Analytical Mechanics
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
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. -
Chapter 6 the Equations of Fluid Motion
Chapter 6 The equations of fluid motion In order to proceed further with our discussion of the circulation of the at- mosphere, and later the ocean, we must develop some of the underlying theory governing the motion of a fluid on the spinning Earth. A differen- tially heated, stratified fluid on a rotating planet cannot move in arbitrary paths. Indeed, there are strong constraints on its motion imparted by the angular momentum of the spinning Earth. These constraints are profoundly important in shaping the pattern of atmosphere and ocean circulation and their ability to transport properties around the globe. The laws governing the evolution of both fluids are the same and so our theoretical discussion willnotbespecifictoeitheratmosphereorocean,butcanandwillbeapplied to both. Because the properties of rotating fluids are often counter-intuitive and sometimes difficult to grasp, alongside our theoretical development we will describe and carry out laboratory experiments with a tank of water on a rotating table (Fig.6.1). Many of the laboratory experiments we make use of are simplified versions of ‘classics’ that have become cornerstones of geo- physical fluid dynamics. They are listed in Appendix 13.4. Furthermore we have chosen relatively simple experiments that, in the main, do nor require sophisticated apparatus. We encourage you to ‘have a go’ or view the atten- dant movie loops that record the experiments carried out in preparation of our text. We now begin a more formal development of the equations that govern the evolution of a fluid. A brief summary of the associated mathematical concepts, definitions and notation we employ can be found in an Appendix 13.2. -
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. -
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 . -
Lagrange As a Historian of Mechanics
Advances in Historical Studies 2013. Vol.2, No.3, 126-130 Published Online September 2013 in SciRes (http://www.scirp.org/journal/ahs) http://dx.doi.org/10.4236/ahs.2013.23016 Lagrange as a Historian of Mechanics Agamenon R. E. Oliveira Polytechnic School of Rio de Janeiro, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil Email: [email protected] Received August 5th, 2013; revised September 6th, 2013; accepted September 15th, 2013 Copyright © 2013 Agamenon R. E. Oliveira. This is an open access article distributed under the Creative Com- mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, pro- vided the original work is properly cited. In the first and second parts of his masterpiece, Analytical Mechanics, dedicated to static and dynamics respectively, Lagrange (1736-1813) describes in detail the development of both branches of mechanics from a historical point of view. In this paper this important contribution of Lagrange (Lagrange, 1989) to the history of mechanics is presented and discussed in tribute to the bicentennial year of his death. Keywords: History of Mechanics; Epistemology of Physics; Analytical Mechanics Introduction in the same meaning that is now understood by modern physi- cists. The logical unity of this theory is based on the least action Lagrange was one of the founders of variational calculus, in principle. However, the two dimensions of formalization and which the Euler-Lagrange equations were derived by him. He unification are the main characteristics of Lagrange’s method. also developed the method of Lagrange multipliers which is a manner of finding local maxima and minima of a function sub- jected to constraints. -
The Equations of Motion in a Rotating Coordinate System
The Equations of Motion in a Rotating Coordinate System Chapter 3 Since the earth is rotating about its axis and since it is convenient to adopt a frame of reference fixed in the earth, we need to study the equations of motion in a rotating coordinate system. Before proceeding to the formal derivation, we consider briefly two concepts which arise therein: Effective gravity and Coriolis force Effective Gravity g is everywhere normal to the earth’s surface Ω 2 R 2 R R Ω R g* g g* g effective gravity g effective gravity on an earth on a spherical earth with a slight equatorial bulge Effective Gravity If the earth were a perfect sphere and not rotating, the only gravitational component g* would be radial. Because the earth has a bulge and is rotating, the effective gravitational force g is the vector sum of the normal gravity to the mass distribution g*, together with a centrifugal force Ω2R, and this has no tangential component at the earth’s surface. gg*R= + Ω 2 The Coriolis force A line that rotates with the roundabout Ω A line at rest in an inertial system Apparent trajectory of the ball in a rotating coordinate system Mathematical derivation of the Coriolis force Representation of an arbitrary vector A(t) Ω A(t) = A (t)i + A (t)j + A (t)k k 1 2 3 k ´ A(t) = A1´(t)i ´ + A2´ (t)j ´ + A3´ (t)k´ inertial j reference system rotating j ´ reference system i ´ i The time derivative of an arbitrary vector A(t) The derivative of A(t) with respect to time d A dA dA dA a =+ijk12 + 3 dt dt dt dt the subscript “a” denotes the derivative in an inertial reference frame ddAdAi′ ′ In the rotating frame a1=++i′′A .. -
Exercises on Analytical Mechanics Problem Set Number 4
Institut fur¨ Physik WS 2012/2013 Friederike Schmid Exercises on Analytical Mechanics Problem set number 4 Questions on the lecture: 27. What do we mean by rigid body? 28. What are the Euler angles? 29. What is the characterizing property of a physical tensor? 30. Give the equation for the inertia tensor of a rigid body with mass distribution ρ(r). 31. What is the relation between the inertia tensor and the moment of inertia? 32. Give the general expressions for the angular momentum and the kinetic energy of a rigid body. Which is the translational contribution? Which the rotational contribution? 33. What are the principal moments of inertia? 34. What are the principal axes of inertia? Problems (To be dropped until 12:00 am on November 11th, 2012 in the red box number 40 at Staudingerweg 7) Problem 10) Jojo (6 Points) Consider a Jojo consisting of a cylinder of length L with radius R and mass M, which is attached to a string at its round side. The Jojo can roll up and down the string. It has the height h. a) Calculate the moment of inertia θ of the cylinder with respect to its long axis. b) Calculate the kinetic energy as a funciton of h˙ . (Don’t forget the rotational contribution!) c) Construct the Lagrange function L and determine the equations of motion. How does the solution look like? Problem 11) Lagrange parameters (6 Points) Use Lagrange parameters to solve the following problems. a) Determine the cuboid with largest volume V which can be fully contained in the ellipsoid x2 y2 z2 + + = 1. -
Chapter 8. Motion in a Noninertial Reference Frame
Chapter 8. Motion in a Noninertial Reference Frame (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 10.) We have so far dealt only with problems situated in inertial reference frame, or if not, problems that could be solved with enough accuracy by ignoring the noninertial nature of the coordinate systems. There are, however, many problems for which it is necessary, or beneficial, to treat the motion of the system at hand in a noninertial reference frame. In this chapter, we will develop the mathematical apparatus that will allow us to deal with such problems, and prepare the way for the study of the motion of rigid bodies that we will tackle in the next chapter. 8.1 Rotating Coordinate Systems Let’s consider two coordinate systems: one that is inertial and for which the axes are fixed, and another whose axes are rotating with respect to the inertial system. We represent the coordinates of the fixed system by xi! and the coordinates of the rotating system by xi . If we choose some point in space P (see Figure 8-1) we have r! = R + r, (8.1) where R locates the origin of the rotating system in the fixed system. We assume that P is at rest in the inertial so that r! is constant. If during an infinitesimal amount of time dt the rotating system undergoes an infinitesimal rotation d! about some axis, then the vector r will vary not only as measured by an observer co-moving with the rotating system, but also when measured in the inertial frame. -
The Origins of Analytical Mechanics in 18Th Century Marco Panza
The Origins of Analytical Mechanics in 18th century Marco Panza To cite this version: Marco Panza. The Origins of Analytical Mechanics in 18th century. H. N. Jahnke. A History of Analysis, American Mathematical Society and London Mathematical Society, pp. 137-153., 2003. halshs-00116768 HAL Id: halshs-00116768 https://halshs.archives-ouvertes.fr/halshs-00116768 Submitted on 29 Nov 2006 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Origins of Analytic Mechanics in the 18th century Marco Panza Last version before publication: H. N. Jahnke (ed.) A History of Analysis, American Math- ematical Society and London Mathematical Society, 2003, pp. 137-153. Up to the 1740s, problems of equilibrium and motion of material systems were generally solved by an appeal to Newtonian methods for the analysis of forces. Even though, from the very beginning of the century—thanks mainly to Varignon (on which cf. [Blay 1992]), Jean Bernoulli, Hermann and Eu- ler—these methods used the analytical language of the differential calculus, and were considerably improved and simplified, they maintained their essen- tial feature. They were founded on the consideration of a geometric diagram representing the mechanical system under examination, and consequently applied only to (simple) particular and explicitly defined systems.