Introduction to Lagrangian and Hamiltonian Mechanics
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THE HOMOGENEITY SCALE of the UNIVERSE 1 Introduction On
THE HOMOGENEITY SCALE OF THE UNIVERSE P. NTELIS Astroparticle and Cosmology Group, University Paris-Diderot 7 , 10 rue A. Domon & Duquet, Paris , France In this study, we probe the cosmic homogeneity with the BOSS CMASS galaxy sample in the redshift region of 0.43 <z<0.7. We use the normalised counts-in-spheres estimator N (<r) and the fractal correlation dimension D2(r) to assess the homogeneity scale of the universe. We verify that the universe becomes homogenous on scales greater than RH 64.3±1.6 h−1Mpc, consolidating the Cosmological Principle with a consistency test of ΛCDM model at the percentage level. Finally, we explore the evolution of the homogeneity scale in redshift. 1 Introduction on Cosmic Homogeneity The standard model of Cosmology, known as the ΛCDM model is based on solutions of the equations of General Relativity for isotropic and homogeneous universes where the matter is mainly composed of Cold Dark Matter (CDM) and the Λ corresponds to a cosmological con- stant. This model shows excellent agreement with current data, be it from Type Ia supernovae, temperature and polarisation anisotropies in the Cosmic Microwave Background or Large Scale Structure. The main assumption of these models is the Cosmological Principle which states that the universe is homogeneous and isotropic on large scales [1]. Isotropy is well tested through various probes in different redshifts such as the Cosmic Microwave Background temperature anisotropies at z ≈ 1100, corresponding to density fluctuations of the order 10−5 [2]. In later cosmic epochs, the distribution of source in surveys the hypothesis of isotropy is strongly supported by in X- ray [3] and radio [4], while the large spectroscopic galaxy surveys such show no evidence for anisotropies in volumes of a few Gpc3 [5].As a result, it is strongly motivated to probe the homogeneity property of our universe. -
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. -
The Lagrangian and Hamiltonian Mechanical Systems
THE LAGRANGIAN AND HAMILTONIAN MECHANICAL SYSTEMS ALEXANDER TOLISH Abstract. Newton's Laws of Motion, which equate forces with the time- rates of change of momenta, are a convenient way to describe mechanical systems in Euclidean spaces with cartesian coordinates. Unfortunately, the physical world is rarely so cooperative|physicists often explore systems that are neither Euclidean nor cartesian. Different mechanical formalisms, like the Lagrangian and Hamiltonian systems, may be more effective at describing such phenomena, as they are geometric rather than analytic processes. In this paper, I shall construct Lagrangian and Hamiltonian mechanics, prove their equivalence to Newtonian mechanics, and provide examples of both non- Newtonian systems in action. Contents 1. The Calculus of Variations 1 2. Manifold Geometry 3 3. Lagrangian Mechanics 4 4. Two Electric Pendula 4 5. Differential Forms and Symplectic Geometry 6 6. Hamiltonian Mechanics 7 7. The Double Planar Pendulum 9 Acknowledgments 11 References 11 1. The Calculus of Variations Lagrangian mechanics applies physics not only to particles, but to the trajectories of particles. We must therefore study how curves behave under small disturbances or variations. Definition 1.1. Let V be a Banach space. A curve is a continuous map : [t0; t1] ! V: A variation on the curve is some function h of t that creates a new curve +h.A functional is a function from the space of curves to the real numbers. p Example 1.2. Φ( ) = R t1 1 +x _ 2dt, wherex _ = d , is a functional. It expresses t0 dt the length of curve between t0 and t1. Definition 1.3. -
Lagrangian Mechanics
Alain J. Brizard Saint Michael's College Lagrangian Mechanics 1Principle of Least Action The con¯guration of a mechanical system evolving in an n-dimensional space, with co- ordinates x =(x1;x2; :::; xn), may be described in terms of generalized coordinates q = (q1;q2;:::; qk)inak-dimensional con¯guration space, with k<n. The Principle of Least Action (also known as Hamilton's principle) is expressed in terms of a function L(q; q_ ; t)knownastheLagrangian,whichappears in the action integral Z tf A[q]= L(q; q_ ; t) dt; (1) ti where the action integral is a functional of the vector function q(t), which provides a path from the initial point qi = q(ti)tothe¯nal point qf = q(tf). The variational principle ¯ " à !# ¯ Z d ¯ tf @L d @L ¯ j 0=±A[q]= A[q + ²±q]¯ = ±q j ¡ j dt; d² ²=0 ti @q dt @q_ where the variation ±q is assumed to vanish at the integration boundaries (±qi =0=±qf), yields the Euler-Lagrange equation for the generalized coordinate qj (j =1; :::; k) à ! d @L @L = ; (2) dt @q_j @qj The Lagrangian also satis¯esthesecond Euler equation à ! d @L @L L ¡ q_j = ; (3) dt @q_j @t and thus for time-independent Lagrangian systems (@L=@t =0)we¯nd that L¡q_j @L=@q_j is a conserved quantity whose interpretationwill be discussed shortly. The form of the Lagrangian function L(r; r_; t)isdictated by our requirement that Newton's Second Law m Är = ¡rU(r;t)describing the motion of a particle of mass m in a nonuniform (possibly time-dependent) potential U(r;t)bewritten in the Euler-Lagrange form (2). -
Lagrangian Mechanics - Wikipedia, the Free Encyclopedia Page 1 of 11
Lagrangian mechanics - Wikipedia, the free encyclopedia Page 1 of 11 Lagrangian mechanics From Wikipedia, the free encyclopedia Lagrangian mechanics is a re-formulation of classical mechanics that combines Classical mechanics conservation of momentum with conservation of energy. It was introduced by the French mathematician Joseph-Louis Lagrange in 1788. Newton's Second Law In Lagrangian mechanics, the trajectory of a system of particles is derived by solving History of classical mechanics · the Lagrange equations in one of two forms, either the Lagrange equations of the Timeline of classical mechanics [1] first kind , which treat constraints explicitly as extra equations, often using Branches [2][3] Lagrange multipliers; or the Lagrange equations of the second kind , which Statics · Dynamics / Kinetics · Kinematics · [1] incorporate the constraints directly by judicious choice of generalized coordinates. Applied mechanics · Celestial mechanics · [4] The fundamental lemma of the calculus of variations shows that solving the Continuum mechanics · Lagrange equations is equivalent to finding the path for which the action functional is Statistical mechanics stationary, a quantity that is the integral of the Lagrangian over time. Formulations The use of generalized coordinates may considerably simplify a system's analysis. Newtonian mechanics (Vectorial For example, consider a small frictionless bead traveling in a groove. If one is tracking the bead as a particle, calculation of the motion of the bead using Newtonian mechanics) mechanics would require solving for the time-varying constraint force required to Analytical mechanics: keep the bead in the groove. For the same problem using Lagrangian mechanics, one Lagrangian mechanics looks at the path of the groove and chooses a set of independent generalized Hamiltonian mechanics coordinates that completely characterize the possible motion of the bead. -
Ten1perature, Least Action, and Lagrangian Mechanics L
14 Auguq Il)<)5 2) 291 PHYSICS LETTERS A 's. Rev. ELSEVIER Physics Letters A 204 (1995) 133-135 l) 759. 7. to be Ten1perature, least action, and Lagrangian mechanics l. 843. William G. Hoover Department ofApplied Science, Ulliuersity ofCalifornia at Davis / Livermore alld Lawrence Licermore Nalional Laboratory, LiueTlrJOre, CA 94551-7808, USA Received 28 March 1995; revised manuscript received 2 June 1995; accepted for publication 2 June J995 Communicated by A.R. Bishop Abstract Temperature is considered from two different mechanical standpoints. The more approach uses Hamilton's least action principle. From it we derive thermostatting forces identical to those found using Gauss' Principle. This approach applies to equilibrium and nonequilibrium systems. The Lagrangian approach is different, and less useful, applying only at equilibrium. Keywords: Least action principle; Nonequilibrium; Lagrangian mechanics 1. Introduction When the context is clear we use just a single q or q to represent whole sets. Feynman repeatedly emphasized the generality of Lagrangian mechanics is particularly well-suited Hamilton's principle of least action [1], the physical to the treatment of constrained systems. Constraints, law from which both kinds of mechanics, classical used first to represent mechanical linkages and cou and quantum, follow. The principle states that the plings, have more recently been used in molecular observed trajectory {qed} is that which minimizes simulations, where they maintain the shape of a the action integral, polyatomic molecule and reduce the number of dy namical equations to be integrated. The usual ap ofL(q,q,t)dt of(K <P)dt=O, proach is to add a "Lagrange multiplier" to the Lagrangian in such a way as to impose the constraint where K is the kinetic energy, <P is the potential without affecting energy conservation [2,3]. -
Analytical Mechanics: Lagrangian and Hamiltonian Formalism
MATH-F-204 Analytical mechanics: Lagrangian and Hamiltonian formalism Glenn Barnich Physique Théorique et Mathématique Université libre de Bruxelles and International Solvay Institutes Campus Plaine C.P. 231, B-1050 Bruxelles, Belgique Bureau 2.O6. 217, E-mail: [email protected], Tel: 02 650 58 01. The aim of the course is an introduction to Lagrangian and Hamiltonian mechanics. The fundamental equations as well as standard applications of Newtonian mechanics are derived from variational principles. From a mathematical perspective, the course and exercises (i) constitute an application of differential and integral calculus (primitives, inverse and implicit function theorem, graphs and functions); (ii) provide examples and explicit solutions of first and second order systems of differential equations; (iii) give an implicit introduction to differential manifolds and to realiza- tions of Lie groups and algebras (change of coordinates, parametrization of a rotation, Galilean group, canonical transformations, one-parameter subgroups, infinitesimal transformations). From a physical viewpoint, a good grasp of the material of the course is indispensable for a thorough understanding of the structure of quantum mechanics, both in its operatorial and path integral formulations. Furthermore, the discussion of Noether’s theorem and Galilean invariance is done in a way that allows for a direct generalization to field theories with Poincaré or conformal invariance. 2 Contents 1 Extremisation, constraints and Lagrange multipliers 5 1.1 Unconstrained extremisation . .5 1.2 Constraints and regularity conditions . .5 1.3 Constrained extremisation . .7 2 Constrained systems and d’Alembert principle 9 2.1 Holonomic constraints . .9 2.2 d’Alembert’s theorem . 10 2.3 Non-holonomic constraints . -
Shallow Water Waves on the Rotating Sphere
PHYSICAL RBVIEW E VOLUME SI, NUMBER S MAYl99S Shallowwater waves on the rotating sphere D. MUller andJ. J. O'Brien Centerfor Ocean-AtmosphericPrediction Studies,Florida State Univenity, TallahDSSee,Florida 32306 (Received8 September1994) Analytical solutions of the linear wave equation of shallow waters on the rotating, spherical surface have been found for standing as weD as for inertial waves. The spheroidal wave operator is shown to playa central role in this wave equation. Prolate spheroidal angular functions capture the latitude- longitude anisotropy, the east-westasymmetry, and the Yoshida inhomogeneity of wave propagation on the rotating spherical surface. The identification of the fundamental role of the spheroidal wave opera- tor permits an overview over the system's wave number space. Three regimes can be distinguished. High-frequency gravity waves do not experience the Yoshida waveguide and exhibit Cariolis-induced east-westasymmetry. A second, low-frequency regime is exclusively populated by Rossby waves. The familiar .8-plane regime provides the appropriate approximation for equatorial, baroclinic, low- frequency waves. Gravity waves on the .8-plane exhibit an amplified Cariolis-induced east-westasym- metry. Validity and limitations of approximate dispersion relations are directly tested against numeri- caDy calculated solutions of the full eigenvalueproblem. PACSnumber(s): 47.32.-y. 47.3S.+i,92.~.Dj. 92.10.Hm L INTRODUcnON [4] in the framework of the .B-pJaneapproximation. In this Cartesian approximation of the spherical geometry The first theoretical investigation of the large-scale near the equator Matsuno identified Rossby, mixed responseof the ocean-atmospheresystem to external tidal Rossby-gravity, as well as gravity waves, including the forces was conducted by Laplace [I] in terms of the KelVin wave. -
Examples in Lagrangian Mechanics C Alex R
Lecture 18 Friday - October 14, 2005 Written or last updated: October 14, 2005 P441 – Analytical Mechanics - I Examples in Lagrangian Mechanics c Alex R. Dzierba Sample problems using Lagrangian mechanics Here are some sample problems. I will assign similar problems for the next problem set. Example 1 In Figure 1 we show a box of mass m sliding down a ramp of mass M. The ramp moves without friction on the horizontal plane and is located by coordinate x1. The box also slides without friction on the ramp and is located by coordinate x2 with respect to the ramp. x 2 m x1 M θ Figure 1: A box slides down a ramp without friction and the ramp slides along a horizontal surface without friction. The kinetic energy of the ramp TM is given by: 1 2 TM = Mx˙ 1 (1) 2 and the kinetic energy of the box Tm is given by: 1 2 2 Tm = m x˙ 1 +x ˙ 2 + 2x ˙ 1x˙ 1 cos θ (2) 2 The velocity of the box is obviously derived from the vector sum of the velocity relative to the ramp and the velocity of the ramp. The potential energy of the system is just the potential energy of the box and that leads to: U = −mgx2 sin θ (3) 1 and finally the Lagrangian is given by: 1 2 1 2 2 L(x1, x˙ 1, x2, x˙ 2)= TM + Tm − U = Mx˙ 1 + m x˙ 1 +x ˙ 2 + 2x ˙ 1x˙ 1 cos θ + mgx2 sin θ (4) 2 2 The equations of motion are: d ∂L ∂L = (5) dt ∂x˙ 1 ∂x1 and d ∂L ∂L = (6) dt ∂x˙ 2 ∂x2 Equation 5 leads to: d [m(x ˙ 1 +x ˙ 2 cos θ)+ Mx˙ 1] = 0 (7) dt The RHS of equation 7 is zero because the Lagrangian does not explicitly depend on x1. -
Lagrangian Mechanics
Chapter 1 Lagrangian Mechanics Our introduction to Quantum Mechanics will be based on its correspondence to Classical Mechanics. For this purpose we will review the relevant concepts of Classical Mechanics. An important concept is that the equations of motion of Classical Mechanics can be based on a variational principle, namely, that along a path describing classical motion the action integral assumes a minimal value (Hamiltonian Principle of Least Action). 1.1 Basics of Variational Calculus The derivation of the Principle of Least Action requires the tools of the calculus of variation which we will provide now. Definition: A functional S[ ] is a map M S[]: R ; = ~q(t); ~q :[t0; t1] R R ; ~q(t) differentiable (1.1) F! F f ⊂ ! g from a space of vector-valued functions ~q(t) onto the real numbers. ~q(t) is called the trajec- tory of a systemF of M degrees of freedom described by the configurational coordinates ~q(t) = (q1(t); q2(t); : : : qM (t)). In case of N classical particles holds M = 3N, i.e., there are 3N configurational coordinates, namely, the position coordinates of the particles in any kind of coordianate system, often in the Cartesian coordinate system. It is important to note at the outset that for the description of a d classical system it will be necessary to provide information ~q(t) as well as dt ~q(t). The latter is the velocity vector of the system. Definition: A functional S[ ] is differentiable, if for any ~q(t) and δ~q(t) where 2 F 2 F d = δ~q(t); δ~q(t) ; δ~q(t) < , δ~q(t) < , t; t [t0; t1] R (1.2) F f 2 F j j jdt j 8 2 ⊂ g a functional δS[ ; ] exists with the properties · · (i) S[~q(t) + δ~q(t)] = S[~q(t)] + δS[~q(t); δ~q(t)] + O(2) (ii) δS[~q(t); δ~q(t)] is linear in δ~q(t). -
Dark Energy and the Homogeneous Universe 暗能量及宇宙整體之膨脹
Dark Energy and the Homogeneous Universe 暗能量及宇宙整體之膨脹 Lam Hui 許林 Columbia University This is a series of 3 talks. Today: Dark energy and the homogeneous universe July 11: Dark matter and the large scale structure of the universe July 18: Inflation and the early universe Outline Basic concepts: a homogeneous, isotropic and expanding universe. Measurements: how do we measure the universe’s expansion history? Dynamics: what determines the expansion rate? Dark energy: a surprising recent finding. There is by now much evidence for homogeneity and isotropy. There is also firm evidence for an expanding universe. scale ∼ 1023 cm Blandton & Hogg scale ∼ 1024 cm Hubble deep field scale ∼ 1025 cm CfA survey Sloan Digital Sky Survey scale ∼ 1026 cm Ned Wright There is by now much evidence for homogeneity and isotropy. There is also firm evidence for an expanding universe. There is by now much evidence for homogeneity and isotropy. There is also firm evidence for an expanding universe. But how should we think about it? Frequently Asked Questions: 1. What is the universe expanding into? 2. Where is the center of expansion? 3. Is our galaxy itself expanding? Frequently Asked Questions: 1. What is the universe expanding into? Nothing. 2. Where is the center of expansion? Everywhere and nowhere. 3. Is our galaxy itself expanding? No. C C A B A B dAC = 2 dAB ∆d ∆d AC = 2 AB Therefore: ∆t ∆t Hubble law: Velocity ∝ Distance Supernova 1998ba Supernova Cosmology Project (Perlmutter, et al., 1998) (as seen from Hubble Space Telescope) 3 Weeks Supernova Before Discovery (as seen from telescopes on Earth) Difference Supernova 超新星as standard candle FAINTER (Farther) (Further back in time) Perlmutter, et al. -
On Some Consequences of the Canonical Transformation in the Hamiltonian Theory of Water Waves
579 On some consequences of the canonical transformation in the Hamiltonian theory of water waves Peter A.E.M. Janssen Research Department November 2008 Series: ECMWF Technical Memoranda A full list of ECMWF Publications can be found on our web site under: http://www.ecmwf.int/publications/ Contact: [email protected] c Copyright 2008 European Centre for Medium-Range Weather Forecasts Shinfield Park, Reading, RG2 9AX, England Literary and scientific copyrights belong to ECMWF and are reserved in all countries. This publication is not to be reprinted or translated in whole or in part without the written permission of the Director. Appropriate non-commercial use will normally be granted under the condition that reference is made to ECMWF. The information within this publication is given in good faith and considered to be true, but ECMWF accepts no liability for error, omission and for loss or damage arising from its use. On some consequences of the canonical transformation in the Hamiltonian theory of water waves Abstract We discuss some consequences of the canonical transformation in the Hamiltonian theory of water waves (Zakharov, 1968). Using Krasitskii’s canonical transformation we derive general expressions for the second order wavenumber and frequency spectrum, and the skewness and the kurtosis of the sea surface. For deep-water waves, the second-order wavenumber spectrum and the skewness play an important role in understanding the so-called sea state bias as seen by a Radar Altimeter. According to the present approach, but in contrast with results obtained by Barrick and Weber (1977), in deep-water second-order effects on the wavenumber spectrum are relatively small.