DOCSLIB.ORG

Hamiltonian Formulation of Classical Fields with Fractional Derivatives

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hamiltonian Formulation of Classical Fields with Fractional Derivatives Hamiltonian Formulation of Classical Fields with Fractional Derivatives A. A. Diab 1, R. S. Hijjawi 2, J. H. Asad 3, and J. M. Khalifeh 4 1General Studies Department- Yanbu Industrial College- P. O Box 30436- Yanbu Industrial City- KSA 2Dep. of Physics- Mutah University- Karak- Jordan 3Dep. of Physics- Tabuk University- P. O Box 741- Tabuk 71491- Saudi Arabia 4Dep. of Physics-Jordan University- Amman 11942 - Jordan Abstract An investigation of classical fields with fractional derivatives is presented using the fractional Hamiltonian formulation. The fractional Hamilton's equations are obtained for two classical field examples. The formulation presented and the resulting equations are very similar to those appearing in classical field theory. Keywords: Fractional derivatives, Lagrangian and Hamiltonian formulation. 1 1. Introduction Fractional calculus is an extension of classical calculus. In this branch of mathematics, definitions are established for integrals and derivatives of arbitrary non-integer (even complex) order, such /1 2 as d f (t) . It has started in 1695 when Leibniz presented his analysis of dt /1 2 the derivative of order 1/2 . Then it is developed mainly as a theoretical aspect of mathematics, being considered by some of the great names in mathematics such as Euler, Lagrange, and Fourier, among many others. This branch of mathematics has experienced a revival of interest and has been used in very diverse topics such as fractal theory 1viscoelasticy 2, electrodynamics 3,4 , optics 5,6 , and thermodynamics 7. The literature of fractional calculus started with Leibniz and today is growing rapidly 8-14 . Recently fractional derivatives have played a significant role in physics, mathematics, engineering, and pure and applied mathematics in recent years 14-20 . Several attempts have been made to include non conservative forces in the Lagrangian and the Hamiltonian mechanics. Riewe 20,21 presented a new approach to mechanics that allows one to obtain the equations for non conservative systems using fractional derivatives. Eqab et al 22-24 developed a general formula for determining the potentials of arbitrary forces, conservative and non conservative, using the Laplace transform of fractional integrals and fractional derivatives. Recently, they 24 also constructed the Hamiltonian formulation of discrete and continuous fields in terms of fractional derivatives. The fractional Hamiltonian is not uniquely defined; it seems that there are several choices of fractional Hamiltonian giving the same classical limit, i.e., the same classical Hamiltonian. This present paper is a generalization of the above work on Hamilton's equations for classical fields with Riemann- Liouville fractional derivatives. This paper is organized as follows: In Sec.2, the definitions of fractional derivatives are discussed briefly. In Sec.3, the fractional form of Euler- Lagrangian equation in terms of functional derivative of the full Lagrangian is presented. In Sec.4, the fractional form of Euler-Lagrange equation in terms of momentum density is investigated. Section 5 is devoted to the equations of motion in terms of Hamiltonian density in fractional form. In section 6, we introduce two examples of classical fields leading to Schrödinger equation and Dirac equation in fractional 2 derivative forms. The work closes with some concluding remarks (section 7). 2. Definitions of Fractional Derivatives In this section two different definitions of the fractional derivatives (left and right Riemann- Liouville fractional derivatives) are discussed. These definitions are used in the Hamiltonian formulation and the solution of examples leading to the equations of motion of the fractional order. The left Riemann-Liouville fractional derivative (LRLFD) reads as 25: 1 d x f (τ ) Dα f (x) = ( ) n dτ . (1) a x Γ(n −α) dx ∫ (x −τ )α −n+1 a The right Riemann-Liouville fractional derivative (RRLFD) reads as 25: b α 1 d n f (τ ) x Db f (x) = (− ) ∫ α −n+1 dτ . (2) Γ(n −α) dx x (τ − x) Here α is the order of the derivative such that n −1≤ α ≤ n and is not equal to zero. If α is an integer, these derivatives are defined in the usual sense, i.e., d Dα f (x) = ( )α f (x) ; a x dx d Dα f (x) = (− )α f (x) ; α = 2,1 ,... (3) x b dx 3. Fractional form of Euler-Lagrangian Equation in terms of Functional derivative of the full Lagrangian L We start our formalism by taking the Lagrangian density to be a function of field amplitude ψ and its fractional derivatives with respect to space and time as: L L α β α β = (,ψaxD ψ , xb D ψ , at D ψ , tb Dt ψ ,) . (4) Euler-Lagrange equation for such Lagrangian density in fractional form can be given as 26 : ∂∂LLα β ∂ L α ∂ L β ∂ L +xbDα + ax D β + tb D α + at D β = 0 . ∂∂ψaxD ψ ∂ xb D ψ ∂ at D ψ ∂ tb D ψ (5) 3 Now we can write the full Lagrangian L as: 3 L= ∫ L d r . (6) Using the variational principle, we can write: δ∫Ldt= δ ∫∫L d3 rdt = ∫∫ ( δ L ) dtd 3 r = 0 . (7) Using Eq. (4), the variation of is: L L L l L ∂∂α ∂ β ∂ α δ=+ δψα δψ()axD + β δψ () xb D + α δψ () at D + ∂∂ψψaxD ∂ xb D ψ ∂ at D ψ ∂L β β δ(tD b ψ )= 0 . (8) ∂ tD b ψ Substituting Eq. (8) into Eq. (7), and using the following commutation relation, ∂Lα ∂ L α αδ()axD ψ= α ax D () δψ . (9) ∂axDψ ∂ ax D ψ we get, ∂∂L Lα ∂l β ∂ L α ∫∫ [()δψ+αaxD () δψ + β xb D () δψ + α δ () at D ψ + ∂∂ψψaxD ∂ xb D ψ ∂ at D ψ ∂L β 3 β δ(tD b ψ )] d r dt = 0 . (10) ∂ tD b ψ Integrating by parts the second and the third terms in Eq. (10) with respect to space, we get: ∂∂Lα L β ∂l ∂ L α 0[=++∫∫dtx D bα a D x β ][() dτδψ + ∫∫ dt α δa D t ψ + ∂∂ψψψaxD ∂ xb D ∂ at D ψ ∂L β β δ(tD b ψ )] d τ . (11) ∂ tD b ψ Now, we can take the integration over space in the previous equation and convert it into summation, thus: ∂∂LLα β ∂ L ∂ L α ∑++xbD ax D δψ ii δτ + ∑ δ( atii D ψ ) δτ + ∂∂ψψψDα ∂ D β ∂ D α ψ i ax xb ii at i ∂L β +∑ δ(tbiD ψ ) δτ i = 0 . (12) ∂ Dβψ i t b i 4 We can write Eq. (12) in terms of Lagrangian density as: L ∑[δ ]i δτ i = 0 . (13) i Where the left hand side in Eqs.(12 and 13) represents the variation of L (i.e. δL ) which is now produced by independent variations inδψ i , α β α δ ( a Dt ψ i ) and δ (t Db ψ i ) . Suppose now that all δψ i ,δ (a Dt ψ i ) and β δ (t Db ψ i ) are zeros except for a particular δψ j . It is natural to define the functional derivative of the full Lagrangian ( ∂/L) with respect to ψ , ( Dαψ ) and ( D βψ ) for a point in the j-th cell to the ratio of δL to δψ 27. a t t b j ∂/L δL = lim . (14) ∂/ψ δτ j →0 δψ j δτ j Using Eq. (12), and note that the left hand side represents we get: ∂/L ∂Lα ∂ L β ∂ L = +xbDα + ax D β . (15a) ∂/ψψ ∂ ∂axD ψ ∂ xb D ψ ∂/Lδ L ∂ L = = . (15b) αlim α α ∂/atDψδτ j →0 δτ jatj δ( D ψ ) ∂ at D ψ ∂/Lδ L ∂ L = = . (15c) βlim β β ∂/tbDψδτ j →0 δτ jtbj δ( D ψ ) ∂ tb D ψ Now, using Eq. (15) we can rewrite Eq. (5) Euler- Lagrange equation in terms of the full Lagrangian L using functional derivative in fractional form: ∂/L α ∂/L β ∂/L +t Db α + a Dt β = 0. (16) ∂/ψ ∂/ a Dt ψ ∂/ t Db ψ It is worth mentioning that for α, β → 1, Eq. (16) reduces to the usual Euler- Lagrange equation for the classical fields 27 . With the help of Eqs. (12 and 16), we can write the variation of full Lagrangian in terms of α β functional derivatives and variations of ψ , a Dt ψ and t Db ψ as: ∂/L ∂/L α ∂/L β 3 δL = ∫ (δψ ) + α δ (a Dt ψ ) + β δ (t Db ψ )d r . (17) ∂/ψ ∂/ a Dt ψ ∂/ t Db ψ 5 4. Fractional form of Euler-Lagrange Equation in terms of Momentum Density The right side fractional form of momentum can be written as 27 : a δL Pj = α . (18a) δ aDt ψ j Using Esq. (12 and 15b) we get: a ∂L ∂/ L Pj=αδτ j = α δτ j . (18b) ∂atjDψ ∂/ atj D ψ From Eq. (18b), we can define the right side form of momentum density πα as: ∂/L ∂ L (πα ) j =α = α . (19) ∂/atjDψ ∂ atj D ψ Now, taking the left fractional derivative for Eq. (19), one gets: ∂/L Dα (π ) = Dα t b α j t b α . (20) ∂/ a Dt ψ j Repeating same steps above for left side fractional form of momentum density π β , we get: ∂/L ∂ L (π β ) j =β = β . (21) ∂/tbjDψ ∂ tbj D ψ Now, taking the right fractional derivative for Eq. (21), one gets: ∂/L Dβ (π ) = Dβ a t β j a t β . (22) ∂/ t Db ψ j Now, substituting Eqs. (20 and 22) into Eq. (16), we get: ∂/L = −[ D β π + Dαπ ]. (23) ∂/ψ a t β t b α 6 The above equation represents the fractional form of Euler- Lagrange equation in terms of momentum density and the functional derivative of full Lagrangian.
Load more

Recommended publications
  • Branched Hamiltonians and Supersymmetry
    Branched Hamiltonians and Supersymmetry Thomas Curtright, University of Miami Wigner 111 seminar, 12 November 2013 Some examples of branched Hamiltonians are explored, as recently advo- cated by Shapere and Wilczek. These are actually cases of switchback poten- tials, albeit in momentum space, as previously analyzed for quasi-Hamiltonian dynamical systems in a classical context. A basic model, with a pair of Hamiltonian branches related by supersymmetry, is considered as an inter- esting illustration, and as stimulation. “It is quite possible ... we may discover that in nature the relation of past and future is so intimate ... that no simple representation of a present may exist.” – R P Feynman Based on work with Cosmas Zachos, Argonne National Laboratory Introduction to the problem In quantum mechanics H = p2 + V (x) (1) is neither more nor less difficult than H = x2 + V (p) (2) by reason of x, p duality, i.e. the Fourier transform: ψ (x) φ (p) ⎫ ⎧ x ⎪ ⎪ +i∂/∂p ⎪ ⇐⇒ ⎪ ⎬⎪ ⎨⎪ i∂/∂x p − ⎪ ⎪ ⎪ ⎪ ⎭⎪ ⎩⎪ This equivalence of (1) and (2) is manifest in the QMPS formalism, as initiated by Wigner (1932), 1 2ipy/ f (x, p)= dy x + y ρ x y e− π | | − 1 = dk p + k ρ p k e2ixk/ π | | − where x and p are on an equal footing, and where even more general H (x, p) can be considered. See CZ to follow, and other talks at this conference. Or even better, in addition to the excellent books cited at the conclusion of Professor Schleich’s talk yesterday morning, please see our new book on the subject ... Even in classical Hamiltonian mechanics, (1) and (2) are equivalent under a classical canonical transformation on phase space: (x, p) (p, x) ⇐⇒ − But upon transitioning to Lagrangian mechanics, the equivalence between the two theories becomes obscure.
    [Show full text]
  • BRST APPROACH to HAMILTONIAN SYSTEMS Our Starting Point Is the Partition Function
    BRST APPROACH TO HAMILTONIAN SYSTEMS A.K.Aringazin1, V.V.Arkhipov2, and A.S.Kudusov3 Department of Theoretical Physics Karaganda State University Karaganda 470074 Kazakstan Preprint KSU-DTP-10/96 Abstract BRST formulation of cohomological Hamiltonian mechanics is presented. In the path integral approach, we use the BRST gauge fixing procedure for the partition func- tion with trivial underlying Lagrangian to fix symplectic diffeomorphism invariance. Resulting Lagrangian is BRST and anti-BRST exact and the Liouvillian of classical mechanics is reproduced in the ghost-free sector. The theory can be thought of as a topological phase of Hamiltonian mechanics and is considered as one-dimensional cohomological field theory with the target space a symplectic manifold. Twisted (anti- )BRST symmetry is related to global N = 2 supersymmetry, which is identified with an exterior algebra. Landau-Ginzburg formulation of the associated d = 1, N = 2 model is presented and Slavnov identity is analyzed. We study deformations and per- turbations of the theory. Physical states of the theory and correlation functions of the BRST invariant observables are studied. This approach provides a powerful tool to investigate the properties of Hamiltonian systems. PACS number(s): 02.40.+m, 03.40.-t,03.65.Db, 11.10.Ef, 11.30.Pb. arXiv:hep-th/9811026v1 2 Nov 1998 [email protected] [email protected] [email protected] 1 1 INTRODUCTION Recently, path integral approach to classical mechanics has been developed by Gozzi, Reuter and Thacker in a series of papers[1]-[10]. They used a delta function constraint on phase space variables to satisfy Hamilton’s equation and a sort of Faddeev-Popov representation.
    [Show full text]
  • Antibrackets and Supersymmetric Mechanics
    Preprint JINR E2-93-225 Antibrackets and Supersymmetric Mechanics Armen Nersessian 1 Laboratory of Theoretical Physics, JINR Dubna, Head Post Office, P.O.Box 79, 101 000 Moscow, Russia Abstract Using odd symplectic structure constructed over tangent bundle of the symplec- tic manifold, we construct the simple supergeneralization of an arbitrary Hamilto- arXiv:hep-th/9306111v2 29 Jul 1993 nian mechanics on it. In the case, if the initial mechanics defines Killing vector of some Riemannian metric, corresponding supersymmetric mechanics can be refor- mulated in the terms of even symplectic structure on the supermanifold. 1E-MAIL:[email protected] 1 Introduction It is well-known that on supermanifolds M the Poisson brackets of two types can be defined – even and odd ones, in correspondence with their Grassmannian grading 1. That is defined by the expression ∂ f ∂ g {f,g} = r ΩAB(z) l , (1.1) κ ∂zA κ ∂zB which satisfies the conditions p({f,g}κ)= p(f)+ p(g) + 1 (grading condition), (p(f)+κ)(p(g)+κ) {f,g}κ = −(−1) {g, f}κ (”antisymmetrisity”), (1.2) (p(f)+κ)(p(h)+κ) (−1) {f, {g, h}1}1 + cycl.perm.(f, g, h) = 0 (Jacobi id.), (1.3) l A ∂r ∂ where z are the local coordinates on M, ∂zA and ∂zA denote correspondingly the right and the left derivatives, κ = 0, 1 denote correspondingly the even and the odd Poisson brackets. Obviously, the even Poisson brackets can be nondegenerate only if dimM = (2N.M), and the odd one if dimM =(N.N). With nondegenerate Poisson bracket one can associate the symplectic structure A B Ωκ = dz Ω(κ)ABdz , (1.4) BC C where Ω(κ)AB Ωκ = δA .
    [Show full text]
  • Classical Hamiltonian Field Theory
    Classical Hamiltonian Field Theory Alex Nelson∗ Email: [email protected] August 26, 2009 Abstract This is a set of notes introducing Hamiltonian field theory, with focus on the scalar field. Contents 1 Lagrangian Field Theory1 2 Hamiltonian Field Theory2 1 Lagrangian Field Theory As with Hamiltonian mechanics, wherein one begins by taking the Legendre transform of the Lagrangian, in Hamiltonian field theory we \transform" the Lagrangian field treatment. So lets review the calculations in Lagrangian field theory. Consider the classical fields φa(t; x¯). We use the index a to indicate which field we are talking about. We should think of x¯ as another index, except it is continuous. We will use the confusing short hand notation φ for the column vector φ1; : : : ; φn. Consider the Lagrangian Z 3 L(φ) = L(φ, @µφ)d x¯ (1.1) all space where L is the Lagrangian density. Hamilton's principle of stationary action is still used to determine the equations of motion from the action Z S[φ] = L(φ, @µφ)dt (1.2) where we find the Euler-Lagrange equations of motion for the field d @L @L µ a = a (1.3) dx @(@µφ ) @φ where we note these are evil second order partial differential equations. We also note that we are using Einstein summation convention, so there is an implicit sum over µ but not over a. So that means there are n independent second order partial differential equations we need to solve. But how do we really know these are the correct equations? How do we really know these are the Euler-Lagrange equations for classical fields? We can obtain it directly from the action S by functional differentiation with respect to the field.
    [Show full text]
  • 15. Hamiltonian Mechanics Gerhard Müller University of Rhode Island, [email protected] Creative Commons License
    University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 15. Hamiltonian Mechanics Gerhard Müller University of Rhode Island, [email protected] Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. Follow this and additional works at: http://digitalcommons.uri.edu/classical_dynamics Abstract Part fifteen of course materials for Classical Dynamics (Physics 520), taught by Gerhard Müller at the University of Rhode Island. Documents will be updated periodically as more entries become presentable. Recommended Citation Müller, Gerhard, "15. Hamiltonian Mechanics" (2015). Classical Dynamics. Paper 7. http://digitalcommons.uri.edu/classical_dynamics/7 This Course Material is brought to you for free and open access by the Physics Course Materials at DigitalCommons@URI. It has been accepted for inclusion in Classical Dynamics by an authorized administrator of DigitalCommons@URI. For more information, please contact [email protected]. Contents of this Document [mtc15] 15. Hamiltonian Mechanics • Legendre transform [tln77] • Hamiltonian and canonical equations [mln82] • Lagrangian from Hamiltonian via Legendre transform [mex188] • Can you find the Hamiltonian of this system? [mex189] • Variational principle in phase space [mln83] • Properties of the Hamiltonian [mln87] • When does the Hamiltonian represent the total energy? [mex81] • Hamiltonian: conserved quantity or total energy? [mex77] • Bead sliding on rotating rod in vertical plane [mex78] • Use of cyclic coordinates in Lagrangian and Hamiltonian mechanics [mln84] • Velocity-dependent potential energy [mln85] • Charged particle in electromagnetic field [mln86] • Velocity-dependent central force [mex76] • Charged particle in a uniform magnetic field [mex190] • Particle with position-dependent mass moving in 1D potential [mex88] • Pendulum with string of slowly increasing length [mex89] • Librations between inclines [mex259] Legendre transform [tln77] Given is a function f(x) with monotonic derivative f 0(x).
    [Show full text]
  • (Limited Distribution) International Atomic Energy Agency And
    IC/91/51 ;E INTERNAL REPORT REI (Limited Distribution) International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS N=2, D=4 SUPERSYMMETRIC er-MODELS AND HAMILTONIAN MECHANICS * A. Galperin The Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, United Kingdom and V. Ogievetsky ** International Centre for Theoretical Physics, Trieste, Italy. ABSTRACT A deep similarity is established between the Hamiltonian mechanics of point particle and supersymmetric N - 2, D = 4 cx-models formulated within harmonic superspace. An essential pan of the latter, the sphere S2, comes out as a couterpart of the time variable. MIRAMARE - TRIESTE May 1991 * To be submitted for publication. Also appeared as IMPERIAIVTP/9O-91/15. ** Permanent address: Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Head Post Office P.O. Box 79,101000 Moscow, USSR. 1 Introduction It is well known that customary a-models, i.e. two-derivative self-couplings of scalar fields A m SN=0 = \ j<?x 9AB(4>) dm<t> d <f>» (1) have a profound relationship with Ritmannian geometry, gA.B(<f>) being in- terpreted as the metric tensor. In our language these are N = 0 a models. Adding fermions and requiring supersymmetry leads to restrictions on the metric that have been understood as the existence of complex structures on the target manifolds. For instance, the target manifold of the N = 1 su- persymmetric <7-models in four dimensions should be Kahler (one complex structure) [lj, while for N = 2 supersymmetric a-models it is hyperKahler (three complex structures) [2].
    [Show full text]
  • ℤ2×ℤ2-Graded Mechanics
    Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 967 (2021) 115426 www.elsevier.com/locate/nuclphysb Z2 × Z2-graded mechanics: The quantization ∗ N. Aizawa a, Z. Kuznetsova b, F. Toppan c, a Department of Physical Sciences, Graduate School of Science, Osaka Prefecture University, Nakamozu Campus, Sakai, Osaka 599-8531 Japan b UFABC, Av. dos Estados 5001, Bangu, CEP 09210-580, Santo André (SP), Brazil c CBPF, Rua Dr. Xavier Sigaud 150, Urca, CEP 22290-180, Rio de Janeiro (RJ), Brazil Received 10 December 2020; received in revised form 23 March 2021; accepted 25 April 2021 Available online 28 April 2021 Editor: Hubert Saleur Abstract In a previous paper we introduced the notion of Z2 × Z2-graded classical mechanics and presented a general framework to construct, in the Lagrangian setting, the worldline sigma models invariant under a Z2 × Z2-graded superalgebra. In this work we discuss at first the classical Hamiltonian formulation for a class of these models and later present their canonical quantization. As the simplest application of the construction we recover the Z2 × Z2-graded quantum Hamiltonian in- troduced by Bruce and Duplij. We prove that this is just the first example of a large class of Z2 × Z2-graded quantum models. We derive in particular interacting multiparticle quantum Hamiltonians given by Hermi- tian, matrix, differential operators. The interacting terms appear as non-diagonal entries in the matrices. The construction of the Noether charges, both classical and quantum, is presented. A comprehensive discussion of the different Z2 × Z2-graded symmetries possessed by the quantum Hamiltonians is given.
    [Show full text]
  • Gauge Functions in Classical Mechanics: from Undriven to Driven Dynamical Systems
    Article Gauge Functions in Classical Mechanics: From Undriven to Driven Dynamical Systems Zdzislaw E. Musielak * , Lesley C. Vestal, Bao D. Tran and Timothy B. Watson Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA; [email protected] (L.C.V.); [email protected] (B.D.T.); [email protected] (T.B.W.) * Correspondence: [email protected] Received: 11 August 2020; Accepted: 3 September 2020; Published: 9 September 2020 Abstract: Novel gauge functions are introduced to non-relativistic classical mechanics and used to define forces. The obtained results show that the gauge functions directly affect the energy function and allow for converting an undriven physical system into a driven one. This is a novel phenomenon in dynamics that resembles the role of gauges in quantum field theories. Keywords: Lagrangian formalism; null Lagrangians; gauge functions; forces; classical mechanics 1. Introduction The background space and time of non-relativistic classical mechanics (CM) is described by the 2 2 2 2 2 2 Galilean metrics ds1 = dt and ds2 = dx + dy + dz , where t is time and x, y, and z are Cartesian coordinates that are associated with an inertial frame of reference [1]. The metrics are invariant with respect to rotations, translations and boots, which form the Galilean transformations. In Newtonian dynamics, the Galilean transformations induce a gauge transformation [2], which is called the Galilean gauge [3]. The presence of this gauge guarantees that the Newton’s law of inertia is invariant with respect to the Galilean transformations, but it also shows that its Lagrangian is not [2,3].
    [Show full text]
  • Hamiltonian Formalism: Hamilton's Equations. Conservation Laws
    Hamiltonian Formalism: Hamilton's equations. Conservation laws. Reduction. Poisson Brackets. Physics 6010, Fall 2010 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]
  • Linear Dynamics, Lecture 1
    Linear Dynamics, Lecture 1 Review of Hamiltonian Mechanics Andy Wolski University of Liverpool, and the Cockcroft Institute, Daresbury, UK. November, 2012 Introduction Joseph John Thomson, 1856-1940 Early accelerators were fairly straightforward. Linear Dynamics, Lecture 1 1 Hamiltonian Mechanics Introduction Modern accelerators are more sophisticated. Linear Dynamics, Lecture 1 2 Hamiltonian Mechanics Introduction For modern accelerators to operate properly, the beam dynamics must be modelled and understood with very high precision. There are many effects that are important, including synchrotron radiation, interactions between the particles, interactions with the residual gas in the vacuum chamber and with the vacuum chamber itself, etc. However, everything starts with understanding the motion of individual particles through the fields from the magnets and the RF cavities. There are several possible approaches. We shall develop an approach starting from the fundamentals of classical mechanics. This requires more initial effort to derive the equations of motion in a form appropriate for accelerator physics; but has the benefit of providing a rigorous framework for modelling the dynamics with the precision required for modern accelerators. Linear Dynamics, Lecture 1 3 Hamiltonian Mechanics Course Outline Part I (Lectures 1 – 5): Dynamics of a relativistic charged particle in the electromagnetic field of an accelerator beamline. 1. Review of Hamiltonian mechanics 2. The accelerator Hamiltonian in a straight coordinate system 3. The Hamiltonian for a relativistic particle in a general electromagnetic field using accelerator coordinates 4. Dynamical maps for linear elements 5. Three loose ends: edge focusing; chromaticity; beam rigidity. Linear Dynamics, Lecture 1 4 Hamiltonian Mechanics Course Outline Part II (Lectures 6 – 10): Description of beam dynamics using optical lattice functions.
    [Show full text]
  • Supersymmetry and Supercoherent States of a Nonrelativistic Free Particle
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server SUPERSYMMETRY AND SUPERCOHERENT STATES OF A NONRELATIVISTIC FREE PARTICLE Boris F. Samsonov 1 Tomsk State University, 36 Lenin Avenue, 634050 Tomsk, Russia (Received 20 October 1996; accepted for publication 16 May 1997) Coordinate atypical representation of the orthosymplectic superalgebra osp(2/2) in a Hilbert superspace of square integrable functions constructed in a special way is given. The quantum nonrelativistic free particle Hamiltonian is an element of this superalgebra which turns out to be a dynamical superalgebra for this system. The supercoherent states, defined by means of a supergroup displacement operator, are explicitly constructed. These are the coordinate representation of the known atypical abstract super group OSp(2/2) coherent states. We interpret obtained results from the classical mechanics viewpoint as a model of classical particle which is immovable in the even sector of the phase superspace and is in rectilinear movement (in the appropriate coordinate system) in its odd sector. c 1997 American Institute of Physics. [S0022- 2488(97)00809-8] I. INTRODUCTION The supersymmetry in physics has been introduced in the quantum field theory for unifying of interactions of different kinds in a unique construction [1]. Supersymmetric formulation of quantum mechanics is due to the problem of spontaneous supersymmetry breaking [2]. Ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems since, and now there are no doubts that the supersymmetric quantum mechanics has its own right to exist (see for Ref. [3] a recent review).
    [Show full text]
  • Quantum Mechanics for Mathematicians Leon A. Takhtajan
    Quantum Mechanics for Mathematicians Leon A. Takhtajan Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651, USA E-mail address: [email protected] To my teacher Ludwig Dmitrievich Faddeev with admiration and gratitude. Contents Preface xiii Part 1. Foundations Chapter 1. Classical Mechanics 3 1. Lagrangian Mechanics 4 § 1.1. Generalized coordinates 4 1.2. The principle of the least action 4 1.3. Examples of Lagrangian systems 8 1.4. Symmetries and Noether’s theorem 15 1.5. One-dimensional motion 20 1.6. The motion in a central field and the Kepler problem 22 1.7. Legendre transform 26 2. Hamiltonian Mechanics 31 § 2.1. Hamilton’s equations 31 2.2. The action functional in the phase space 33 2.3. The action as a function of coordinates 35 2.4. Classical observables and Poisson bracket 38 2.5. Canonical transformations and generating functions 39 2.6. Symplectic manifolds 42 2.7. Poisson manifolds 51 2.8. Hamilton’s and Liouville’s representations 56 3. Notes and references 60 § Chapter 2. Basic Principles of Quantum Mechanics 63 1. Observables, states, and dynamics 65 § 1.1. Mathematical formulation 66 vii viii Contents 1.2. Heisenberg’s uncertainty relations 74 1.3. Dynamics 75 2. Quantization 79 § 2.1. Heisenberg commutation relations 80 2.2. Coordinate and momentum representations 86 2.3. Free quantum particle 93 2.4. Examples of quantum systems 98 2.5. Old quantum mechanics 102 2.6. Harmonic oscillator 103 2.7. Holomorphic representation and Wick symbols 111 3. Weyl relations 118 § 3.1.
    [Show full text]