DOCSLIB.ORG

Phase Space. Liouville Equation and Theorem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Phase Space. Liouville Equation and Theorem Liouville Equation In this section we will build a bridge from Classical Mechanics to Statistical Physics. The bridge is Liouville equation. We start with the Hamiltonian formalism of the Classical Mechanics, where the state of a system with m degrees of freedom is described by m pairs of conjugated variables called (generalized) coordinates and momenta fqs; psg, s = 1; 2; : : : ; m. The equations of motion are generated with the Hamiltonian function, H(fqs; psg), by the following rule @H q_s = ; (1) @ps @H p_s = ¡ : (2) @qs Fore example, if we have N three-dimensional particles of mass M interact- ing with each other via a pair potential U, and also interacting with some external potential V , then the Hamiltonian for this system reads: XN p2 XN X H = j + V (r ) + U(r ¡ r ) ; (3) 2M j i j j=1 j=1 i<j where rj and pj are the radius-vector and momentum of the j-th particle, respectively. In this example, m = 3N: each component of each radius- vector represents a separate degree of freedom. The following property of Eqs. (1)-(2) will be crucial for us. If we need to describe time evolution of some function A(fqs; psg) due to the evolution of coordinates and momenta, then the following relation takes place A_ = fH; Ag ; (4) where the symbol in the r.h.s. is a shorthand notation|called Poisson bracket|for the following expression X @H @A @H @A fH; Ag = ¡ : (5) s @ps @qs @qs @ps [The proof is straightforward. The chain rule for dA(fqs(t); ps(t)g)=dt and then Eqs. (1)-(2) forq _s andp _s.] 1 Hence, any quantity A(fqs; psg) is a constant of motion if, and only if, its Poisson bracket with the Hamiltonian is zero. In particular, the Hamiltonian itself is a constant of motion, since fH; Hg = 0, and this is nothing else than the conservation of energy, because the physical meaning of the Hamiltonian function is energy expressed in terms of coordinates and momenta. De¯nition: The phase space is a 2m-dimensional space of points, or, equivalently, vectors of the following form: X = (q1; q2; : : : ; qm; p1; p2; : : : ; pm) : (6) Each point/vector in the phase space represents a state of the mechanical system. If we know X at some time moment, say, t = 0, then the further evolution of X|the trajectory X(t) in the phase space|is unambiguously given by Eqs. (1)-(2), since these are the ¯rst-order di®erential equations with respect to the vector function X(t). (For the same reason di®erent trajectories cannot intersect!) The phase space is convenient for statistical description of mechanical system. Suppose that the initial state for a system is known only with a certain ¯nite accuracy. This means that actually we know only the proba- bility density W0(X) of having the point X somewhere in the phase space. If the initial condition is speci¯ed in terms of probability density, then the subsequent evolution should be also described probabilistically, that is we have to work with the distribution W (X; t), which should be somehow re- lated to the initial condition W (X; 0) = W0(X). Our goal is to establish this relation. We introduce a notion of a statistical ensemble. Instead of dealing with probability density, we will work with a quantity which is proportional to it, and is much transparent. Namely, we simultaneously take some large number Nens of identical and independent systems distributed in accordance with W (X; t). We call this set of systems statistical ensemble. The j-th member of the ensemble is represented by its point Xj in the phase space. The crucial observation is that the quantity Nens W (X; t) gives the concentration of the points fXjg. Hence, to ¯nd the evolution of W we just need to describe the evolution of the concentration of the points Xj, which is intuitively easier, since each Xj obeys the Hamiltonian equation of motion. A toy model. To get used to the ensemble description, and also to obtain some important insights, consider the following dynamical model with just one degree of freedom: H = (1=4)(p2 + q2)2 : (7) 2 The equations of motion are: q_ = (p2 + q2) p ; (8) p_ = ¡(p2 + q2) q : (9) The quantity ! = p2 + q2 (10) is a constant of motion, since, up to a numeric factor, it is a square root of energy. We thus have a linear system of equations q_ = !p ; (11) p_ = ¡!q ; (12) which is easily solved: q(t) = q0 cos !t + p0 sin !t ; (13) p(t) = p0 cos !t ¡ q0 sin !t ; (14) 2 2 where q0 ´ q(0), p0 ´ p(0), and ! = p0 + q0. We see that our system is a non-linear harmonic oscillator. It performs harmonic oscillations, but in contrast to a linear harmonic oscillator, the frequency of oscillations is a function of energy. Now we take Nens = 1000 replicas of our system and uniformly dis- tribute them within the square 0:75 · q · 1:25, ¡0:25 · p · 0:25 of the two-dimensional phase space. Then we apply the equations of motion (13)- (14) to each points and trace the evolution. Some characteristic snapshots are presented in Fig. 1. In accordance with the equationsq of motion, each 2 2 point rotates along corresponding circle of the radius p0 + q0. Since our oscillators are non-linear, points with larger radii rotate faster, and this leads to the formation of the spiral structure. The number of the spiral windings increases with time. With a ¯xed number of points in the ensemble, at some large enough time it becomes simply impossible to resolve the spiral struc- ture. For all practical purposes, it means that instead of dealing with the actual distribution W (X; t), which is beyond our \experimental" resolution, we can work with an e®ective distribution We® (X; t) obtained by slightly smearing W (X; t). [Actually, this or that sort of \smearing" (either ex- plicit or implicit) is an unavoidable ingredient of any Statistical-Mechanical description!] In contrast to the genuine distribution W (X; t) that keeps in- creasing the number of spiral windings, the smeared distribution We® (X; t) 3 saturates to a certain equilibrium (=time-independent) function, perfectly describing our ensemble at large times (see the plot for t = 1000). With our equations of motion, we see that the generic structure of our equilibrium 2 2 We® (X) (no matter what is the initial distribution) is We® (X) = f(p + q ), the particular form of the function f coming from the initial distribution. Indeed, with respect to an individual member of the ensemble, the evolu- tion is a kind of roulette that randomizes the positionp of corresponding phase 2 2 space point Xj along the circle of the radius p + q . Below we will see how this property is generalized to any equilibrium ensemble of Hamiltonian systems. 4 Figure 1: Evolution of the ensemble of 1000 systems described by the Hamil- tonian (7). After playing with a toy model, we are ready to consider a general case. From now on we normalize the function W (X; t) to the number of the en- semble members. Correspondingly, the number of points in the phase space volume ­0 at the time t is given by the integral Z N­0 (t) = W (X; t) d­ ; (15) ­0 where d­ = dq1 : : : dqm dp1 : : : dpm is the element of the phase space volume; the integration is over the volume ­0. To characterize the rate of variation of the number of points within the volume ­0, we use the following time derivative Z @ _ N­0 = W (X; t) d­ : (16) ­0 @t 5 By the de¯nition of the function W (X; t), its variable X does not depend on time, so that the time derivative deals only with the variable t. _ There is an alternative way of calculating N­0 . We may count the num- ber of points that cross the surface of the volume ­0 per unit time: Z _ N­0 = ¡ J ¢ dS : (17) surface of ­0 Here J is the flux of the points [number of points per unit (and perpendicular to velocity) surface per unit time]; dS = n dS, where n is the unit normal vector at a surface point, and dS is the surface element. We assume that n is directed outwards and thus write the sign minus in the right-hand side of (17). In accordance with the known theorem of calculus, the surface integral (17) can be converted into the bulk integral Z Z J ¢ dS = r ¢ J d­ ; (18) surface of ­0 ­0 where r is the vector di®erential operator µ ¶ @ @ @ @ r = ;:::; ; ;:::; : (19) @q1 @qm @p1 @pm We arrive at the equality Z Z @ W (X; t) d­ = ¡ r ¢ J d­ : (20) ­0 @t ­0 Since Eq. (20) is true for an arbitrary ­0, including an in¯nitesimally small one, we actually have @ W (X; t) = ¡ r ¢ J : (21) @t This is a quite general relation, known as the continuity equation. It arises in theories describing flows of conserving quantities (say, particles of fluids and gases). The dimensionality of the problem does not matter. Now we are going to independently relate the flux J to W (X; t) and thus end up with a closed equation in terms of W (X; t). By the de¯nition of J we have J = W (X; t) X_ ; (22) 6 because the flux of particles is always equal to their concentration times velocity.
Load more

Recommended publications
  • 1 the Basic Set-Up 2 Poisson Brackets
    MATHEMATICS 7302 (Analytical Dynamics) YEAR 2016–2017, TERM 2 HANDOUT #12: THE HAMILTONIAN APPROACH TO MECHANICS These notes are intended to be read as a supplement to the handout from Gregory, Classical Mechanics, Chapter 14. 1 The basic set-up I assume that you have already studied Gregory, Sections 14.1–14.4. The following is intended only as a succinct summary. We are considering a system whose equations of motion are written in Hamiltonian form. This means that: 1. The phase space of the system is parametrized by canonical coordinates q =(q1,...,qn) and p =(p1,...,pn). 2. We are given a Hamiltonian function H(q, p, t). 3. The dynamics of the system is given by Hamilton’s equations of motion ∂H q˙i = (1a) ∂pi ∂H p˙i = − (1b) ∂qi for i =1,...,n. In these notes we will consider some deeper aspects of Hamiltonian dynamics. 2 Poisson brackets Let us start by considering an arbitrary function f(q, p, t). Then its time evolution is given by n df ∂f ∂f ∂f = q˙ + p˙ + (2a) dt ∂q i ∂p i ∂t i=1 i i X n ∂f ∂H ∂f ∂H ∂f = − + (2b) ∂q ∂p ∂p ∂q ∂t i=1 i i i i X 1 where the first equality used the definition of total time derivative together with the chain rule, and the second equality used Hamilton’s equations of motion. The formula (2b) suggests that we make a more general definition. Let f(q, p, t) and g(q, p, t) be any two functions; we then define their Poisson bracket {f,g} to be n def ∂f ∂g ∂f ∂g {f,g} = − .
    [Show full text]
  • Hamilton's Equations. Conservation Laws. Reduction. Poisson Brackets
    Hamiltonian Formalism: Hamilton's equations. Conservation laws. Reduction. Poisson Brackets. Physics 6010, Fall 2016 Hamiltonian Formalism: Hamilton's equations. Conservation laws. Reduction. Poisson Brackets. Relevant Sections in Text: 8.1 { 8.3, 9.5 The Hamiltonian Formalism We now return to formal developments: a study of the Hamiltonian formulation of mechanics. This formulation of mechanics is in many ways more powerful than the La- grangian formulation. Among the advantages of Hamiltonian mechanics we note that: it leads to powerful geometric techniques for studying the properties of dynamical systems, it allows for a beautiful expression of the relation between symmetries and conservation laws, and it leads to many structures that can be viewed as the macroscopic (\classical") imprint of quantum mechanics. Although the Hamiltonian form of mechanics is logically independent of the Lagrangian formulation, it is convenient and instructive to introduce the Hamiltonian formalism via transition from the Lagrangian formalism, since we have already developed the latter. (Later I will indicate how to give an ab initio development of the Hamiltonian formal- ism.) The most basic change we encounter when passing from Lagrangian to Hamiltonian methods is that the \arena" we use to describe the equations of motion is no longer the configuration space, or even the velocity phase space, but rather the momentum phase space. Recall that the Lagrangian formalism is defined once one specifies a configuration space Q (coordinates qi) and then the velocity phase space Ω (coordinates (qi; q_i)). The mechanical system is defined by a choice of Lagrangian, L, which is a function on Ω (and possible the time): L = L(qi; q_i; t): Curves in the configuration space Q { or in the velocity phase space Ω { satisfying the Euler-Lagrange (EL) equations, @L d @L − = 0; @qi dt @q_i define the dynamical behavior of the system.
    [Show full text]
  • Poisson Structures and Integrability
    Poisson Structures and Integrability Peter J. Olver University of Minnesota http://www.math.umn.edu/ olver ∼ Hamiltonian Systems M — phase space; dim M = 2n Local coordinates: z = (p, q) = (p1, . , pn, q1, . , qn) Canonical Hamiltonian system: dz O I = J H J = − dt ∇ ! I O " Equivalently: dpi ∂H dqi ∂H = = dt − ∂qi dt ∂pi Lagrange Bracket (1808): n ∂pi ∂qi ∂qi ∂pi [ u , v ] = ∂u ∂v − ∂u ∂v i#= 1 (Canonical) Poisson Bracket (1809): n ∂u ∂v ∂u ∂v u , v = { } ∂pi ∂qi − ∂qi ∂pi i#= 1 Given functions u , . , u , the (2n) (2n) matrices with 1 2n × respective entries [ u , u ] u , u i, j = 1, . , 2n i j { i j } are mutually inverse. Canonical Poisson Bracket n ∂F ∂H ∂F ∂H F, H = F T J H = { } ∇ ∇ ∂pi ∂qi − ∂qi ∂pi i#= 1 = Poisson (1809) ⇒ Hamiltonian flow: dz = z, H = J H dt { } ∇ = Hamilton (1834) ⇒ First integral: dF F, H = 0 = 0 F (z(t)) = const. { } ⇐⇒ dt ⇐⇒ Poisson Brackets , : C∞(M, R) C∞(M, R) C∞(M, R) { · · } × −→ Bilinear: a F + b G, H = a F, H + b G, H { } { } { } F, a G + b H = a F, G + b F, H { } { } { } Skew Symmetric: F, H = H, F { } − { } Jacobi Identity: F, G, H + H, F, G + G, H, F = 0 { { } } { { } } { { } } Derivation: F, G H = F, G H + G F, H { } { } { } F, G, H C∞(M, R), a, b R. ∈ ∈ In coordinates z = (z1, . , zm), F, H = F T J(z) H { } ∇ ∇ where J(z)T = J(z) is a skew symmetric matrix. − The Jacobi identity imposes a system of quadratically nonlinear partial differential equations on its entries: ∂J jk ∂J ki ∂J ij J il + J jl + J kl = 0 ! ∂zl ∂zl ∂zl " #l Given a Poisson structure, the Hamiltonian flow corresponding to H C∞(M, R) is the system of ordinary differential equati∈ons dz = z, H = J(z) H dt { } ∇ Lie’s Theory of Function Groups Used for integration of partial differential equations: F , F = G (F , .
    [Show full text]
  • Advanced Quantum Theory AMATH473/673, PHYS454
    Advanced Quantum Theory AMATH473/673, PHYS454 Achim Kempf Department of Applied Mathematics University of Waterloo Canada c Achim Kempf, September 2017 (Please do not copy: textbook in progress) 2 Contents 1 A brief history of quantum theory 7 1.1 The classical period . 7 1.2 Planck and the \Ultraviolet Catastrophe" . 7 1.3 Discovery of h ................................ 8 1.4 Mounting evidence for the fundamental importance of h . 9 1.5 The discovery of quantum theory . 9 1.6 Relativistic quantum mechanics . 11 1.7 Quantum field theory . 12 1.8 Beyond quantum field theory? . 14 1.9 Experiment and theory . 16 2 Classical mechanics in Hamiltonian form 19 2.1 Newton's laws for classical mechanics cannot be upgraded . 19 2.2 Levels of abstraction . 20 2.3 Classical mechanics in Hamiltonian formulation . 21 2.3.1 The energy function H contains all information . 21 2.3.2 The Poisson bracket . 23 2.3.3 The Hamilton equations . 25 2.3.4 Symmetries and Conservation laws . 27 2.3.5 A representation of the Poisson bracket . 29 2.4 Summary: The laws of classical mechanics . 30 2.5 Classical field theory . 31 3 Quantum mechanics in Hamiltonian form 33 3.1 Reconsidering the nature of observables . 34 3.2 The canonical commutation relations . 35 3.3 From the Hamiltonian to the equations of motion . 38 3.4 From the Hamiltonian to predictions of numbers . 42 3.4.1 Linear maps . 42 3.4.2 Choices of representation . 43 3.4.3 A matrix representation . 44 3 4 CONTENTS 3.4.4 Example: Solving the equations of motion for a free particle with matrix-valued functions .
    [Show full text]
  • Hamilton-Poisson Formulation for the Rotational Motion of a Rigid Body In
    Hamilton-Poisson formulation for the rotational motion of a rigid body in the presence of an axisymmetric force field and a gyroscopic torque Petre Birtea,∗ Ioan Ca¸su, Dan Com˘anescu Department of Mathematics, West University of Timi¸soara Bd. V. Pˆarvan, No 4, 300223 Timi¸soara, Romˆania E-mail addresses: [email protected]; [email protected]; [email protected] Keywords: Poisson bracket, Casimir function, rigid body, gyroscopic torque. Abstract We give sufficient conditions for the rigid body in the presence of an axisymmetric force field and a gyroscopic torque to admit a Hamilton-Poisson formulation. Even if by adding a gyroscopic torque we initially lose one of the conserved Casimirs, we recover another conservation law as a Casimir function for a modified Poisson structure. We apply this frame to several well known results in the literature. 1 Introduction The generalized Euler-Poisson system, which is a dynamical system for the rotational motion of a rigid body in the presence of an axisymmetric force field, admits a Hamilton-Poisson formulation, where the bracket is the ”−” Kirillov-Kostant-Souriau bracket on the dual Lie algebra e(3)∗. The Hamiltonian function is of the type kinetic energy plus potential energy. The Hamiltonian function and the two Casimir functions for the K-K-S Poisson structure are conservation laws for the dynamic. Adding a certain type of gyroscopic torque the Hamiltonian and one of the Casimirs remain conserved along the solutions of the new dynamical system. In this paper we give a technique for finding a third conservation law in the presence of gyroscopic torque.
    [Show full text]
  • An Introduction to Free Quantum Field Theory Through Klein-Gordon Theory
    AN INTRODUCTION TO FREE QUANTUM FIELD THEORY THROUGH KLEIN-GORDON THEORY JONATHAN EMBERTON Abstract. We provide an introduction to quantum field theory by observing the methods required to quantize the classical form of Klein-Gordon theory. Contents 1. Introduction and Overview 1 2. Klein-Gordon Theory as a Hamiltonian System 2 3. Hamilton's Equations 4 4. Quantization 5 5. Creation and Annihilation Operators and Wick Ordering 9 6. Fock Space 11 7. Maxwell's Equations and Generalized Free QFT 13 References 15 Acknowledgments 16 1. Introduction and Overview The phase space for classical mechanics is R3 × R3, where the first copy of R3 encodes position of an object and the second copy encodes momentum. Observables on this space are simply functions on R3 × R3. The classical dy- namics of the system are then given by Hamilton's equations x_ = fh; xg k_ = fh; kg; where the brackets indicate the canonical Poisson bracket and jkj2 h(x; k) = + V (x) 2m is the classical Hamiltonian. To quantize this classical field theory, we set our quantized phase space to be L2(R3). Then the observables are self-adjoint operators A on this space, and in a state 2 L2(R3), the mean value of A is given by hAi := hA ; i: The dynamics of a quantum system are then described by the quantized Hamil- ton's equations i i x_ = [H; x]_p = [H; p] h h 1 2 JONATHAN EMBERTON where the brackets indicate the commutator and the Hamiltonian operator H is given by jpj2 2 H = h(x; p) = − + V (x) = − ~ ∆ + V (x): 2m 2m A question that naturally arises from this construction is then, how do we quan- tize other classical field theories, and is this quantization as straight-forward as the canonical construction? For instance, how do Maxwell's Equations behave on the quantum scale? This happens to be somewhat of a difficult problem, with many steps involved.
    [Show full text]
  • Physics 550 Problem Set 1: Classical Mechanics
    Problem Set 1: Classical Mechanics Physics 550 Physics 550 Problem Set 1: Classical Mechanics Name: Instructions / Notes / Suggestions: • Each problem is worth five points. • In order to receive credit, you must show your work. • Circle your final answer. • Unless otherwise specified, answers should be given in terms of the variables in the problem and/or physical constants and/or cartesian unit vectors (e^x; e^x; e^z) or radial unit vector ^r. Problem Set 1: Classical Mechanics Physics 550 Problem #1: Consider the pendulum in Fig. 1. Find the equation(s) of motion using: (a) Newtonian mechanics (b) Lagrangian mechanics (c) Hamiltonian mechanics Answer: Not included. Problem Set 1: Classical Mechanics Physics 550 Problem #2: Consider the double pendulum in Fig. 2. Find the equations of motion. Note that you may use any formulation of classical mechanics that you want. Answer: Multiple answers are possible. Problem Set 1: Classical Mechanics Physics 550 Problem #3: In Problem #1, you probably found the Lagrangian: 1 L = ml2θ_2 + mgl cos θ 2 Using the Legendre transformation (explicitly), find the Hamiltonian. Answer: Recall the Legendre transform: X H = q_ipi − L i where: @L pi = @q_i _ In the present problem, q = θ andq _ = θ. Using the expression for pi: 2 _ pi = ml θ We can therefore write: _ 2 _ 1 2 _2 H =θml θ − 2 ml θ − mgl cos θ 1 2 _2 = 2 ml θ − mgl cos θ Problem Set 1: Classical Mechanics Physics 550 Problem #4: Consider a dynamical system with canonical coordinate q and momentum p vectors.
    [Show full text]
  • 2.3. Classical Mechanics in Hamiltonian Formulation
    2.3. CLASSICAL MECHANICS IN HAMILTONIAN FORMULATION 23 expressions that involve all those positions and momentum variables? To this end, we need to define what the Poisson brackets in between positions and momenta of different particles should be. They are defined to be simply zero. Therefore, to summarize, we define the basic Poisson brackets of n masses as (r) (s) {xi , p j } = δi,j δr,s (2.16) (r) (s) {xi , x j } =0 (2.17) (r) (s) {pi , p j } =0 (2.18) where r, s ∈ { 1, 2, ..., n } and i, j ∈ { 1, 2, 3}. The evaluation rules of Eqs.2.9-2.13 are defined to stay just the same. Exercise 2.6 Mathematically, the set of polynomials in positions and momenta is an example of what is called a Poisson algebra. A general Poisson algebra is a vector space with two extra multiplications: One multiplication which makes the vector space into an associative algebra, and one (non-associative) multiplication {, }, called the Lie bracket, which makes the vector space into what is called a Lie algebra. If the two multiplications are in a certain sense compatible then the set is said to be a Poisson algebra. Look up and state the axioms of a) a Lie algebra, b) an associative algebra and c) a Poisson algebra. 2.3.3 The Hamilton equations Let us recall why we introduced the Poisson bracket: A technique that uses the Poisson bracket is supposed to allow us to derive all the differential equations of motion of a system from the just one piece of information, namely from the expression of the total energy of the system, i.e., from its Hamiltonian.
    [Show full text]
  • PHY411 Lecture Notes Part 2
    PHY411 Lecture notes Part 2 Alice Quillen October 2, 2018 Contents 1 Canonical Transformations 2 1.1 Poisson Brackets . 2 1.2 Canonical transformations . 3 1.3 Canonical Transformations are Symplectic . 5 1.4 Generating Functions for Canonical Transformations . 6 1.4.1 Example canonical transformation - action angle coordinates for the harmonic oscillator . 7 2 Some Geometry 9 2.1 The Tangent Bundle . 9 2.1.1 Differential forms and Wedge product . 9 2.2 The Symplectic form . 12 2.3 Generating Functions Geometrically . 13 2.4 Vectors generate Flows and trajectories . 15 2.5 The Hamiltonian flow and the symplectic two-form . 15 2.6 Extended Phase space . 16 2.7 Hamiltonian flows preserve the symplectic two-form . 17 2.8 The symplectic two-form and the Poisson bracket . 19 2.8.1 On connection to the Lagrangian . 21 2.8.2 Discretized Systems and Symplectic integrators . 21 2.8.3 Surfaces of section . 21 2.9 The Hamiltonian following a canonical transformation . 22 2.10 The Lie derivative . 23 3 Examples of Canonical transformations 25 3.1 Orbits in the plane of a galaxy or around a massive body . 25 3.2 Epicyclic motion . 27 3.3 The Jacobi integral . 31 3.4 The Shearing Sheet . 32 1 4 Symmetries and Conserved Quantities 37 4.1 Functions that commute with the Hamiltonian . 37 4.2 Noether's theorem . 38 4.3 Integrability . 39 1 Canonical Transformations It is straightforward to transfer coordinate systems using the Lagrangian formulation as minimization of the action can be done in any coordinate system.
    [Show full text]
  • Hamilton's Principle and Symmetries
    Hamiltionians Poisson Brackets Angular Momentum Algebra Keywords and References Hamilton’s principle and Symmetries Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2011 August 18, 2011 Sourendu Gupta Classical Mechanics 2011: Lecture 5 Hamiltionians Poisson Brackets Angular Momentum Algebra Keywords and References The Hamiltonian The change in the Lagrangian due to a virtual change of coordinates is dL = p˙ k dqk + pk dq˙ k . k X Using this, one can define a function, called the Hamiltonian, H(pk , qk ), by eliminatingq ˙ k to write dH = d pk q˙ k − L = q˙ k dpk − p˙ k dqk . " k # k X X Since pk and qk are independent variables, one has Hamilton’s equations ∂H ∂H q˙ k = and p˙ k = − . ∂pk ∂qk Sourendu Gupta Classical Mechanics 2011: Lecture 5 Hamiltionians Poisson Brackets Angular Momentum Algebra Keywords and References An example For a single particle moving in a potential, one can write 1 ∂L L = m|x˙|2 − V (x), and p = = mx˙. 2 ∂x˙ Eliminating x˙, the Hamiltonian is 1 H = p · x˙ − L = |p|2 + V (x). 2m Hamilton’s equations are 1 x˙ = p, and p˙ = −∇V (x). m These are exactly the usual equations of motion. Sourendu Gupta Classical Mechanics 2011: Lecture 5 Hamiltionians Poisson Brackets Angular Momentum Algebra Keywords and References Poisson brackets Take any two functions on phase space, f (qk , pk ) and g(qk , pk ). Then we define the Poisson bracket of these two functions as ∂f ∂g ∂g ∂f [f , g]= − . ∂qk ∂pk ∂qk ∂pk k X Clearly, interchanging f and g in the expression on the right changes the sign.
    [Show full text]
  • Hamiltonian Dynamics CDS 140B
    Hamiltonian Dynamics CDS 140b Joris Vankerschaver [email protected] CDS 01/22/08, 01/24/08 Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 1 / 36 Outline for this week 1. Introductory concepts; 2. Poisson brackets; 3. Integrability; 4. Perturbations of integrable systems. 4.1 The KAM theorem; 4.2 Melnikov's method. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 2 / 36 References Books I J. Marsden and T. Ratiu: Introduction to Mechanics and Symmetry. I V. Arnold: Mathematical Methods of Classical Mechanics. I M. Tabor: Chaos and Integrability in Nonlinear Dynamics. I P. Newton: The N-vortex problem. I F. Verhulst. Papers I J. D. Meiss: Visual Exploration of Dynamics: the Standard Map. See http://arxiv.org/abs/0801.0883 (has links to software used to make most of the plots in this lecture) I J. D. Meiss: Symplectic maps, variational principles, and transport. Rev. Mod. Phys. 64 (1992), no. 3, pp. 795{848. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 3 / 36 References Software I GniCodes: symplectic integration of 2nd order ODEs. Similar in use as Matlab's ode suite. See http://www.unige.ch/∼hairer/preprints/gnicodes.html I StdMap: Mac program to explore the dynamics of area preserving maps. See http://amath.colorado.edu/faculty/jdm/stdmap.html Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 4 / 36 Introduction Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 5 / 36 Transition to the Hamiltonian framework I Consider a mechanical system with n degrees of freedom and described by generalised coordinates (q1;:::; qn).
    [Show full text]
  • Supersymmetric Classical Mechanics*
    Pram,~.na- J. Phys., Vol. 27, Nos I & 2, July & August 1986, pp. 117-127. @') Printed in India. Supersymmetric classical mechanics* S N BISWAS and S K SONI Department of Physics and Astrophysics,University of Delhi, Delhi 110007, India Abstract. The purpose of the paper is to construct a supersymmetricLagrangian within the framework of classical mechanics which would be regarded as a candidate for passage to supcrsymmetric quantum mechanics. Keywords. Supersymmetry;classical mechanics;generalized Poisson bracket; equations of motion; canonical transformation;classical Lagrangian. PACS No. 1. Introduction Supersymmetric quantum mechanics was originally proposed by Witten 1981 to study the breakdown of supersymmetry by non-perturbative effects. Cooper and Freedman (1982; see also Salmonson and Van Holten 1981) explored non-perturbative dynamical breaking of supersymmetry in various supersymmetric quantum-mechanical models and in addition they also considered the formulation of a supersymmetric Lagrangian at the classical level. To derive this they started from a supersymmetric field theory based on the superfield formalism of Salam and Strathdee (1975) by letting the dimension to become 1. The corresponding Hamiltonian for the supersymmetric quantum theory takes the form =½p2 + ½W2(q)_½[~bl-, ~b] --,dW (1) dq where [q, p] = ih and {~,*, ~,} = 1. Here q, p are the conventional bosonic operators while ~b* and ~, are the fermionic operators. The combinations (p-iW)d/ and (p + i W) @* are called supersymmetric charges designated by Q and Q* respectively. The Q and Q* satisfy the properties, namely {Q*,Q} = 2H and furthermore Q and Q* commute with H. The purpose of the paper is to construct a supersymmetric classical Lagrangian which in the quantum limit leads to the same Hamiltonian as given in (1).
    [Show full text]