# Complex Variables in Classical Hamiltonian Mechanics

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Complex Variables in Classical Hamiltonian Mechanics In the classical Hamiltonian formalism, the state of a mechanical system is given by a set of pairs of conjugated variables, of variables {q j, p j }, (j = 1, 2,...) referred to as coordinates and momenta, respectively. The equations of motion are generated with the Hamiltonian function, H({q j, p j }), by the following axiom q j = p j, (1) ṗ j = q j. (2) Some times it proves reasonable to introduce complex variables {a j } related to {q j, p j } by a j = (αq j + iβp j ), (3) where α and β are some complex numbers (which can also depend on j, but we are not interested here in such a generalization). The equations of motion then acquire the form iλȧ j =, (4) with λ = 1/(αβ + α β). That is each pair of real equations (1)-(2) is replaced with one complex equation (4). One may equivalently use the complex-conjugated equation iλȧ j = /. Here we take into account that H is real. Poisson bracket, {A, B}, is a function of coordinates and momenta constructed from the two other functions, A({q j, p j }) and B({q j, p j }) by the following prescription: {A, B} = j A B p j q j A q j B p j. (5) As is seen from (1)-(2) and this is the most important property of the Poisson brackets, for any function of coordinate and momenta, A({q j, p j }, its time derivative during the evolution can be expressed as Ȧ = {H, A}. (6) In particular, if {H, A} 0, then the quantity A is a constant of motion. 1

2 In complex variables, the definition (5) acquires the form {A, B} = (i/λ) j A B A B. (7) Problem A1. Check this. Make sure that {a j, a k } = {a j, a k } = 0, {a j, a k } = (i/λ)δ j,k. Problem A2. (a) Derive and solve complex equation of motion for the system H = ω a 2 (harmonic oscillator). (Set λ = 1. It s just a matter of units in which one measures time.) (b) Derive complex equations of motion for the system of two (non-trivially) coupled oscillators: H = ω 1 a ω 2 a γ[(a 1) 2 a (a 2) 2 a 2 1]. Do not solve it, but use it to show that the system features the constant of motion N = a a 2 2 ; show the same with Poisson bracket technique. The constant of motion N in the Problem A2(b) is actually related to some special symmetry of corresponding Hamiltonian. This symmetry known as global U(1) symmetry implies invariance of the Hamiltonian with respect to global U(1) transformation, which is simultaneously shifting the phases of all complex variables a j by one and the same angle θ 0 : a j e iθ 0 a j. If the global U(1) symmetry takes place, then the following quadratic form of the complex variables N = a j 2 (8) j is a constant of motion: {H, N} = 0. Problem A3. Prove this statement. Hint. Start with noticing that in the limit of θ 0 0, the global U(1) transformation reduces to H H +θ 0 j /θ j, so that global U(1) invariance implies j /θ j 0. Don t forget about the relation: where z = x + iy = z e iθ. /θ = x /y y /x = i(z /z z /z ), (9) Note. Complex variables {a j } prove very convenient when finding a quantum counterpart of a classical system. In this procedure, known as canonical quantization, the variables a j and a j are simply replaced with annihilation and creation operators, respectively. The parameter λ now cannot be arbitrarily chosen. It is one and the same for all quantum systems, and is nothing else than the Planks s constant, h. Upon the quantization procedure, 2

3 former Poisson brackets turn out to correspond to commutators between corresponding operators: {A, B} (i/ h)[a, B]. The quantity N in the quantum system is nothing else than the total number of quanta (particles); it is conserved by U(1)-symmetric Hamiltonians. Canonical Transformation Suppose we have a set of complex canonical variables, {a j }, and would like to consider another set of variables, {b s }, b s b s ({a j }). How do the equations of motion look like, if expressed in terms of new variables? The answer is given by the general relation iḃs = i{h, b s } = j [ bs b s ]. (10) The next question is whether it is possible to select new variables in such a way independently of the particular form of H that they would be also canonical, that is (10) would be actually equivalent to iḃs =, (11) for any H. The answer is positive. To arrive at the necessary and sufficient conditions for the new variables to be canonical we note that basically we require that [ bs a j j a b ] s j a j b [ a j s b j s a + a ] j j b. (12) s Since the differential operators {/, / } are linear independent, Eq. (12) immediately leads to b s = a j, b s =. (13) These are the necessary and sufficient conditions for {b s } to be canonical. Problem A4. Show that the following two elementary transformations dealing with only one variable, a j, are canonical. (a) Shift of the variable: a j a j + c j, where c j is a complex constant. 3

4 (b) Shift of the variable s phase: a j e iϕj a j, where ϕ j is a real constant. (c) Show that the linear transformation b s = j u sj a j is canonical, if and only if the matrix u is unitary. A remarkable fact readily following from (13) is that the Poisson brackets are invariant with respect to the canonical transformation. That is the result for {A, B} is the same for any set of canonical variables. Problem A5. Prove this. Actually, the invariance of the Poisson brackets under some transformation of variables is a sufficient condition for that transformation to be canonical. Indeed, noticing that for any quantity A we have A = i{a, a j }, A = i{a j, A}, (14) and requiring that the Poisson brackets be invariant under our transformation of variables, we immediately arrive at (13): b s = i{a j, b s } = a j, (15) b s = i{b s, a j } = i{a j, b s } =. (16) Bilinear Hamiltonian. Bogoliubov Transformation Given a Hamiltonian function, one can bilinearize it in the vicinity of its (local) minimum to study the normal modes. The general form of the Hamiltonian after bilinearization is (each variable is now reckoned from its equilibrium value a shift of a variable is a canonical transformation): H = ( A ij a i a j B ij a i a j + 1 ) 2 B ij a i a j, (17) ij where without loss of generality we can assume that (the requirement for the Hamiltonian to be real and symmetrization) A ij = A j i, B ij = B j i. (18) If a Hamiltonian of the form (17) features a (local) minimum at the point a 1 = a 2 = a 3 =... = 0, then it can be diagonalized by linear canonical 4

5 transformation (Bogoliubov transformation), b s = j (u sj a j + v sj a j). (19) By diagonalized we mean that in terms of the new canonical variables {b s } the Hamiltonian will read H = s E s b sb s, (20) that is will be equivalent to a set of non-interacting harmonic oscillators (normal modes). Before we show how to relate the u and v coefficients to the matrices A and B, let us first establish their general properties. From the requirement that the transformation (19) be a canonical we get [differentiate both sides with respect to a j, a j, b r, and b r and use (13)]: [u sj u rj v sj vrj] = δ sr, (21) j [u sj v rj v sj u rj ] = 0. (22) j Now we would like to make sure that the conditions (21)-(22) are not only necessary, but also sufficient for the transformation to be canonical. To this end we prove a useful Lemma. If coefficients u and v satisfy (21)-(22), then the inverse transformation is given by a j = (ũ j s b s + ṽ j s b s), (23) s where ũ j s = u sj, ṽ j s = v sj. (24) Problem A6. Prove this lemma by substituting a j from (23) into the right hand side of (19). Now when we know how to express a s in terms of b s, we just need to make sure that Eqs. (13) are satisfied. And this is easily seen from (24). The relations (21)-(22) for the transformation (23) read [ũ js ũ ks ṽ js ṽks] = δ jk, (25) s 5

6 [ũ js ṽ ks ṽ js ũ ks ] = 0. (26) s With (24) these can be rewritten as [u sj u sk v sj vsk] = δ jk, (27) s s [u sj v sk v sj u sk] = 0. (28) Now we obtain the equations for the u and v coefficients. Eq. (20) implies E s b s =, (29) which becomes non-trivial if H is expressed in terms of a s. Differentiating both sides with respect to a j and a j, and taking into account b s u sj, b s v sj, (30) we get E s u sj = E s v sj = = =, (31). (32) Note that here it is legitimate to change the order of operators (/b) s and (/a) s, because our transformation is linear and, say, (/b) s are just linear combinations of (/a) s: = j [ aj + a j ] = j [ ṽ js + ũ js ]. (33) Calculating the derivatives = i [A ij a i + B ij a i ], (34) and observing that = i [A j i a i + B ij a i ], (35) a i ṽ is = v si, a i ũ is = u si, (36) 6

7 we finally arrive at the system of equations: [A ij u si B ji v si ] = E s u sj, (37) i Using the vector notation i [B ji u si A ji v si ] = E s v sj. (38) u s = (u s,1, u s,2, u s,3,...), v s = (v s,1, v s,2, v s,3,...), (39) and taking into account (18), we write this system as A u s B v s = E s u s, (40) B u s A v s = E s v s. (41) We can also write this system in terms of ũ s and ṽ s. Introducing the vectors we have ũ s = (ũ 1,s, ũ 2,s, ũ 3,s,...), ṽ s = (ṽ 1,s, ṽ 2,s, ṽ 3,s,...), (42) A ũ s + B ṽ s = E s ũ s, (43) B ũ s + A ṽ s = E s ṽ s. (44) The energies of the normal modes, E s s, arise as the eigenvalues of the problem. The relations (21)-(22), which in the vector notation read are automatically guaranteed at E r E s. Problem A7. Make sure that this is the case. u r u s v r v s = δ sr, (45) v r u s u r v s = 0, (46) Eqs. (40)-(41) are invariant with respect to the transformation u s v s, v s u s, E s E s. (47) This means that each solution has its counterpart of the opposite energy. However, only one solution of the pair is physically relevant. Namely, the one with u s u s v s v s > 0. (48) 7

8 For its counterpart we then have u s u s v s v s < 0, (49) which means that it is impossible to normalize it to unity, in contradiction with (45). Problem A8. Consider the one-mode Hamiltonian H = a a [ γ a a + γ a a ]. (50) (a) Show that without a loss of generality (up to a simple canonical transformation) one may choose γ to be real and positive. (b) Explore the possibility of solving for the dynamics of this model by Bogoliubov transformation. (c) In the region of γ where Bogoliubov transformation is not helpful, directly solve the equation of motion. Evolution as a Canonical Transformation Remarkably, the evolution of a dynamical system can be considered as a canonical transformation. For the given set of variables {a j } we introduce another set {b j }, where, by definition, the value of b j is equal to the value of a j after some fixed period of evolution, t. First we note that the definition is consistent, because given the particular form of the Hamiltonian, fixed time period t, and particular initial state {a j }, we unambiguously fix {a j (t)} by the equations of motion. Hence, each b j = a j (t) can be considered as just a function of all variables {a j }, the particular form of the function being related to the form of the Hamiltonian and the time period t. Now we make sure that our transformation is canonical. It is enough to do it in the limit t 0, because then the case of finite t can be viewed as just a chain of infinitesimal canonical transformations. Assuming that t is arbitrarily small, from the equation of motion we have and, correspondingly, b j = a j it + O(t 2 ), (51) b j a k = δ jk it 2 H a k + O(t 2 ), (52) 8

9 b j a k 2 H = it a k a j + O(t 2 ). (53) To check that the relations (13) do really take place (up to the terms t 2 ), we note that = + O(t). (54) b j Problem A9. Finish the proof. If the Hamiltonian is bilinear, then the evolution equation is linear. Correspondingly, {a j (t)} is related to {a j (0)} by a linear transformation, which can be written in the vector notation, a (a 1, a 2, a 3,...), as a(t) = U(t) a(0) + V (t) a (0). (55) Here U and V are some time-dependent matrices the form of which is defined by the Hamiltonian only not by the initial state a(0). The above-proven theorem states that the transformation (55) is canonical. But the linear canonical transformation is nothing else than the Bogoliubov transformation. Hence, the evolution of a system with bilinear Hamiltonian is given by time-dependent Bogoliubov transformation. Problem A9. Suppose some bilinear Hamiltonian is diagonalizable by the Bogoliubov transformation (19). Express the matrix elements U jk (t) and V jk (t) in the equation (55) in terms of Bogoliubov matrices, u sj and v sj, and eigenvalues, E s. Problem A10. In the case of a single-mode system with a bilinear Hamiltonian, Eq. (55) reads a(t) = U(t) a(0) + V (t) a (0), (56) where U(t) and V (t) are some functions of time. Find U(t) and V (t) for the Hamiltonian (50) of the Problem A8. Stability of Equilibrium Solution By equilibrium solution of the equation of motion we mean the solution which does not evolve in time: s, ȧ s 0. This immediately implies that the equilibrium solution corresponds to such a point, {a (0) s }, in the space of variables {a s }, where s : a s = a s = 0 (at a s = a (0) s ). (57) 9

10 We will also refer to the equilibrium solution as equilibrium point. Eq. (57) means that the equilibrium points are the points of extremal behavior of the Hamiltonian. In particular, the previously discussed points of local minima are the equilibrium points. At the equilibrium point {a (0) s }, a natural question arises of what happens if the state of the system is slightly perturbed. The general way of answering this question is to expand the Hamiltonian in the vicinity of the equilibrium in terms of the shifted variables a s a s a (0) s (assuming that their absolute values are small enough). The resulting Hamiltonian is of the bilinear form (17), with A ij = 2 H a i, B ij = 2 H a i (at a s = a (0) s ). (58) Linear terms of the expansion are zero because of Eq. (57) and the constant term is of no interest. The equation of motion for the Hamiltonian (17) is i t a = A a + B a. (59) Since this is a linear equation, its general solution can be written as a sum of elementary solutions (the necessity of the term with complex conjugated ω comes from the term with complex conjugated a) a = e iωt f + e iω t g. (60) Substituting this into (59) and separating the terms with e iωt and e iω t, which are linear independent at Re ω 0, we get If Re ω = 0, we write and obtain A f + B g = ω f, (61) B f + A g = ω g. (62) a = e λt f, (63) A f + B f = iλ f. (64) The form of the system of equations (61)-(62) is the same as that of (43)-(44). And this is not a coincidence. In the case when ω s are real the elementary solutions (60) correspond to the normal modes found by Bogoliubov 10

11 transformation. If at least one of the frequencies ω is not real, then the equilibrium state is dynamically unstable with respect to small perturbations. There will be a mode (with Im ω > 0) that will be increasing exponentially. But how does one derive the existence of Im ω > 0 from the fact of existence of a complex ω? Why is it impossible to have Im ω < 0 for all complex ω s? Actually, this follows from the fact that the evolution of the system can be viewed as the Bogoliubov transformation. Suppose we have some complex ω = ω 0 + iλ: { } a(t) = e iω0t f + e iω0t g e λt. (65) On the other hand, inverting the Bogoliubov transformation (55), we have where, in accordance with (24), Combining (65) and (66), we get a(0) = a(0) = Ũ a(t) + Ṽ a (t), (66) Ũ = U, Ṽ = V T. (67) { } e iω0t Ũ f + e iω0t Ũ g + e iω0t Ṽ f + e iω0t Ṽ g e λt. (68) The left-hand side of Eq. (68) is t-independent, while the right-hand side contains the global exponential factor e λt. Hence, either Ũ, or Ṽ, or both should contain the compensating factor e λt. But then, by Eq. (67), the same factor should be present in U, or V, or both. Hence, there should exist an elementary solution containing this factor. Apart from the dynamic instability, when the solution grows exponentially, there can also be a statistical instability. In the case of statistical instability, all ω s are real, but not all of them are positive. The Hamiltonian thus can be diagonalized by the Bogoliubov transformation, and the dynamics within the bilinear Hamiltonian is just the superposition of oscillating normal modes. However, in a typical real situation, when there are higher order terms in the Hamiltonian (say, an interaction with a heat bath) the amplitudes of all the normal modes will be gradually drifting to higher and higher values, since this leads to the increasing entropy without violating energy conservation: The positive contribution to the total energy from the positive-e modes is compensated by negative contribution from the negative-e modes. 11

### Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability

S. Widnall 16.07 Dynamics Fall 2009 Version 1.0 Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability Vibration, Instability An important class of problems in dynamics concerns the free

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### Theta Functions. Lukas Lewark. Seminar on Modular Forms, 31. Januar 2007

Theta Functions Lukas Lewark Seminar on Modular Forms, 31. Januar 007 Abstract Theta functions are introduced, associated to lattices or quadratic forms. Their transformation property is proven and the

### Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo

Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for

### Lecture 1: Schur s Unitary Triangularization Theorem

Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections

### 5.3 The Cross Product in R 3

53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

### Class XI Chapter 5 Complex Numbers and Quadratic Equations Maths. Exercise 5.1. Page 1 of 34

Question 1: Exercise 5.1 Express the given complex number in the form a + ib: Question 2: Express the given complex number in the form a + ib: i 9 + i 19 Question 3: Express the given complex number in

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### Quantum Mechanics and Representation Theory

Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30

### 1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as

Chapter 3 (Lecture 4-5) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series

### ASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1

19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A Two-DOF Mass-Spring-Dashpot Dynamic System Consider the lumped-parameter, mass-spring-dashpot dynamic system shown in the Figure. It has two point

### CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

### 1 Review of complex numbers

1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

### Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

### The Phase Plane. Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations

The Phase Plane Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations Phase Portraits of Linear Systems Consider a systems of linear differential

### 1 Eigenvalues and Eigenvectors

Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

### Physics in the Laundromat

Physics in the Laundromat Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (Aug. 5, 1997) Abstract The spin cycle of a washing machine involves motion that is stabilized

### PHY411. PROBLEM SET 3

PHY411. PROBLEM SET 3 1. Conserved Quantities; the Runge-Lenz Vector The Hamiltonian for the Kepler system is H(r, p) = p2 2 GM r where p is momentum, L is angular momentum per unit mass, and r is the

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### LS.6 Solution Matrices

LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

### The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

### Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

### Unit 21 Influence Coefficients

Unit 21 Influence Coefficients Readings: Rivello 6.6, 6.13 (again), 10.5 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Have considered the vibrational behavior of

### Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.

ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?

### Lecture 2. Observables

Lecture 2 Observables 13 14 LECTURE 2. OBSERVABLES 2.1 Observing observables We have seen at the end of the previous lecture that each dynamical variable is associated to a linear operator Ô, and its expectation

### University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

### EC9A0: Pre-sessional Advanced Mathematics Course

University of Warwick, EC9A0: Pre-sessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Pre-sessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond

### Three Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009

Three Pictures of Quantum Mechanics Thomas R. Shafer April 17, 2009 Outline of the Talk Brief review of (or introduction to) quantum mechanics. 3 different viewpoints on calculation. Schrödinger, Heisenberg,

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

### Elasticity Theory Basics

G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold

### Understanding Poles and Zeros

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function

### Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,

### v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with

### Chapter 12 Modal Decomposition of State-Space Models 12.1 Introduction The solutions obtained in previous chapters, whether in time domain or transfor

Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter

### Tensors on a vector space

APPENDIX B Tensors on a vector space In this Appendix, we gather mathematical definitions and results pertaining to tensors. The purpose is mostly to introduce the modern, geometrical view on tensors,

### Orthogonal Diagonalization of Symmetric Matrices

MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

### ELEMENTS OF VECTOR ALGEBRA

ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions

### Numerically integrating equations of motion

Numerically integrating equations of motion 1 Introduction to numerical ODE integration algorithms Many models of physical processes involve differential equations: the rate at which some thing varies

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### F ij = Gm im j r i r j 3 ( r j r i ).

Physics 3550, Fall 2012 Newton s Third Law. Multi-particle systems. Relevant Sections in Text: 1.5, 3.1, 3.2, 3.3 Newton s Third Law. You ve all heard this one. Actioni contrariam semper et qualem esse

### Low Dimensional Representations of the Loop Braid Group LB 3

Low Dimensional Representations of the Loop Braid Group LB 3 Liang Chang Texas A&M University UT Dallas, June 1, 2015 Supported by AMS MRC Program Joint work with Paul Bruillard, Cesar Galindo, Seung-Moon

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### Math 504, Fall 2013 HW 3

Math 504, Fall 013 HW 3 1. Let F = F (x) be the field of rational functions over the field of order. Show that the extension K = F(x 1/6 ) of F is equal to F( x, x 1/3 ). Show that F(x 1/3 ) is separable

### Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

### C 1 x(t) = e ta C = e C n. 2! A2 + t3

Matrix Exponential Fundamental Matrix Solution Objective: Solve dt A x with an n n constant coefficient matrix A x (t) Here the unknown is the vector function x(t) x n (t) General Solution Formula in Matrix

### L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014

L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 Unfortunately, no one can be told what the Matrix is. You have to see it for yourself. -- Morpheus Primary concepts:

### Chapter 9 Unitary Groups and SU(N)

Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three

### 7 Gaussian Elimination and LU Factorization

7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method

### 3. Reaction Diffusion Equations Consider the following ODE model for population growth

3. Reaction Diffusion Equations Consider the following ODE model for population growth u t a u t u t, u 0 u 0 where u t denotes the population size at time t, and a u plays the role of the population dependent

### Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses

University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,

### Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

The Hadamard Product Elizabeth Million April 12, 2007 1 Introduction and Basic Results As inexperienced mathematicians we may have once thought that the natural definition for matrix multiplication would

### 4. MATRICES Matrices

4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

### Introduction to Matrix Algebra I

Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### [1] Diagonal factorization

8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:

### The Quantum Harmonic Oscillator Stephen Webb

The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems

### Factorization Theorems

Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization

### 1 Scalars, Vectors and Tensors

DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical

### Lecture L3 - Vectors, Matrices and Coordinate Transformations

S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

### DIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents

DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition

### 1 Lecture 3: Operators in Quantum Mechanics

1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators

### Lecture 2: Essential quantum mechanics

Department of Physical Sciences, University of Helsinki http://theory.physics.helsinki.fi/ kvanttilaskenta/ p. 1/46 Quantum information and computing Lecture 2: Essential quantum mechanics Jani-Petri Martikainen

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### T ( a i x i ) = a i T (x i ).

Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

### Lecture L22-2D Rigid Body Dynamics: Work and Energy

J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

### 2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

### Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### Math 497C Sep 9, Curves and Surfaces Fall 2004, PSU

Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 15 sometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics We say

### Chapter 6. Orthogonality

6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be

### 4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

### Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

### We seek a factorization of a square matrix A into the product of two matrices which yields an

LU Decompositions We seek a factorization of a square matrix A into the product of two matrices which yields an efficient method for solving the system where A is the coefficient matrix, x is our variable

### 5 Homogeneous systems

5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m

### Change of Continuous Random Variable

Change of Continuous Random Variable All you are responsible for from this lecture is how to implement the Engineer s Way (see page 4) to compute how the probability density function changes when we make

### 3.1 State Space Models

31 State Space Models In this section we study state space models of continuous-time linear systems The corresponding results for discrete-time systems, obtained via duality with the continuous-time models,

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### These axioms must hold for all vectors ū, v, and w in V and all scalars c and d.

DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms

### WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course

### The Inverse of a Square Matrix

These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### VON-NEUMANN STABILITY ANALYSIS

university-logo VON-NEUMANN STABILITY ANALYSIS School of Mathematics Semester 1 2008 OUTLINE 1 REVIEW 2 EXPONENTIAL GROWTH Stability under exponential growth 3 VON-NEUMANN STABILITY ANALYSIS 4 SUMMARY