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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} = − . -
Integrals of Motion Key Point #15 from Bachelor Course
KeyKey12-10-07 see PointPoint http://www.strw.leidenuniv.nl/˜ #15#10 fromfrom franx/college/ BachelorBachelor mf-sts-07-c4-5 Course:Course:12-10-07 see http://www.strw.leidenuniv.nl/˜ IntegralsIntegrals ofof franx/college/ MotionMotion mf-sts-07-c4-6 4.2 Constants and Integrals of motion Integrals (BT 3.1 p 110-113) Much harder to define. E.g.: 1 2 Energy (all static potentials): E(⃗x,⃗v)= 2 v + Φ First, we define the 6 dimensional “phase space” coor- dinates (⃗x,⃗v). They are conveniently used to describe Lz (axisymmetric potentials) the motions of stars. Now we introduce: L⃗ (spherical potentials) • Integrals constrain geometry of orbits. • Constant of motion: a function C(⃗x,⃗v, t) which is constant along any orbit: examples: C(⃗x(t1),⃗v(t1),t1)=C(⃗x(t2),⃗v(t2),t2) • 1. Spherical potentials: E,L ,L ,L are integrals of motion, but also E,|L| C is a function of ⃗x, ⃗v, and time t. x y z and the direction of L⃗ (given by the unit vector ⃗n, Integral of motion which is defined by two independent numbers). ⃗n de- • : a function I(x, v) which is fines the plane in which ⃗x and ⃗v must lie. Define coor- constant along any orbit: dinate system with z axis along ⃗n I[⃗x(t1),⃗v(t1)] = I[⃗x(t2),⃗v(t2)] ⃗x =(x1,x2, 0) I is not a function of time ! Thus: integrals of motion are constants of motion, ⃗v =(v1,v2, 0) but constants of motion are not always integrals of motion! → ⃗x and ⃗v constrained to 4D region of the 6D phase space. -
STUDENT HIGHLIGHT – Hassan Najafi Alishah - “Symplectic Geometry, Poisson Geometry and KAM Theory” (Mathematics)
SEPTEMBER OCTOBER // 2010 NUMBER// 28 STUDENT HIGHLIGHT – Hassan Najafi Alishah - “Symplectic Geometry, Poisson Geometry and KAM theory” (Mathematics) Hassan Najafi Alishah, a CoLab program student specializing in math- The Hamiltonian function (or energy function) is constant along the ematics, is doing advanced work in the Mathematics Departments of flow. In other words, the Hamiltonian function is a constant of motion, Instituto Superior Técnico, Lisbon, and the University of Texas at Austin. a principle that is sometimes called the energy conservation law. A This article provides an overview of his research topics. harmonic spring, a pendulum, and a one-dimensional compressible fluid provide examples of Hamiltonian Symplectic and Poisson geometries underlie all me- systems. In the former, the phase spaces are symplec- chanical systems including the solar system, the motion tic manifolds, but for the later phase space one needs of a rigid body, or the interaction of small molecules. the more general concept of a Poisson manifold. The origins of symplectic and Poisson structures go back to the works of the French mathematicians Joseph-Louis A Hamiltonian system with enough suitable constants Lagrange (1736-1813) and Simeon Denis Poisson (1781- of motion can be integrated explicitly and its orbits can 1840). Hermann Weyl (1885-1955) first used “symplectic” be described in detail. These are called completely in- with its modern mathematical meaning in his famous tegrable Hamiltonian systems. Under appropriate as- monograph “The Classical groups.” (The word “sym- sumptions (e.g., in a compact symplectic manifold) plectic” itself is derived from a Greek word that means the motions of a completely integrable Hamiltonian complex.) Lagrange introduced the concept of a system are quasi-periodic. -
Lecture 8 Symmetries, Conserved Quantities, and the Labeling of States Angular Momentum
Lecture 8 Symmetries, conserved quantities, and the labeling of states Angular Momentum Today’s Program: 1. Symmetries and conserved quantities – labeling of states 2. Ehrenfest Theorem – the greatest theorem of all times (in Prof. Anikeeva’s opinion) 3. Angular momentum in QM ˆ2 ˆ 4. Finding the eigenfunctions of L and Lz - the spherical harmonics. Questions you will by able to answer by the end of today’s lecture 1. How are constants of motion used to label states? 2. How to use Ehrenfest theorem to determine whether the physical quantity is a constant of motion (i.e. does not change in time)? 3. What is the connection between symmetries and constant of motion? 4. What are the properties of “conservative systems”? 5. What is the dispersion relation for a free particle, why are E and k used as labels? 6. How to derive the orbital angular momentum observable? 7. How to check if a given vector observable is in fact an angular momentum? References: 1. Introductory Quantum and Statistical Mechanics, Hagelstein, Senturia, Orlando. 2. Principles of Quantum Mechanics, Shankar. 3. http://mathworld.wolfram.com/SphericalHarmonic.html 1 Symmetries, conserved quantities and constants of motion – how do we identify and label states (good quantum numbers) The connection between symmetries and conserved quantities: In the previous section we showed that the Hamiltonian function plays a major role in our understanding of quantum mechanics using it we could find both the eigenfunctions of the Hamiltonian and the time evolution of the system. What do we mean by when we say an object is symmetric?�What we mean is that if we take the object perform a particular operation on it and then compare the result to the initial situation they are indistinguishable.�When one speaks of a symmetry it is critical to state symmetric with respect to which operation. -
Conservation of Total Momentum TM1
L3:1 Conservation of total momentum TM1 We will here prove that conservation of total momentum is a consequence Taylor: 268-269 of translational invariance. Consider N particles with positions given by . If we translate the whole system with some distance we get The potential energy must be una ected by the translation. (We move the "whole world", no relative position change.) Also, clearly all velocities are independent of the translation Let's consider to be in the x-direction, then Using Lagrange's equation, we can rewrite each derivative as I.e., total momentum is conserved! (Clearly there is nothing special about the x-direction.) L3:2 Generalized coordinates, velocities, momenta and forces Gen:1 Taylor: 241-242 With we have i.e., di erentiating w.r.t. gives us the momentum in -direction. generalized velocity In analogy we de✁ ne the generalized momentum and we say that and are conjugated variables. Note: Def. of gen. force We also de✁ ne the generalized force from Taylor 7.15 di ers from def in HUB I.9.17. such that generalized rate of change of force generalized momentum Note: (generalized coordinate) need not have dimension length (generalized velocity) velocity momentum (generalized momentum) (generalized force) force Example: For a system which rotates around the z-axis we have for V=0 s.t. The angular momentum is thus conjugated variable to length dimensionless length/time 1/time mass*length/time mass*(length)^2/time (torque) energy Cyclic coordinates = ignorable coordinates and Noether's theorem L3:3 Cyclic:1 We have de ned the generalized momentum Taylor: From EL we have 266-267 i.e. -
Newtonian Mechanics Is Most Straightforward in Its Formulation and Is Based on Newton’S Second Law
CLASSICAL MECHANICS D. A. Garanin September 30, 2015 1 Introduction Mechanics is part of physics studying motion of material bodies or conditions of their equilibrium. The latter is the subject of statics that is important in engineering. General properties of motion of bodies regardless of the source of motion (in particular, the role of constraints) belong to kinematics. Finally, motion caused by forces or interactions is the subject of dynamics, the biggest and most important part of mechanics. Concerning systems studied, mechanics can be divided into mechanics of material points, mechanics of rigid bodies, mechanics of elastic bodies, and mechanics of fluids: hydro- and aerodynamics. At the core of each of these areas of mechanics is the equation of motion, Newton's second law. Mechanics of material points is described by ordinary differential equations (ODE). One can distinguish between mechanics of one or few bodies and mechanics of many-body systems. Mechanics of rigid bodies is also described by ordinary differential equations, including positions and velocities of their centers and the angles defining their orientation. Mechanics of elastic bodies and fluids (that is, mechanics of continuum) is more compli- cated and described by partial differential equation. In many cases mechanics of continuum is coupled to thermodynamics, especially in aerodynamics. The subject of this course are systems described by ODE, including particles and rigid bodies. There are two limitations on classical mechanics. First, speeds of the objects should be much smaller than the speed of light, v c, otherwise it becomes relativistic mechanics. Second, the bodies should have a sufficiently large mass and/or kinetic energy. -
Chapter 04 Rotational Motion
Chapter 04 Rotational Motion P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 04: Rotational Motion Angular Momentum Angular momentum of particle with respect to origin, O, is given by l⃗ = ⃗r × p⃗ Rate of change of angular momentum is given z by cross product of ⃗r with applied force. p m dl⃗ dp⃗ = ⃗r × = ⃗r × F⃗ = ⃗휏 r dt dt O y Cross product is defined as applied torque, ⃗휏. x Unlike linear momentum, angular momentum depends on origin choice. P. J. Grandinetti Chapter 04: Rotational Motion Conservation of Angular Momentum Consider system of N Particles z m5 m 2 Rate of change of angular momentum is m3 ⃗ ∑N l⃗ ∑N ⃗ m1 dL d 훼 dp훼 = = ⃗r훼 × dt dt dt 훼=1 훼=1 y which becomes m4 x ⃗ ∑N dL ⃗ net = ⃗r훼 × F dt 훼 Total angular momentum is 훼=1 ∑N ∑N ⃗ ⃗ L = l훼 = ⃗r훼 × p⃗훼 훼=1 훼=1 P. J. Grandinetti Chapter 04: Rotational Motion Conservation of Angular Momentum ⃗ ∑N dL ⃗ net = ⃗r훼 × F dt 훼 훼=1 Taking an earlier expression for a system of particles from chapter 1 ∑N ⃗ net ⃗ ext ⃗ F훼 = F훼 + f훼훽 훽=1 훽≠훼 we obtain ⃗ ∑N ∑N ∑N dL ⃗ ext ⃗ = ⃗r훼 × F + ⃗r훼 × f훼훽 dt 훼 훼=1 훼=1 훽=1 훽≠훼 and then obtain 0 > ⃗ ∑N ∑N ∑N dL ⃗ ext ⃗ rd ⃗ ⃗ = ⃗r훼 × F + ⃗r훼 × f훼훽 double sum disappears from Newton’s 3 law (f = *f ) dt 훼 12 21 훼=1 훼=1 훽=1 훽≠훼 P. -
Analytical Mechanics
A Guided Tour of Analytical Mechanics with animations in MAPLE Rouben Rostamian Department of Mathematics and Statistics UMBC [email protected] December 2, 2018 ii Contents Preface vii 1 An introduction through examples 1 1.1 ThesimplependulumàlaNewton ...................... 1 1.2 ThesimplependulumàlaEuler ....................... 3 1.3 ThesimplependulumàlaLagrange.. .. .. ... .. .. ... .. .. .. 3 1.4 Thedoublependulum .............................. 4 Exercises .......................................... .. 6 2 Work and potential energy 9 Exercises .......................................... .. 12 3 A single particle in a conservative force field 13 3.1 The principle of conservation of energy . ..... 13 3.2 Thescalarcase ................................... 14 3.3 Stability....................................... 16 3.4 Thephaseportraitofasimplependulum . ... 16 Exercises .......................................... .. 17 4 TheKapitsa pendulum 19 4.1 Theinvertedpendulum ............................. 19 4.2 Averaging out the fast oscillations . ...... 19 4.3 Stabilityanalysis ............................... ... 22 Exercises .......................................... .. 23 5 Lagrangian mechanics 25 5.1 Newtonianmechanics .............................. 25 5.2 Holonomicconstraints............................ .. 26 5.3 Generalizedcoordinates .......................... ... 29 5.4 Virtual displacements, virtual work, and generalized force....... 30 5.5 External versus reaction forces . ..... 32 5.6 The equations of motion for a holonomic system . ... -
Fundamental Governing Equations of Motion in Consistent Continuum Mechanics
Fundamental governing equations of motion in consistent continuum mechanics Ali R. Hadjesfandiari, Gary F. Dargush Department of Mechanical and Aerospace Engineering University at Buffalo, The State University of New York, Buffalo, NY 14260 USA [email protected], [email protected] October 1, 2018 Abstract We investigate the consistency of the fundamental governing equations of motion in continuum mechanics. In the first step, we examine the governing equations for a system of particles, which can be considered as the discrete analog of the continuum. Based on Newton’s third law of action and reaction, there are two vectorial governing equations of motion for a system of particles, the force and moment equations. As is well known, these equations provide the governing equations of motion for infinitesimal elements of matter at each point, consisting of three force equations for translation, and three moment equations for rotation. We also examine the character of other first and second moment equations, which result in non-physical governing equations violating Newton’s third law of action and reaction. Finally, we derive the consistent governing equations of motion in continuum mechanics within the framework of couple stress theory. For completeness, the original couple stress theory and its evolution toward consistent couple stress theory are presented in true tensorial forms. Keywords: Governing equations of motion, Higher moment equations, Couple stress theory, Third order tensors, Newton’s third law of action and reaction 1 1. Introduction The governing equations of motion in continuum mechanics are based on the governing equations for systems of particles, in which the effect of internal forces are cancelled based on Newton’s third law of action and reaction. -
Energy and Equations of Motion in a Tentative Theory of Gravity with a Privileged Reference Frame
1 Energy and equations of motion in a tentative theory of gravity with a privileged reference frame Archives of Mechanics (Warszawa) 48, N°1, 25-52 (1996) M. ARMINJON (GRENOBLE) Abstract- Based on a tentative interpretation of gravity as a pressure force, a scalar theory of gravity was previously investigated. It assumes gravitational contraction (dilation) of space (time) standards. In the static case, the same Newton law as in special relativity was expressed in terms of these distorted local standards, and was found to imply geodesic motion. Here, the formulation of motion is reexamined in the most general situation. A consistent Newton law can still be defined, which accounts for the time variation of the space metric, but it is not compatible with geodesic motion for a time-dependent field. The energy of a test particle is defined: it is constant in the static case. Starting from ”dust‘, a balance equation is then derived for the energy of matter. If the Newton law is assumed, the field equation of the theory allows to rewrite this as a true conservation equation, including the gravitational energy. The latter contains a Newtonian term, plus the square of the relative rate of the local velocity of gravitation waves (or that of light), the velocity being expressed in terms of absolute standards. 1. Introduction AN ATTEMPT to deduce a consistent theory of gravity from the idea of a physically privileged reference frame or ”ether‘ was previously proposed [1-3]. This work is a further development which is likely to close the theory. It is well-known that the concept of ether has been abandoned at the beginning of this century. -
Relativistic Corrections to the Central Force Problem in a Generalized
Relativistic corrections to the central force problem in a generalized potential approach Ashmeet Singh∗ and Binoy Krishna Patra† Department of Physics, Indian Institute of Technology Roorkee, India - 247667 We present a novel technique to obtain relativistic corrections to the central force problem in the Lagrangian formulation, using a generalized potential energy function. We derive a general expression for a generalized potential energy function for all powers of the velocity, which when made a part of the regular classical Lagrangian can reproduce the correct (relativistic) force equation. We then go on to derive the Hamiltonian and estimate the corrections to the total energy of the system up to the fourth power in |~v|/c. We found that our work is able to provide a more comprehensive understanding of relativistic corrections to the central force results and provides corrections to both the kinetic and potential energy of the system. We employ our methodology to calculate relativistic corrections to the circular orbit under the gravitational force and also the first-order corrections to the ground state energy of the hydrogen atom using a semi-classical approach. Our predictions in both problems give reasonable agreement with the known results. Thus we feel that this work has pedagogical value and can be used by undergraduate students to better understand the central force and the relativistic corrections to it. I. INTRODUCTION correct relativistic formulation which is Lorentz invari- ant, but it is seen that the complexity of the equations The central force problem, in which the ordinary po- is an issue, even for the simplest possible cases like that tential function depends only on the magnitude of the of a single particle. -
Chapter 4 Orbital Angular Momentum and the Hydrogen Atom
Chapter 4 Orbital angular momentum and the hydrogen atom If I have understood correctly your point of view then you would gladly sacrifice the simplicity [of quantum mechan- ics] to the principle of causality. Perhaps we could comfort ourselves that the dear Lord could go beyond [quantum me- chanics] and maintain causality. -Werner Heisenberg responds to Einstein In quantum mechanics degenerations of energy eigenvalues typically are due to symmetries. The symmetries, in turn, can be used to simplify the Schr¨odinger equation, for example, by a separation ansatz in appropriate coordinates. In the present chapter we will study rotationally symmetric potentials und use the angular momentum operator to compute the energy spectrum of hydrogen-like atoms. 4.1 The orbital angular momentum According to Emmy Noether’s first theorem continuous symmetries of dynamical systems im- ply conservation laws. In turn, the conserved quantities (called charges in general, or energy and momentum for time and space translations, respectively) can be shown to generate the respective symmetry transformations via the Poisson brackets. These properties are inherited by quantum mechanics, where Poisson brackets of phase space functions are replaced by com- mutators. According to the Schr¨odinger equation i~∂tψ = Hψ, for example, time evolution is generated by the Hamiltonian. Similarly, the momentum operator P~ = ~ ~ generates (spatial) i ∇ translations. A (Hermitian) charge operator Q is conserved if it commutes with the Hamil- tonian [H, Q] = 0. This equation can also be interpreted as invariance of the Hamiltonian 77 CHAPTER 4. ORBITAL ANGULAR MOMENTUM AND THE HYDROGEN ATOM 78 1 H = Uλ− HUλ under the unitary 1-parameter transformation group of finite transformations Uλ = exp(iλQ) that is generated by the infinitesimal transformation Q.