectors Nature consders all possble acceleratons A ~x = b whch comply wth the constrants 3 Nature pcks the one possble acceleraton whch mnmzes the Gaussan T G(~x) = ~x x u M ~x a, e, Nature s solvng a global mnmzaton problem at each nstant of tme or Nature takes the mnmum devaton from the unconstraned moton

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

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

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

11. Dynamics in Rotating Frames of Reference

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

More information

Lagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013

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

More information

Lecture 20: Noether s Theorem

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

More information

LAGRANGIAN MECHANICS

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,

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

Mechanics Physics 151

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 ( )

More information

coordinates. Then, the position vectors are described by

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,

More information

Poisson brackets and canonical transformations

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

More information

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force.

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,

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

Mechanics Physics 151

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

More information

10. Canonical Transformations Michael Fowler

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

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

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

PHYSICS 231 Review problems for midterm 2

PHYSICS 231 Review problems for midterm 2 PHYSICS 31 Revew problems for mdterm Topc 5: Energy and Work and Power Topc 6: Momentum and Collsons Topc 7: Oscllatons (sprng and pendulum) Topc 8: Rotatonal Moton The nd exam wll be Wednesday October

More information

Physics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints

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

More information

PHYS 705: Classical Mechanics. Hamilton-Jacobi Equation

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

More information

A Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph

A Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph A Tale o Frcton Basc Rollercoaster Physcs Fahrenhet Rollercoaster, Hershey, PA max heght = 11 t max speed = 58 mph PLAY PLAY PLAY PLAY Rotatonal Movement Knematcs Smlar to how lnear velocty s dened, angular

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

Lagrangian Field Theory

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,

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

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

Lesson 5: Kinematics and Dynamics of Particles

Lesson 5: Kinematics and Dynamics of Particles Lesson 5: Knematcs and Dynamcs of Partcles hs set of notes descrbes the basc methodology for formulatng the knematc and knetc equatons for multbody dynamcs. In order to concentrate on the methodology and

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

Classical Mechanics ( Particles and Biparticles )

Classical 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 information

Integrals and Invariants of

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

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

Rigid body simulation

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

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

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

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

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

Study Guide For Exam Two

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

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

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

Conservation of Angular Momentum = "Spin"

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

More information

Physics 111: Mechanics Lecture 11

Physics 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 information

Chapter 11: Angular Momentum

Chapter 11: Angular Momentum Chapter 11: ngular Momentum Statc Equlbrum In Chap. 4 we studed the equlbrum of pontobjects (mass m) wth the applcaton of Newton s aws F 0 F x y, 0 Therefore, no lnear (translatonal) acceleraton, a0 For

More 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

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

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

More information

Classical Mechanics Virtual Work & d Alembert s Principle

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

More information

Solutions to exam in SF1811 Optimization, Jan 14, 2015

Solutions to exam in SF1811 Optimization, Jan 14, 2015 Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O -5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable

More information

PHYS 705: Classical Mechanics. Canonical Transformation II

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

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

Physics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.

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

More information

6. Hamilton s Equations

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

More information

Analytical classical dynamics

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

More information

1 What is a conservation law?

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,

More information

Spin-rotation coupling of the angularly accelerated rigid body

Spin-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 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

Lorentz Group. Ling Fong Li. 1 Lorentz group Generators Simple representations... 3

Lorentz Group. Ling Fong Li. 1 Lorentz group Generators Simple representations... 3 Lorentz Group Lng Fong L ontents Lorentz group. Generators............................................. Smple representatons..................................... 3 Lorentz group In the dervaton of Drac

More information

CHEMICAL REACTIONS AND DIFFUSION

CHEMICAL REACTIONS AND DIFFUSION CHEMICAL REACTIONS AND DIFFUSION A.K.A. NETWORK THERMODYNAMICS BACKGROUND Classcal thermodynamcs descrbes equlbrum states. Non-equlbrum thermodynamcs descrbes steady states. Network thermodynamcs descrbes

More information

So far: simple (planar) geometries

So 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 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

Solutions HW #2. minimize. Ax = b. Give the dual problem, and make the implicit equality constraints explicit. Solution.

Solutions HW #2. minimize. Ax = b. Give the dual problem, and make the implicit equality constraints explicit. Solution. Solutons HW #2 Dual of general LP. Fnd the dual functon of the LP mnmze subject to c T x Gx h Ax = b. Gve the dual problem, and make the mplct equalty constrants explct. Soluton. 1. The Lagrangan s L(x,

More information

Physics 141. Lecture 14. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 14, Page 1

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

More information

ENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15

ENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15 NGN 40 ynamcs and Vbratons Homework # 7 ue: Frday, Aprl 15 1. Consder a concal hostng drum used n the mnng ndustry to host a mass up/down. A cable of dameter d has the mass connected at one end and s wound/unwound

More information

Linear Momentum. Center of Mass.

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

More information

Army Ants Tunneling for Classical Simulations

Army Ants Tunneling for Classical Simulations Electronc Supplementary Materal (ESI) for Chemcal Scence. Ths journal s The Royal Socety of Chemstry 2014 electronc supplementary nformaton (ESI) for Chemcal Scence Army Ants Tunnelng for Classcal Smulatons

More information

Please initial the statement below to show that you have read it

Please initial the statement below to show that you have read it EN40: Dynamcs and Vbratons Mdterm Examnaton Thursday March 5 009 Dvson of Engneerng rown Unversty NME: Isaac Newton General Instructons No collaboraton of any knd s permtted on ths examnaton. You may brng

More information

Quantum Mechanics I Problem set No.1

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

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

SELECTED SOLUTIONS, SECTION (Weak duality) Prove that the primal and dual values p and d defined by equations (4.3.2) and (4.3.3) satisfy p d.

SELECTED SOLUTIONS, SECTION (Weak duality) Prove that the primal and dual values p and d defined by equations (4.3.2) and (4.3.3) satisfy p d. SELECTED SOLUTIONS, SECTION 4.3 1. Weak dualty Prove that the prmal and dual values p and d defned by equatons 4.3. and 4.3.3 satsfy p d. We consder an optmzaton problem of the form The Lagrangan for ths

More information

Chapter 3 and Chapter 4

Chapter 3 and Chapter 4 Chapter 3 and Chapter 4 Chapter 3 Energy 3. Introducton:Work Work W s energy transerred to or rom an object by means o a orce actng on the object. Energy transerred to the object s postve work, and energy

More information

Chapter 8. Potential Energy and Conservation of Energy

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

More information

Physics 207: Lecture 20. Today s Agenda Homework for Monday

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

More information

CHAPTER 10 ROTATIONAL MOTION

CHAPTER 10 ROTATIONAL MOTION CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n x-y plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The

More information

Chapter 11 Angular Momentum

Chapter 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 information

Spring Force and Power

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

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

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

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

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

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

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

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

where the sums are over the partcle labels. In general H = p2 2m + V s(r ) V j = V nt (jr, r j j) (5) where V s s the sngle-partcle potental and V nt

where the sums are over the partcle labels. In general H = p2 2m + V s(r ) V j = V nt (jr, r j j) (5) where V s s the sngle-partcle potental and V nt Physcs 543 Quantum Mechancs II Fall 998 Hartree-Fock and the Self-consstent Feld Varatonal Methods In the dscusson of statonary perturbaton theory, I mentoned brey the dea of varatonal approxmaton schemes.

More information

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13 CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 13 GENE H GOLUB 1 Iteratve Methods Very large problems (naturally sparse, from applcatons): teratve methods Structured matrces (even sometmes dense,

More information

The classical spin-rotation coupling

The 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 information

10/24/2013. PHY 113 C General Physics I 11 AM 12:15 PM TR Olin 101. Plan for Lecture 17: Review of Chapters 9-13, 15-16

10/24/2013. PHY 113 C General Physics I 11 AM 12:15 PM TR Olin 101. Plan for Lecture 17: Review of Chapters 9-13, 15-16 0/4/03 PHY 3 C General Physcs I AM :5 PM T Oln 0 Plan or Lecture 7: evew o Chapters 9-3, 5-6. Comment on exam and advce or preparaton. evew 3. Example problems 0/4/03 PHY 3 C Fall 03 -- Lecture 7 0/4/03

More information

Symmetric Lie Groups and Conservation Laws in Physics

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.

More information

Part C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis

Part C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis Part C Dynamcs and Statcs of Rgd Body Chapter 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta

More information

Formulas for the Determinant

Formulas 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 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

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

Lecture 23: Newton-Euler Formulation. Vaibhav Srivastava

Lecture 23: Newton-Euler Formulation. Vaibhav Srivastava Lecture 23: Newton-Euler Formulaton Based on Chapter 7, Spong, Hutchnson, and Vdyasagar Vabhav Srvastava Department of Electrcal & Computer Engneerng Mchgan State Unversty Aprl 10, 2017 ECE 818: Robotcs

More information

STATISTICAL MECHANICS

STATISTICAL MECHANICS STATISTICAL MECHANICS Thermal Energy Recall that KE can always be separated nto 2 terms: KE system = 1 2 M 2 total v CM KE nternal Rgd-body rotaton and elastc / sound waves Use smplfyng assumptons KE of

More information

Physics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2

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

More information

A Review of Analytical Mechanics

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

More information

Advanced Quantum Mechanics

Advanced 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 information

In this section is given an overview of the common elasticity models.

In 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 information

NEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system).

NEWTON 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 information

Angular Momentum and Fixed Axis Rotation. 8.01t Nov 10, 2004

Angular Momentum and Fixed Axis Rotation. 8.01t Nov 10, 2004 Angular Momentum and Fxed Axs Rotaton 8.01t Nov 10, 2004 Dynamcs: Translatonal and Rotatonal Moton Translatonal Dynamcs Total Force Torque Angular Momentum about Dynamcs of Rotaton F ext Momentum of a

More information

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS

8.4 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 information

Which Separator? Spring 1

Which Separator? Spring 1 Whch Separator? 6.034 - Sprng 1 Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng 3 Margn of a pont " # y (w $ + b) proportonal

More information

Iterative General Dynamic Model for Serial-Link Manipulators

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

More information

Report on Image warping

Report on Image warping Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.

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