Nonlinear physics (solitons, chaos, discrete breathers)


 Erika Logan
 1 years ago
 Views:
Transcription
1 Nonlinear physics (solitons, chaos, discrete breathers) N. Theodorakopoulos Konstanz, June 2006
2 Contents Foreword vi 1 Background: Hamiltonian mechanics Lagrangian formulation of dynamics Hamiltonian dynamics Canonical momenta Poisson brackets Equations of motion Canonical transformations Point transformations HamiltonJacobi theory HamiltonJacobi equation Relationship to action Conservative systems Separation of variables Periodic motion. Actionangle variables Complete integrability Symmetries and conservation laws Homogeneity of time Homogeneity of space Galilei invariance Isotropy of space (rotational symmetry of Lagrangian) Continuum field theories Lagrangian field theories in 1+1 dimensions Symmetries and conservation laws Perturbations of integrable systems Background: Statistical mechanics Scope Formulation Phase space Liouville s theorem Averaging over time Ensemble averaging Equivalence of ensembles Ergodicity The FPU paradox The harmonic crystal: dynamics The harmonic crystal: thermodynamics The FPU numerical experiment i
3 Contents 4 The Korteweg  de Vries equation Shallow water waves Background: hydrodynamics Statement of the problem; boundary conditions Satisfying the bottom boundary condition Euler equation at top boundary A solitary wave Is the solitary wave a physical solution? KdV as a limiting case of anharmonic lattice dynamics KdV as a field theory KdV Lagrangian Symmetries and conserved quantities KdV as a Hamiltonian field theory Solving KdV by inverse scattering Isospectral property Lax pairs Inverse scattering transform: the idea The inverse scattering transform The direct problem Time evolution of scattering data Reconstructing the potential from scattering data (inverse scattering problem) IST summary Application of the IST: reflectionless potentials A single bound state Multiple bound states Integrals of motion Lemma: a useful representation of a(k) Asymptotic expansions of a(k) IST as a canonical transformation to actionangle variables Solitons in anharmonic lattice dynamics: the Toda lattice The model The dual lattice A pulse soliton Complete integrability Thermodynamics Chaos in low dimensional systems Visualization of simple dynamical systems Two dimensional phase space dimensional phase space dimensional phase space; nonautonomous systems with one degree of freedom Small denominators revisited: KAM theorem Chaos in area preserving maps Twist maps Local stability properties PoincaréBirkhoff theorem Chaos diagnostics The standard map The Arnold cat map The baker map; Bernoulli shifts ii
4 Contents The circle map. Frequency locking Topology of chaos: stable and unstable manifolds, homoclinic points Solitons in scalar field theories Definitions and notation Lagrangian, continuum field equations Static localized solutions (general KG class) General properties Specific potentials Intrinsic Properties of kinks Linear stability of kinks Special properties of the SG field The SineGordon breather Complete Integrability Atoms on substrates: the FrenkelKontorova model The CommensurateIncommensurate transition The continuum approximation The special case ɛ = 0: kinks and antikinks The general case ɛ > 0: the soliton lattice Breaking of analyticity FK ground state as minimizing periodic orbit of the standard map Small amplitude motion Free end boundary conditions Metastable states: spatial chaos as a model of glassy structure Solitons in magnetic chains Introduction Classical spin dynamics Spin Poisson brackets An alternative representation Solitons in ferromagnetic chains The continuum approximation The classical, isotropic, ferromagnetic chain The easyplane ferromagnetic chain in an external field Solitons in antiferromagnets Continuum dynamics The isotropic antiferromagnetic chain Easy axis anisotropy Easy plane anisotropy Easy plane anisotropy and symmetrybreaking field Solitons in conducting polymers Peierls instability Electrons decoupled from the lattice Electronphonon coupling; dimerization Solitons and polarons in (CH) x A continuum approximation Dimerization The soliton The polaron iii
5 Contents 12 Solitons in nonlinear optics Background: Interaction of light with matter, MaxwellBloch equations Semiclassical theoretical framework and notation Dynamics Propagation at resonance. Selfinduced transparency Slow modulation of the optical wave Further simplifications: Selfinduced transparency Selffocusing offresonance Offresonance limit of the MB equations Nonlinear terms Spacetime dependence of the modulation: the nonlinear Schrödinger equation Soliton solutions Solitons in BoseEinstein Condensates The GrossPitaevskii equation Propagating solutions. Dark solitons Unbinding the double helix A nonlinear lattice dynamics approach Mesoscopic modeling of DNA Thermodynamics Nonlinear structures (domain walls) and DNA melting Local equilibria Thermodynamics of domain walls Pulse propagation in nerve cells: the HodgkinHuxley model Background The HodgkinHuxley model The axon membrane as an array of electrical circuit elements Ion transport via distinct ionic channels Voltage clamping Ionic channels controlled by gates Membrane activation is a threshold phenomenon A qualitative picture of ion transport during nerve activation Pulse propagation Localization and transport of energy in proteins: The Davydov soliton Background. Model Hamiltonian Energy storage in C=O stretching modes. Excitonic Hamiltonian Coupling to lattice vibrations. Analogy to polaron BornOppenheimer dynamics Quantum (excitonic) dynamics Lattice motion Coupled excitonphonon dynamics The Davydov soliton The heavy ion limit. Static Solitons Moving solitons iv
6 Contents 17 Nonlinear localization in translationally invariant systems: discrete breathers The SieversTakeno conjecture Numerical evidence of localization Diagnostics of energy localization Internal dynamics Towards exact discrete breathers A Impurities, disorder and localization 164 A.1 Definitions A.1.1 Electrons A.1.2 Phonons A.2 A single impurity A.2.1 An exact result A.2.2 Numerical results A.3 Disorder A.3.1 Electrons in disordered onedimensional media A.3.2 Vibrational spectra of onedimensional disordered lattices Bibliography 173 v
7 Foreword The fact that most fundamental laws of physics, notably those of electrodynamics and quantum mechanics, have been formulated in mathematical language as linear partial differential equations has resulted historically in a preferred mode of thought within the physics community  a linear theoretical bias. The Fourier decomposition  an admittedly powerful procedure of describing an arbitrary function in terms of sines and cosines, but nonetheless a mathematical tool  has been firmly embedded in the conceptual framework of theoretical physics. Photons, phonons, magnons are prime examples of how successive generations of physicists have learned to describe properties of light, lattice vibrations, or the dynamics of magnetic crystals, respectively, during the last 100 years. This conceptual bias notwithstanding, engineers or physicists facing specific problems in classical mechanics, hydrodynamics or quantum mechanics were never shy of making particular approximations which led to nonlinear ordinary, or partial differential equations. Therefore, by the 1960 s, significant expertise had been accumulated in the field of nonlinear differential and/or integral equations; in addition, major breakthroughs had occurred on some fundamental issues related to chaos in classical mechanics (Poincaré, Birkhoff, KAM theorems). Due to the underlying linear bias however, this substantial progress took unusually long to find its way to the core of physical theory. This changed rapidly with the advent of electronic computation and the new possibilities of numerical visualization which accompanied it. Computer simulations became instrumental in catalyzing the birth of nonlinear science. This set of lectures does not even attempt to cover all areas where nonlinearity has proved to be of importance in modern physics. I will however try to describe some of the basic concepts mainly from the angle of condensed matter / statistical mechanics, an area which provided an impressive list of nonlinearly governed phenomena over the last fifty years  starting with the FermiPastaUlam numerical experiment and its subsequent interpretation by Zabusky and Kruskal in terms of solitons ( paradox turned discovery, in the words of J. Ford). There is widespread agreement that both solitons and chaos have achieved the status of theoretical paradigm. The third concept introduced here, localization in the absence of disorder, is a relatively recent breakthrough related to the discovery of independent (nonlinear) localized modes (ILMs), a.k.a. discrete breathers. Since neither the development of the field nor its present state can be described in terms of a unique linear narrative, both the exact choice of topics and the digressions necessary to describe the wider context are to a large extent arbitrary. The latter are however necessary in order to provide a selfcontained presentation which will be useful for the nonexpert, i.e. typically the advanced undergraduate student with an elementary knowledge of quantum mechanics and statistical physics. Konstanz, June 2006 vi
8 1 Background: Hamiltonian mechanics Consider a mechanical system with s degrees of freedom. The state of the mechanical system at any instant of time is described by the coordinates {Q i (t), i = 1, 2,, s} and the corresponding velocities { Q i (t)}. In many applications that I will deal with, this may be a set of N point particles which are free to move in one spatial dimension. In that particular case s = N and the coordinates are the particle displacements. The rules for temporal evolution, i.e. for the determination of particle trajectories, are described in terms of Newton s law  or, in the more general Lagrangian and Hamiltonian formulations. The more general formulations are necessary in order to develop and/or make contact with fundamental notions of statistical and/or quantum mechanics. 1.1 Lagrangian formulation of dynamics The Lagrangian is given as the difference between kinetic and potential energies. particle system interacting by velocityindependent forces For a L({Q i, Q i }) = T V (1.1) T = 1 s m i Q 2 i 2 i=1 V = V ({Q i }, t). where an explicit dependence of the potential energy on time has been allowed. Lagrangian dynamics derives particle trajectories by determining the conditions for which the action integral t S(t, t 0 ) = dτl({q i, Q i, τ}) (1.2) t 0 has an extremum. The result is d L dt Q = L (1.3) i Q i which for Lagrangians of the type (1.2) becomes i.e. Newton s law. m i Qi = V Q i (1.4) 1.2 Hamiltonian dynamics Canonical momenta Hamiltonian mechanics, uses instead of velocities, the canonical momenta conjugate to the coordinates {Q i }, defined as P i = L Q i. (1.5) 1
9 1 Background: Hamiltonian mechanics In the case of (1.2) it is straightforward to express the Hamiltonian function (the total energy) H = T + V in terms of P s and Q s. The result is H({P i, Q i }) = s I=1 P 2 i 2m i + V ({Q i }). (1.6) Poisson brackets Hamiltonian dynamics is described in terms of Poisson brackets {A, B} = s i=1 { A B A } B Q i P i P i Q i (1.7) where A, B are any functions of the coordinates and momenta. The momenta are canonically conjugate to the coordinates because they satisfy the relationships Equations of motion According to Hamiltonian dynamics, the time evolution of any function A({P i, Q i }, t) is determined by the linear differential equations Ȧ da dt = {A, H} + A t. (1.8) where the second term denotes any explicit dependence of A on the time t. Application of (1.8) to the cases A = P i and A = Q i respectively leads to P i = {P i, H} Q i = {Q i, H} (1.9) which can be shown to be equivalent to (1.4). The time evolution of the Hamiltonian itself is governed by dh dt = H ( = V ). (1.10) t t Canonical transformations Hamiltonian formalism important because the symplectic structure of equations of motion (from Greek συµπλɛκω = crosslink  of momenta & coordinate variables ) remains invariant under a class of transformations obtained by a suitable generating function ( canonical transformations). Example, transformation from old coordinates & momenta {P, Q} to new ones {p, q}, via a generating function F 1 (Q, q, t) which depends on old and new coordinates (but not on old and new momenta  NB there are three more forms of generating functions  ): P i = F 1(q, Q, t) Q i p i = F 1(q, Q, t) q i K = H + F 1 t (1.11) 2
10 1 Background: Hamiltonian mechanics new coordinates are obtained by solving the first of the above eqs., and new momenta by introducing the solution in the second. It is straightforward to verify that the dynamics remains forminvariant in the new coordinate system, i.e. ṗ i = {p i, K} q i = {q i, K} (1.12) and dk(p, q, t) K(p, q, t) =. (1.13) dt t Note that if there is no explicit dependence of F 1 on time, the new Hamiltonian K is equal to the old H Point transformations A special case of canonical transformations are point transformations, generated by F 2 (Q, p, t) = i f i (Q, t)p i ; (1.14) New coordinates depend only on old coordinates  not on old momenta; in general new momenta depend on both old coordinates and momenta. A special case of point transformations are orthogonal transformations, generated by F 2 (Q, p) = i,k a ik Q k p i (1.15) where a is an orthogonal matrix. It follows that q i = k p i = k a ik Q k a ik P k. (1.16) Note that, in the case of orthogonal transformations, coordinates transform among themselves; so do the momenta. Normal mode expansion is an example of (1.16). 1.3 HamiltonJacobi theory HamiltonJacobi equation Hamiltonian dynamics consists of a system of 2N coupled firstorder linear differential equations. In general, a complete integration would involve 2N constants (e.g. the initial values of coordinates and momenta). Canonical transformations enable us to play the following game: 1 Look for a transformation to a new set of canonical coordinates where the new Hamiltonian is zero and hence all new coordinates and momenta are constants of the motion. 2 Let (p, q) be the set of original momenta and coordinates in eqs of previous section, 1 HamiltonJacobi theory is not a recipe for integration of the coupled ODEs; nor does it in general lead to a more tractable mathematical problem. However, it provides fresh insight to the general problem, including important links to quantum mechanics and practical applications on how to deal with mechanical perturbations of a known, solved system. 2 Does this seem like too many constants? We will later explore what independent constants mean in mechanics, but at this stage let us just note that the original mathematical problem of integrating the 2N Hamiltonian equations does indeed involve 2N constants. 3
11 1 Background: Hamiltonian mechanics (α, β) the set of new constant momenta and coordinates generated by the generating function F 2 (q, α, t) which depends on the original coordinates and the new momenta. The choice of K 0 in (1.11) means that F 2 t + H(q 1, q s ; F 2 q 1,, F 2 q s ; t) = 0. (1.17) Suppose now that you can [miraculously] obtain a solution of the firstorder in general nonlinearpde (1.17), F 2 = S(q, α, t). Note that the solution in general involves s constants {α i, i = 1,, s}. The s + 1st constant involved in the problem is a trivial one, because if S is a solution, so is S + A, where A is an arbitrary constant. It is now possible to use the defining equation of the generating function F 2 β i = S(q, α, t) α i (1.18) to obtain the new [constant] coordinates {β i, i = 1,, s}; finally, turning inside out (1.18) yields the trajectories q j = q j (α, β, t). (1.19) In other words, a solution of the HamiltonJacobi equation (1.17) provides a solution of the original dynamical problem Relationship to action It can be easily shown that the solution of the HamiltonJacobi equation satisfies or S(q, α, t) S(q, α, t 0 ) = ds dt = L, (1.20) t t 0 dτ L(q, q, τ) (1.21) where the r.h.s involves the actual particle trajectories; this shows that the solution of the HamiltonJacobi equation is indeed the extremum of the action function used in Lagrangian mechanics Conservative systems If the Hamiltonian does not depend explicitly on time, it is possible to separate out the time variable, i.e. S(q, α, t) = W (q, α) λ 0 t (1.22) where now the timeindependent function W (q) (Hamilton s characteristic function) satisfies ( H q 1, q s ; W,, W ) = λ 0, (1.23) q 1 q s and involves s 1 independent constants, more precisely, the s constants α 1, α s depend on λ 0. 4
12 1.3.4 Separation of variables 1 Background: Hamiltonian mechanics The previous example separated out the time coordinate from the rest of the variables of the HJ function. Suppose q 1 and W q 1 enter the Hamiltonian only in the combination φ 1 (q 1, W q 1 ). The Ansatz W = W 1 (q 1 ) + W (q 2,, q s ) (1.24) in (1.23) yields H (q 2, q s ; W,, W ; φ 1 q 2 q s ( q 1, W ) ) 1 = λ 0 ; (1.25) q 1 since (1.25) must hold identically for all q, we have H ( φ 1 q 1, W ) 1 q 1 ) (q 2, q s ; W,, W ; λ 1 q 2 q s = λ 1 = λ 0. (1.26) The process can be applied recursively if the separation condition holds. Note that cyclic coordinates lead to a special case of separability; if q 1 is cyclic, then φ 1 = W q 1 = W 1 q 1, and hence W 1 (q 1 ) = λ 1 q 1. This is exactly how the time coordinate separates off in conservative systems (1.23). Complete separability occurs if we can write Hamilton s characteristic function  in some set of canonical variables  in the form W (q, α) = i W i (q i, α 1,, α s ). (1.27) Periodic motion. Actionangle variables Consider a completely separable system in the sense of (1.27). The equation p i = S q i = W i(q i, α 1,, α s ) q i (1.28) provides the phase space orbit in the subspace (q i, p i ). Now suppose that the motion in all subspaces {(q i, p i ), i = 1,, s} is periodic  not necessarily with the same period. Note that this may mean either a strict periodicity of p i, q i as a function of time (such as occurs in the bounded motion of a harmonic oscillator), or a motion of the freely rotating pendulum type, where the angle coordinate is physically significant only mod 2π. The action variables are defined as J i = 1 2π p i dq i = 1 2π dq i W i (q i, α 1,, α s ) q i (1.29) and therefore depend only on the integration constants, i.e. they are constants of the motion. If we can turn inside out (1.29), we can express W as a function of the J s instead of the α s. Then we can use the function W as a generating function of a canonical transformation to a new set of variables with the J s as new momenta, and new angle coordinates θ i = W J i = W i(q i, J 1,, J s ) J i. (1.30) 5
13 1 Background: Hamiltonian mechanics In the new set of canonical variables, Hamilton s equations of motion are J i = 0 θ i = H(J) J i ω i (J). (1.31) Note that the Hamiltonian cannot depend on the angle coordinates, since the action coordinates, the J s, are  by construction  all constants of the motion. In the set of actionangle coordinates, the motion is as trivial as it can get: J i = const θ i = ω i (J) t + const. (1.32) Complete integrability A system is called completely integrable in the sense of Liouville if it can be shown to have s independent conserved quantities in involution (this means that their Poisson brackets, taken in pairs, vanish identically). If this is the case, one can always perform a canonical transformation to actionangle variables. 1.4 Symmetries and conservation laws A change of coordinates, if it reflects an underlying symmetry of physical laws, will leave the form of the equations of motion invariant. Because Lagrangian dynamics is derived from an action principle, any such infinitesimal change which changes the particle coordinates and adds a total time derivative to the Lagrangian, i.e. q i q i = q i + ɛf i (q, t) q i q i = q i + ɛf i (q, t) (1.33) L = L + ɛ df dt, (1.34) will leave the equations of motion invariant. On the other hand, the transformed Lagrangian will generally be equal to and therefore the quantity will be conserved. L ({q i, q i}) = L({q i, q i}) = L({q i, q i }) + = L({q i, q i }) + = L({q i, q i }) + s i=1 s [ L ɛf i + L ] ɛf q i q i i s [ ( ) d L ɛf i + L ] ɛf dt q i q i i s ( ) d L f i dt q i i=1 i=1 i=1 Such underlying symmetries of classical mechanics are: L q i f i F (1.35) 6
14 1.4.1 Homogeneity of time 1 Background: Hamiltonian mechanics L = L(t + ɛ) = L(t) + ɛdl/dt, i.e. F = L; furthermore, q i = q i(t + ɛ) = q i + ɛ q i, i.e. f i = q i. As a result, the quantity s L H = q i L (1.36) q i (Hamiltonian) is conserved. i= Homogeneity of space The transformation q i q i + ɛ (hence f i = 1) leaves the Lagrangian invariant (F = 0). The conserved quantity is s L P = (1.37) q i (total momentum). i= Galilei invariance The transformation q i q i ɛt (hence f i = t) does not generally change the potential energy (if it depends only on relative particle positions). It adds to the kinetic energy a term ɛp, i.e. F = m i q i. The conserved quantity is (uniform motion of the center of mass). s m i q i P t (1.38) i= Isotropy of space (rotational symmetry of Lagrangian) Let the position of the ith particle in space be represented by the vector coordinate q i. Rotation around an axis parallel to the unit vector ˆn is represented by the transformation q i q i + ɛ f i where f i = ˆn q i. The change in kinetic energy is ɛ i q i f i = 0. If the potential energy is a function of the interparticle distances only, it too remains invariant under a rotation. Since the Lagrangian is invariant, the conserved quantity (1.35) is s i=1 L q i f i = s m i qi (ˆn q i ) = ˆn I, i=1 where is the total angular momentum. s I = m i ( q i q i ) (1.39) i=1 7
15 1.5 Continuum field theories 1 Background: Hamiltonian mechanics Lagrangian field theories in 1+1 dimensions Given a Lagrangian in 1+1 dimensions, L = dxl(φ, φ x, φ t ) (1.40) where the Lagrangian density L depends only on the field φ and first space and time derivatives, the equations of motion can be derived by minimizing the total action S = dtdxl (1.41) and have the form d dt ( L φ t ) + d ( ) L L dx φ x φ = 0. (1.42) Symmetries and conservation laws The form (1.42) remains invariant under a transformation which adds to the Lagrangian density a term of the form ɛ µ J µ (1.43) where the implied summation is over µ = 0, 1, because this adds only surface boundary terms to the action integral. If the transformation changes the field by δφ, and the derivatives by δφ x, δφ t, the same argument as in discrete systems leads us to conclude that L L δφ + δφ x + L ( dj0 δφ t = ɛ φ φ x φ t dt + dj ) 1 (1.44) dx which can be transformed, using the equations of motion, to ( ) d L δφ + L δφ t + d ( ) L δφ + L ( dj0 δφ x = ɛ dt φ t φ t dx φ x φ x dt + dj ) 1 dx (1.45) Examples: 1. homogeneity of space (translational invariance) x x + ɛ δφ = φ(x + ɛ) φ(x) = φ x ɛ δφ t = φ t (x + ɛ) φ t (x) = φ xt ɛ δφ x = φ x (x + ɛ) φ x (x) = φ xx ɛ δl = dl dl δx = dx dx ɛ J 1 = L, J 0 = 0. (1.46) Eq. (1.45) becomes ( ) d L φ x + L φ xt + d ( ) L φ x + L φ xx = dl dt φ t φ t dx φ x φ x dx or (1.47) ( ) d L φ x + d ( ) L φ x L = 0 ; (1.48) dt φ t dx φ x 8
16 1 Background: Hamiltonian mechanics integrating over all space, this gives dx L φ x P φ t (1.49) i.e. the total momentum is a constant. 2. homogeneity of time t t + ɛ δφ = φ(t + ɛ) φ(t) = φ t ɛ δφ t = φ t (t + ɛ) φ t (t) = φ tt ɛ δφ x = φ x (t + ɛ) φ x (t) = φ xt ɛ δl = dl dl δt = dt dt ɛ J 0 = L, J 1 = 0. (1.50) Eq. (1.45) becomes ( ) d L φ t + L φ tt + d ( ) L φ t + L φ tx = dl dt φ t φ t dx φ x φ x dt or (1.51) ( ) d L φ t L + d ( ) L φ t = 0 ; (1.52) dt φ t dx φ x integrating over all space, this gives [ ] L dx φ t L H (1.53) φ t i.e. the total energy is a constant. 3. Lorentz invariance 1.6 Perturbations of integrable systems Consider a conservative Hamiltonian system H 0 (J) which is completely integrable, i.e. it possesses s independent integrals of motion. Note that I use the actionangle coordinates, so that H 0 is a function of the (conserved) action coordinates J j. The angles θ j are cyclic variables, so they do not appear in H 0. Suppose now that the system is slightly perturbed, by a timeindependent perturbation Hamiltonian µh 1 (µ 1) A sensible question to ask is: what exactly happens to the integrals of motion? We know of course that the energy of the perturbed system remains constant  since H 1 has been assumed to be time independent. But what exactly happens to the other s 1 constants of motion? The question was first addressed by Poincaré in connection with the stability of the planetary system. He succeeded in showing that there are no analytic invariants of the perturbed system, i.e. that it is not possible, starting from a constant Φ 0 of the unperturbed system, to construct quantities Φ = Φ 0 (J) + µφ 1 (J, θ) + µ 2 Φ 2 (J, θ), (1.54) where the Φ n s are analytic functions of J, θ, such that {Φ, H} = 0 (1.55) 9
17 1 Background: Hamiltonian mechanics holds, i.e. Φ is a constant of motion of the perturbed system. The proof of Poincaré s theorem is quite general. The only requirement on the unperturbed Hamiltonian is that it should have functionally independent frequencies ω j = H 0 / J j. Although the proof itself is lengthy and I will make no attempt to reproduce it, it is fairly straightforward to see where the problem with analytic invariants lies. To second order in µ, the requirement (1.55) implies {Φ 0 + µφ 1 + µ 2 Φ 2, H 0 + µh 1 } = 0 {Φ 0, H 0 } + µ ({Φ 1, H 0 } + {Φ 0, H 1 }) + µ 2 ({Φ 2, H 0 } + {Φ 1, H 1 }) = 0. The coefficients of all powers must vanish. Note that the zeroth order term vanishes by definition. The higher order terms will do so, provided {Φ 1, H 0 } = {Φ 0, H 1 } (1.56) {Φ 2, H 0 } = {Φ 1, H 1 }. The process can be continued iteratively to all orders, by requiring {Φ n, H 0 } = {Φ n+1, H 1 }. (1.57) Consider the lowestorder term generated by (1.57). Writing down the Poisson brackets gives s ( Φ1 H 0 Φ ) 1 H 0 s ( Φ0 H 1 = Φ ) 0 H 1. (1.58) θ i J i J i θ i θ i J i J i θ i j=1 The second term on the left hand side and the first term on the righthand side vanish because the θ s are cyclic coordinates in the unperturbed system. The rest can be rewritten as s j=1 ω i (J) Φ 1 θ i = j=1 s j=1 Φ 0 J i H 1 θ i. (1.59) For notational simplicity, let me now restrict myself to the case of two degrees of freedom. The perturbed Hamiltonian can be written in a double Fourier series H 1 = n 1,n 2 A n1,n 2 (J 1, J 2 ) cos(n 1 θ 1 + n 2 θ 2 ). (1.60) Similarly, one can make a double Fourier series ansatz for Φ 1, Φ 1 = n 1,n 2 B n1,n 2 (J 1, J 2 ) cos(n 1 θ 1 + n 2 θ 2 ). (1.61) Now apply (1.59) to the case Φ 0 (J) = J 1. Using the double Fourier series I obtain B (J n 1) 1 n 1,n 2 = A n1,n n 1 ω 1 + n 2 ω 2, (1.62) 2 which in principle determines the firstorder term in the µ expansion of the constant of motion J 1 which should replace J 1 in the new system. It is straightforward to show, using the same process for J 2, that the perturbed Hamiltonian can be written in terms of the new constants J 1 as H = H 0 (J 1, J 2) + O(µ 2 ). (1.63) Unfortunately, what looks like the beginning of a systematic expansion suffers from a fatal flaw. If the frequencies are functionally independent, the denominator in (1.62) will in general vanish on a denumerably infinite number of surfaces in phase space. This however means that Φ 1 cannot be an analytic function of J 1, J 2. Analytic invariants are not possible. All integrals of motion  other than the energy  are irrevocably destroyed by the perturbation. 10
18 2 Background: Statistical mechanics 2.1 Scope Classical statistical mechanics attempts to establish a systematic connection between microscopic theory which governs the dynamical motion of individual entities (atoms, molecules, local magnetic moments on a lattice) and the macroscopically observed behavior of matter. Microscopic motion is described  depending on the particular scale of the problem  either by classical or quantum mechanics. The rules of macroscopically observed behavior under conditions of thermal equilibrium have been codified in the study of thermodynamics. Thermodynamics will tell you which processes are macroscopically allowed, and can establish relationships between material properties. In principle, it can reduce everything  everything which can be observed under varying control parameters ( temperature, pressure or other external fields) to the equation of state which describes one of the relevant macroscopic observables as a function of the control parameters. Deriving the form of the equation of state is beyond thermodynamics. It needs a link to microscopic theory  i.e. to the underlying mechanics of the individual particles. This link is provided by equilibrium statistical mechanics. A more general theory of nonequilibrium statistical mechanics is necessary to establish a link between nonequilibrium macroscopic behavior (e.g. a steady state flow) and microscopic dynamics. Here I will only deal with equilibrium statistical mechanics. 2.2 Formulation A statistical description always involves some kind of averaging. Statistical mechanics is about systematically averaging over hopefully nonessential details. What are these details and how can we show that they are nonessential? In order to decide this you have to look first at a system in full detail and then decide what to throw out  and how to go about it consistently Phase space An Hamiltonian system with s degrees of freedom is fully described at any given time if we know all coordinates and momenta, i.e. a total of 2s quantities (=6N if we are dealing with point particles moving in threedimensional space). The microscopic state of the system can be viewed as a point, a vector in 2s dimensional space. The dynamical evolution of the system in time can be viewed as a motion of this point in the 2s dimensional space (phase space). I will use the shorthand notation Γ (q i, p i, i = 1, s) to denote a point in phase space. More precisely, Γ(t) will denote a trajectory in phase space with the initial condition Γ(t 0 ) = Γ Note that trajectories in phase space do not cross. A history of a Hamiltonian system is determined by differential equations which are firstorder in time, and is therefore reversible  and hence unique. 11
19 2.2.2 Liouville s theorem 2 Background: Statistical mechanics Consider an element of volume dσ 0 in phase space; the set of trajectories starting at time t 0 at some point Γ 0 dσ 0 lead, at time t to points Γ dσ. Liouville s theorem asserts that dσ = dσ 0. (invariance of phase space volume). The proof consists of showing that the Jacobi determinant (q, p) D(t, t 0 ) (q 0, p 0 (2.1) ) corresponding to the coordinate transformation (q 0, p 0 ) (q, p), is equal to unity. Using general properties of Jacobians (q, p) (q, p) (q 0, p 0 = ) (q 0, p) (q 0, p) (q 0, p 0 ) = (q) (p) (q 0 ) p=const (p 0 ) (2.2) q=const and D(t, t 0 t = t=t0 s i=1 ( qi + ṗ ) i = q i p i t=t0 s i=1 ( 2 H q i p i ) 2 H = 0, (2.3) p i q i and noting that D(t 0, t 0 ) = 1, it follows that D(t, t 0 ) = 1 at all times Averaging over time Consider a function A(Γ) of all coordinates and momenta. If you want to compute its longtime average under conditions of thermal equilibrium, you need to follow the state of the system over a long time, record it, evaluate the function A at each instant of time, and take a suitable average. Following the trajectory of the point in phase space allows us to define a longtime average 1 T Ā = lim dta[γ(t)]. (2.4) T T 0 Since the system is followed over infinite time this can then be regarded as a true equilibrium average. More on this later Ensemble averaging On the other hand, we could consider an ensemble of identically prepared systems and attempt a series of observations. One system could be in the state Γ 1, another in the state Γ 2. Then perhaps we could determine the distribution of states ρ(γ), i.e. the probability ρ(γ)δγ, that the state vector is in the neighborhood (Γ, Γ + δγ). The average of A in this case would be < A >= dγρ(γ)a(γ) (2.5) Note that since ρ is a probability distribution, its integral over all phase space should be normalized to unity: dγρ(γ) = 1 (2.6) A distribution in phase space must obey further restrictions. Liouville s theorem states that if we view the dynamics of a Hamiltonian system as a flow in phase space, elements of volume are invariant  in other words the fluid is incompressible: d dt ρ(γ, t) = {ρ, H} + ρ(γ, t) = 0. (2.7) t 12
20 2 Background: Statistical mechanics For a stationary distribution ρ(γ)  as one expects to obtain for a system at equilibrium  {ρ, H} = 0, (2.8) i.e. ρ can only depend on the energy 2. This is a very severe restriction on the forms of allowed distribution functions in phase space. Nonetheless it still allows for any functional dependence on the energy. A possible choice (Boltzmann) is to assume that any point on the phase space hypersurface defined by H(Γ) = E may occur with equal probability. This corresponds to ρ(γ) = 1 δ {H(Γ) E} (2.9) Ω(E) where Ω(E) = dγ δ {H(Γ) E} (2.10) is the volume of the hypersurface H(Γ) = E. This is the microcanonical ensemble. Other choices are possible  e.g. the canonical (Gibbs) ensemble defined as ρ(γ) = 1 Z(β) e βh(γ) (2.11) where the control parameter β can be identified with the inverse temperature and Z(β) = dγe βh(γ) (2.12) is the classical partition function Equivalence of ensembles The choice of ensemble, although it may appear arbitrary, is meant to reflect the actual experimental situation. For example, the Gibbs ensemble may be derived  in the sense that it can be shown to correspond to a small (but still macroscopic) system in contact with a much larger reservoir of energy  which in effect holds the smaller system at a fixed temperature T = 1/β. Ensembles must  and to some extent can  be shown to be equivalent, in the sense that the averages computed using two different ensembles coincide if the control parameters are appropriately chosen. For example a microcanonical average of a function A(Γ) over the energy surface H(Γ) = ɛ will be equal with the canonical average at a certain temperature T if we choose ɛ to be equal to the canonical average of the energy at that temperature, i.e. < A(Γ) > micro ɛ =< A(Γ) > canon T if ɛ =< H(Γ) > canon T. If ensembles can be shown to be equivalent to each other in this sense, we do not need to perform the actual experiment of waiting and observing the realization of a large number of identical systems as postulated in the previous section. We can simply use the most convenient ensemble for the problem at hand as a theoretical tool for calculating averages. In general one uses the canonical ensemble, which is designed for computing average quantities as functions of temperature Ergodicity The usage of ensemble averages  and therefore of the whole edifice of classical statistical mechanics  rests on the implicit assumption that they somehow coincide with the more physical time averages. Since the various ensembles can be shown to be equivalent (cf. 2 or  in principle  on other conserved quantities; in dealing with large systems it may well be necessary to account for other macroscopically conserved quantities in defining a proper distribution function. 13
N 1. (q k+1 q k ) 2 + α 3. k=0
Teoretisk Fysik Handin problem B, SI1142, Spring 2010 In 1955 Fermi, Pasta and Ulam 1 numerically studied a simple model for a one dimensional chain of nonlinear oscillators to see how the energy distribution
More information(Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7)
Chapter 4. Lagrangian Dynamics (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7 4.1 Important Notes on Notation In this chapter, unless otherwise stated, the following
More informationMASTER OF SCIENCE IN PHYSICS MASTER OF SCIENCES IN PHYSICS (MS PHYS) (LIST OF COURSES BY SEMESTER, THESIS OPTION)
MASTER OF SCIENCE IN PHYSICS Admission Requirements 1. Possession of a BS degree from a reputable institution or, for nonphysics majors, a GPA of 2.5 or better in at least 15 units in the following advanced
More informationMODULE VII LARGE BODY WAVE DIFFRACTION
MODULE VII LARGE BODY WAVE DIFFRACTION 1.0 INTRODUCTION In the wavestructure interaction problems, it is classical to divide into two major classification: slender body interaction and large body interaction.
More informationContinuous Groups, Lie Groups, and Lie Algebras
Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... Bertrand Russell The groups
More informationThe 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
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationExponentially small splitting of separatrices for1 1 2 degrees of freedom Hamiltonian Systems close to a resonance. Marcel Guardia
Exponentially small splitting of separatrices for1 1 2 degrees of freedom Hamiltonian Systems close to a resonance Marcel Guardia 1 1 1 2 degrees of freedom Hamiltonian systems We consider close to completely
More informationHarmonic Oscillator Physics
Physics 34 Lecture 9 Harmonic Oscillator Physics Lecture 9 Physics 34 Quantum Mechanics I Friday, February th, 00 For the harmonic oscillator potential in the timeindependent Schrödinger equation: d ψx
More information7. DYNAMIC LIGHT SCATTERING 7.1 First order temporal autocorrelation function.
7. DYNAMIC LIGHT SCATTERING 7. First order temporal autocorrelation function. Dynamic light scattering (DLS) studies the properties of inhomogeneous and dynamic media. A generic situation is illustrated
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More information1 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
More informationChapter 18 Statistical mechanics of classical systems
Chapter 18 Statistical mechanics of classical systems 18.1 The classical canonical partition function When quantum effects are not significant, we can approximate the behavior of a system using a classical
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationIntroduction to SME and Scattering Theory. Don Colladay. New College of Florida Sarasota, FL, 34243, U.S.A.
June 2012 Introduction to SME and Scattering Theory Don Colladay New College of Florida Sarasota, FL, 34243, U.S.A. This lecture was given at the IUCSS summer school during June of 2012. It contains a
More informationThe Role of Electric Polarization in Nonlinear optics
The Role of Electric Polarization in Nonlinear optics Sumith Doluweera Department of Physics University of Cincinnati Cincinnati, Ohio 45221 Abstract Nonlinear optics became a very active field of research
More informationRotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve
QUALITATIVE THEORY OF DYAMICAL SYSTEMS 2, 61 66 (2001) ARTICLE O. 11 Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve Alexei Grigoriev Department of Mathematics, The
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationOrbits of the LennardJones Potential
Orbits of the LennardJones Potential Prashanth S. Venkataram July 28, 2012 1 Introduction The LennardJones potential describes weak interactions between neutral atoms and molecules. Unlike the potentials
More informationBlasius solution. Chapter 19. 19.1 Boundary layer over a semiinfinite flat plate
Chapter 19 Blasius solution 191 Boundary layer over a semiinfinite flat plate Let us consider a uniform and stationary flow impinging tangentially upon a vertical flat plate of semiinfinite length Fig
More information* Biot Savart s Law Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B. PPT No.
* Biot Savart s Law Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B PPT No. 17 Biot Savart s Law A straight infinitely long wire is carrying
More informationLecture 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
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationCoefficient of Potential and Capacitance
Coefficient of Potential and Capacitance Lecture 12: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We know that inside a conductor there is no electric field and that
More informationEXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL
EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist
More informationNumerically 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
More informationChapter 7. Lyapunov Exponents. 7.1 Maps
Chapter 7 Lyapunov Exponents Lyapunov exponents tell us the rate of divergence of nearby trajectories a key component of chaotic dynamics. For one dimensional maps the exponent is simply the average
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationElasticity Theory Basics
G22.3033002: 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
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationSystems with Persistent Memory: the Observation Inequality Problems and Solutions
Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +
More informationA Theoretical Model for Mutual Interaction between Coaxial Cylindrical Coils Lukas Heinzle
A Theoretical Model for Mutual Interaction between Coaxial Cylindrical Coils Lukas Heinzle Page 1 of 15 Abstract: The wireless power transfer link between two coils is determined by the properties of the
More informationChapter 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
More informationPlate waves in phononic crystals slabs
Acoustics 8 Paris Plate waves in phononic crystals slabs J.J. Chen and B. Bonello CNRS and Paris VI University, INSP  14 rue de Lourmel, 7515 Paris, France chen99nju@gmail.com 41 Acoustics 8 Paris We
More informationTill now, almost all attention has been focussed on discussing the state of a quantum system.
Chapter 13 Observables and Measurements in Quantum Mechanics Till now, almost all attention has been focussed on discussing the state of a quantum system. As we have seen, this is most succinctly done
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More information1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as
Chapter 3 (Lecture 45) 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
More informationReview of Vector Analysis in Cartesian Coordinates
R. evicky, CBE 6333 Review of Vector Analysis in Cartesian Coordinates Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers.
More informationLecture L222D 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 L3 for
More informationSome Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)
Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving
More informationPHY411. PROBLEM SET 3
PHY411. PROBLEM SET 3 1. Conserved Quantities; the RungeLenz 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
More informationThe Essentials of Quantum Mechanics
The Essentials of Quantum Mechanics Prof. Mark Alford v7, 2008Oct22 In classical mechanics, a particle has an exact, sharply defined position and an exact, sharply defined momentum at all times. Quantum
More informationAssessment Plan for Learning Outcomes for BA/BS in Physics
Department of Physics and Astronomy Goals and Learning Outcomes 1. Students know basic physics principles [BS, BA, MS] 1.1 Students can demonstrate an understanding of Newton s laws 1.2 Students can demonstrate
More informationAP1 Oscillations. 1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false?
1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The
More informationFigure 2.1: Center of mass of four points.
Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would
More information develop a theory that describes the wave properties of particles correctly
Quantum Mechanics Bohr's model: BUT: In 192526: by 1930s:  one of the first ones to use idea of matter waves to solve a problem  gives good explanation of spectrum of single electron atoms, like hydrogen
More informationDynamic Analysis. Mass Matrices and External Forces
4 Dynamic Analysis. Mass Matrices and External Forces The formulation of the inertia and external forces appearing at any of the elements of a multibody system, in terms of the dependent coordinates that
More informationChapter 11 Equilibrium
11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of
More informationTHEORETICAL MECHANICS
PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents
More information3. Derive the partition function for the ideal monoatomic gas. Use Boltzmann statistics, and a quantum mechanical model for the gas.
Tentamen i Statistisk Fysik I den tjugosjunde februari 2009, under tiden 9.0015.00. Lärare: Ingemar Bengtsson. Hjälpmedel: Penna, suddgummi och linjal. Bedömning: 3 poäng/uppgift. Betyg: 03 = F, 46
More informationCONTROLLABILITY. 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
More information6 J  vector electric current density (A/m2 )
Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J  vector electric current density (A/m2 ) M  vector magnetic current density (V/m 2 ) Some problems
More informationTime dependence in quantum mechanics Notes on Quantum Mechanics
Time dependence in quantum mechanics Notes on Quantum Mechanics http://quantum.bu.edu/notes/quantummechanics/timedependence.pdf Last updated Thursday, November 20, 2003 13:22:3705:00 Copyright 2003 Dan
More informationarxiv:1111.4354v2 [physics.accph] 27 Oct 2014
Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK arxiv:1111.4354v2 [physics.accph] 27 Oct 2014 Abstract We discuss the theory of electromagnetic
More informationThe Kinetic Theory of Gases Sections Covered in the Text: Chapter 18
The Kinetic Theory of Gases Sections Covered in the Text: Chapter 18 In Note 15 we reviewed macroscopic properties of matter, in particular, temperature and pressure. Here we see how the temperature and
More informationMidterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m
Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.
More informationASEN 3112  Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1
19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A TwoDOF MassSpringDashpot Dynamic System Consider the lumpedparameter, massspringdashpot dynamic system shown in the Figure. It has two point
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationarxiv:physics/0004029v1 [physics.edph] 14 Apr 2000
arxiv:physics/0004029v1 [physics.edph] 14 Apr 2000 Lagrangians and Hamiltonians for High School Students John W. Norbury Physics Department and Center for Science Education, University of WisconsinMilwaukee,
More information2.2 Magic with complex exponentials
2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or
More informationLecture 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
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationChapter 24 Physical Pendulum
Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...
More informationNotes on Elastic and Inelastic Collisions
Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationCBE 6333, R. Levicky 1 Differential Balance Equations
CBE 6333, R. Levicky 1 Differential Balance Equations We have previously derived integral balances for mass, momentum, and energy for a control volume. The control volume was assumed to be some large object,
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationSecond Order Linear Partial Differential Equations. Part I
Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables;  point boundary value problems; Eigenvalues and Eigenfunctions Introduction
More informationQuantum 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
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationBasic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics. Indian Institute of Technology, Delhi
Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics. Indian Institute of Technology, Delhi Module No. # 02 Simple Solutions of the 1 Dimensional Schrodinger Equation Lecture No. # 7. The Free
More informationLaws of Motion and Conservation Laws
Laws of Motion and Conservation Laws The first astrophysics we ll consider will be gravity, which we ll address in the next class. First, though, we need to set the stage by talking about some of the basic
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationThe First Law of Thermodynamics: Closed Systems. Heat Transfer
The First Law of Thermodynamics: Closed Systems The first law of thermodynamics can be simply stated as follows: during an interaction between a system and its surroundings, the amount of energy gained
More informationELEMENTS 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
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationLecture Notes. on Mathematical Modelling. in Applied Sciences. Authors. Nicola Bellomo, Elena De Angelis, and Marcello Delitala
Lecture Notes on Mathematical Modelling in Applied Sciences Authors Nicola Bellomo, Elena De Angelis, and Marcello Delitala c 2007 N. Bellomo, E. De Angelis, M. Delitala Nicola Bellomo Department of Mathematics
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationChapter 2. Derivation of the Equations of Open Channel Flow. 2.1 General Considerations
Chapter 2. Derivation of the Equations of Open Channel Flow 2.1 General Considerations Of interest is water flowing in a channel with a free surface, which is usually referred to as open channel flow.
More informationStability Of Structures: Basic Concepts
23 Stability Of Structures: Basic Concepts ASEN 3112 Lecture 23 Slide 1 Objective This Lecture (1) presents basic concepts & terminology on structural stability (2) describes conceptual procedures for
More informationBCM 6200  Protein crystallography  I. Crystal symmetry Xray diffraction Protein crystallization Xray sources SAXS
BCM 6200  Protein crystallography  I Crystal symmetry Xray diffraction Protein crystallization Xray sources SAXS Elastic Xray Scattering From classical electrodynamics, the electric field of the electromagnetic
More informationOscillations. Vern Lindberg. June 10, 2010
Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1
More information4. The Infinite Square Well
4. The Infinite Square Well Copyright c 215 216, Daniel V. Schroeder In the previous lesson I emphasized the free particle, for which V (x) =, because its energy eigenfunctions are so simple: they re the
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationThree 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,
More informationStructure Factors 59553 78
78 Structure Factors Until now, we have only typically considered reflections arising from planes in a hypothetical lattice containing one atom in the asymmetric unit. In practice we will generally deal
More informationElementary Differential Equations
Elementary Differential Equations EIGHTH EDITION Earl D. Rainville Late Professor of Mathematics University of Michigan Phillip E. Bedient Professor Emeritus of Mathematics Franklin and Marshall College
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationThe 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
More informationWAVES AND FIELDS IN INHOMOGENEOUS MEDIA
WAVES AND FIELDS IN INHOMOGENEOUS MEDIA WENG CHO CHEW UNIVERSITY OF ILLINOIS URBANACHAMPAIGN IEEE PRESS Series on Electromagnetic Waves Donald G. Dudley, Series Editor IEEE Antennas and Propagation Society,
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More informationFundamentals of grain boundaries and grain boundary migration
1. Fundamentals of grain boundaries and grain boundary migration 1.1. Introduction The properties of crystalline metallic materials are determined by their deviation from a perfect crystal lattice, which
More informationThe 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
More informationk u (t) = k b + k s t
Chapter 2 Stripe domains in thin films 2.1 Films with perpendicular anisotropy In the first part of this chapter, we discuss the magnetization of films with perpendicular uniaxial anisotropy. The easy
More informationAlgebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard
Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express
More informationA First Course in Elementary Differential Equations. Marcel B. Finan Arkansas Tech University c All Rights Reserved
A First Course in Elementary Differential Equations Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 1 Basic Terminology 4 2 Qualitative Analysis: Direction Field of y = f(t, y)
More informationFLAP P11.2 The quantum harmonic oscillator
F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module P. Opening items. Module introduction. Fast track questions.3 Ready to study? The harmonic oscillator. Classical description of
More informationContents. Microfluidics  Jens Ducrée Physics: NavierStokes Equation 1
Contents 1. Introduction 2. Fluids 3. Physics of Microfluidic Systems 4. Microfabrication Technologies 5. Flow Control 6. Micropumps 7. Sensors 8. InkJet Technology 9. Liquid Handling 10.Microarrays 11.Microreactors
More information