DOCSLIB.ORG

Hamel's Formalism and Variational Integrators on a Sphere

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hamel's Formalism and Variational Integrators on a Sphere 51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA Hamel’s Formalism and Variational Integrators on a Sphere Dmitry V. Zenkov, Melvin Leok, and Anthony M. Bloch Abstract— This paper discusses Hamel’s formalism and its The paper is organized as follows. Hamel’s formalism applications to structure-preserving integration of mechanical and its discretization are briefly discussed in Sections II systems. It utilizes redundant coordinates in order to elim- and III. The dynamics of a spherical pendulum is reviewed inate multiple charts on the configuration space as well as nonphysical artificial singularities induced by local coordinates, in Section IV. The discrete model for the pendulum based on while keeping the minimal possible degree of redundancy and Hamel’s formalism, its comparison to some other discretiza- avoiding integration of differential-algebraic equations. tion techniques, and simulations are given in Sections V, VI, and VII. I. INTRODUCTION This paper introduces a new variational integrator for a II. LAGRANGIAN MECHANICS spherical pendulum. The configuration space for this pendu- A. The Euler–Lagrange Equations lum is a two-dimensional sphere. Calculations in spherical A Lagrangian mechanical system is specified by a smooth coordinates are not a good option because of unavoidable manifold Q called the configuration space and a function artificial singularities introduced by these coordinates at the L : TQ ! R called the Lagrangian. In many cases, poles. In addition, the topology of a sphere makes it impos- the Lagrangian is the kinetic minus potential energy of the sible to use global singularity-free intrinsic coordinates. system, with the kinetic energy defined by a Riemannian In order to avoid the issues mentioned above, an integrator metric and the potential energy being a smooth function on that utilizes the interpretation of a sphere as a homogeneous the configuration manifold Q. If necessary, non-conservative space was introduced in [15]. This integrator performs very forces can be introduced (e.g., gyroscopic forces that are well, but has a somewhat large degree of redundancy. This represented by terms in L that are linear in the velocity), but paper targets the development of an integrator whose perfor- this is not discussed in detail in this paper. mance is similar to that of the integrator in [15] and whose In local coordinates q = (q1; : : : ; qn) on the configuration redundancy is the minimum possible. space Q we write L = L(q; q_). The dynamics is given by Both the present paper and [15] interpret the pendulum the Euler–Lagrange equations as a rotating rigid body. The algorithm introduced in [15] is d @L @L based on the evaluation of the rotation matrix that represents = ; i = 1; : : : ; n: (1) the attitude for this body. The key feature of the dynamics dt @q_i @qi utilized in the present paper is that, in order to capture the These equations were originally derived by Lagrange [14] orientation of a rigid body, it is sufficient to evaluate just in 1788 by requiring that simple force balance F = ma one column of that rotation matrix. The resulting equations be covariant, i.e. expressible in arbitrary generalized coordi- of motion are interpreted as Hamel’s equations written in nates. A variational derivation of the Euler–Lagrange equa- redundant coordinates. tions, namely Hamilton’s principle (also called the principle The general exposition of discrete Hamel’s formalism will of critical action), came later in the work of Hamilton [12] be a subject of a future publication. Here we demonstrate and [13] in 1834/35. For more details, see [4], [18], and the usefulness of some of this formalism by constructing Theorem 2.1 below. an integrator for a spherical pendulum that is energy- and momentum-preserving. The calculations are carried out in B. The Hamel Equations the Cartesian coordinates of the three-dimensional Euclidean In this paragraph we briefly discuss the Hamel equations. space. This allows one to avoid singularities and/or multiple The exposition follows paper [5]. coordinate charts that are inevitable for calculations on a In many cases the Lagrangian and the equations of motion sphere. Hamel’s approach allows one, among other things, have a simpler structure when written using velocity compo- to represent the dynamics in such a way that the length nents measured against a frame that is unrelated to system’s constraint becomes unnecessary. Thus, one avoids the well- local configuration coordinates. An example of such a system known difficulty of numerically solving differential-algebraic is the rigid body. equations. Let q = (q1; : : : ; qn) be local coordinates on the config- uration space Q and ui 2 TQ, i = 1; : : : ; n, be smooth D. V. Zenkov is with the Department of Mathematics, North Carolina independent local vector fields defined in the same coordi- [email protected] State University, Raleigh, NC 27695, USA, 1 M. Leok is with the Department of Mathematics, University of California nate neighborhood. The components of ui relative to the San Diego, La Jolla, CA 92093, USA, [email protected] 1 A. M. Bloch is with the Department of Mathematics, University of In certain cases, some or all of ui can be chosen to be global vector Michigan, Ann Arbor, MI 48109, USA, [email protected] fields on Q. 978-1-4673-2064-1/12/$31.00 ©2012 IEEE 7504 j j basis @=@q will be denoted i ; that is, on the space of curves in Q connecting qa to qb on the @ interval [a; b], where we choose variations of the curve u (q) = j(q) ; q(t) that satisfy δq(a) = δq(b) = 0. i i @qj (ii) The curve q(t) satisfies the Euler–Lagrange equa- where i; j = 1; : : : ; n and where summation on j is under- tions (1). stood. (iii) The curve (q(t); ξ(t)) is a critical point of the functional Let ξ = (ξ1; : : : ; ξn) 2 n be the components of the R Z b velocity vector q_ 2 TQ relative to the basis u1; : : : ; un, i.e., l(q; ξ) dt (7) a q_ = ξiu (q); (2) i with respect to variations δξ, induced by the variations i then δq = η ui(q), and given by i l(q; ξ) := L(q; ξ ui(q)) (3) k k k i j δξ =η _ + cij(q)ξ η : (8) is the Lagrangian of the system written in the local coor- The curve (q(t); ξ(t)) satisfies the Hamel equations dinates (q; ξ) on the tangent bundle TQ. The coordinates (iv) (5) coupled with the equations q_ = ξiu (q): (q; ξ) are Lagrangian analogues of non-canonical variables i in Hamiltonian dynamics. For the proof of Theorem 2.1 and the early development Define the quantities cm(q) by the equations and history of these equations, as well as other variational ij structures associated with Hamel’s equations see [22], [10], m [ui(q); uj(q)] = cij (q)um(q); (4) [5], and [2]. i; j; m = 1; : : : ; n. These quantities vanish if and only if III. THE DISCRETE HAMEL EQUATIONS the vector fields ui(q), i = 1; : : : ; n, commute. Here and A. Discrete Hamilton’s Principle elsewhere, [ · ; · ] is the Jacobi–Lie bracket of vector fields A discrete analogue of Lagrangian mechanics can be on Q. obtained by discretizing Hamilton’s principle; this approach Viewing u as vector fields on TQ whose fiber components i underlies the construction of variational integrators. See equal 0 (that is, taking the vertical lift of these vector fields), Marsden and West [19], and references therein, for a more one defines the directional derivatives u [l] for a function i detailed discussion of discrete mechanics. l : TQ ! by the formula R A key notion is that of the discrete Lagrangian, which is j @l a map Ld : Q×Q ! that approximates the action integral ui[l] = : R i @qj along an exact solution of the Euler–Lagrange equations The evolution of the variables (q; ξ) is governed by the joining the configurations qk; qk+1 2 Q, Hamel equations Z h d L (qk; qk+1) ≈ ext L(q; q_) dt; (9) d @l m @l i q2C([0;h];Q) 0 j = cij m ξ + uj[l]: (5) dt @ξ @ξ where C([0; h];Q) is the space of curves q : [0; h] ! Q with i coupled with equations (2). If ui = @=@q , equations (5) q(0) = qk, q(h) = qk+1, and ext denotes extremum. become the Euler–Lagrange equations (1). In the discrete setting, the action integral of Lagrangian Equations (5) were introduced in [10] (see also [21] for mechanics is replaced by an action sum details and some history). N−1 d X d C. Hamilton’s Principle for Hamel’s Equations S (q0; q1; : : : ; qN ) = L (qk; qk+1); k=0 Let γ :[a; b] ! Q be a smooth curve in the configuration where q 2 Q, k = 0; 1;:::;N, is a finite sequence space. A variation of the curve γ(t) is a smooth map β : k in the configuration space. The equations are obtained by [a; b] × [−"; "] ! Q that satisfies the condition β(t; 0) = the discrete Hamilton’s principle, which extremizes the γ(t). This variation defines the vector field discrete action given fixed endpoints q and q . Taking 0 N @β(t; s) the extremum over q ; : : : ; q gives the discrete Euler– δγ(t) = 1 N−1 @s s=0 Lagrange equations d d along the curve γ(t). D1L (qk; qk+1) + D2L (qk−1; qk) = 0; Theorem 2.1: Let L : TQ ! R be a Lagrangian and for k = 1;:::;N −1. This implicitly defines the update map l : TQ ! R be its representation in local coordinates (q; ξ).
Load more

Recommended publications
  • Hamiltonian Mechanics
    Hamiltonian Mechanics Marcello Seri Bernoulli Institute University of Groningen m.seri (at) rug.nl Version 1.0 May 12, 2020 This work is licensed under a Creative Commons “Attribution-NonCommercial-ShareAlike 4.0 Inter- national” license. Contents 1 Classical mechanics, from Newton to Lagrange and back 3 1.1 Newtonian mechanics .................... 3 1.1.1 Motion in one degree of freedom ........... 5 1.2 Hamilton’s variational principle ................ 7 1.2.1 Dynamics of point particles: from Lagrange back to Newton 13 1.3 Euler-Lagrange equations on smooth manifolds ........ 18 1.4 First steps with conserved quantities ............. 24 1.4.1 Back to one degree of freedom ............ 24 1.4.2 The conservation of energy .............. 25 1.4.3 Back to one degree of freedom, again ......... 27 2 Conservation Laws 34 2.1 Noether theorem ....................... 34 2.1.1 Homogeneity of space: total momentum ....... 38 2.1.2 Isotropy of space: angular momentum ......... 40 2.1.3 Homogeneity of time: the energy ........... 44 2.1.4 Scale invariance: Kepler’s third law .......... 44 2.2 The spherical pendulum ................... 45 2.3 Intermezzo: small oscillations ................. 47 2.3.1 Decomposition into characteristic oscillations ..... 50 2.3.2 The double pendulum ................. 51 2.3.3 The linear triatomic molecule ............. 52 2.3.4 Zero modes ...................... 53 2.4 Motion in a central potential ................. 54 2.4.1 Kepler’s problem ................... 57 2.5 D’Alembert principle and systems with constraints ...... 62 3 Hamiltonian mechanics 67 3.1 The Legendre Transform in the euclidean plane ........ 68 3.2 The Legendre Transform ................... 69 3.3 Poisson brackets and first integrals of motion ........
    [Show full text]
  • Solution Midterm
    Midterm Physics 411 03.09.2018 Mechanics I 1. Consider the following statement: If at all times during a projectile's flight its speed is much les than the terminal speed vter, the effects of the air resistance aew usually very small. (a) Without reference to the explicit equations for the magnitude of vter, explain clearly why this is so. SOLUTION - Without air resistance, a free falling projectile would keep acceler- ating due to gravity, meaning its speed would only get bigger with time. Because there is air resistance, that doesn't happen: as the speed increases so does the air resistance, stalling the projectile. Terminal speed happens when when gravity is matched by the air resistance effects, thus once the projectile achieves vter its speed remains constant, ultimately working as an upper limit for speed. If the speed is much lower than this cap, air resistance effects won't be as relevant, as the drag should be much less than the weight. (b) By examining the explicit equations in the book (2.26) and (2.53) explain why the statement above is even more useful for the case of quadratic drag than for the lienar case. [Hint: Express the ratio f=mg of the drag to the weight in terms of the ratio v=vter.] SOLUTION - Writing the ratio of linear and quadratic drag to the weight, we have 2 jflinearj = bv; jfquadj = cv (1) 2 2 flinear bv v fquad cv v = = ; = = 2 (2) mg mg vter mg mg vter From the ratio, we have that a small value v wil result in a more dramatic change in the quadratic drag given its squared dependence on v, making the statement even more useful - for a v that is far from the terminal velocity the quadratic drag corrections would be even smaller.
    [Show full text]
  • Notes: Spherical Pendulum
    M3-4-5 A16 Notes for Geometric Mechanics: Oct{Nov 2011 Professor Darryl D Holm 25 October 2011 Imperial College London [email protected] http://www.ma.ic.ac.uk/~dholm/ Text for the course: Geometric Mechanics I: Dynamics and Symmetry, by Darryl D Holm World Scientific: Imperial College Press, Singapore, Second edition (2011). ISBN 978-1-84816-195-5 Where are we going in this course? 1. Euler{Poincar´eequation 4. Elastic spherical pendulum 2. Rigid body 3. Spherical pendulum 5. Differential forms e These notes: Spherical pendulum 1 Notes for Geometric Mechanics: Spherical pendulum DD Holm Nov 2011 2 What will we study about the spherical pendulum? 1. Newtonian, Lagrangian and Hamilto- 5. Reduced Poisson bracket in R3 nian formulations of motion equation. 6. Nambu Hamiltonian form 2. S1 symmetry and Noether's theorem 3. Lie symmetry reduction 7. Geometric solution interpretation 4. Reduced Poisson bracket in R3 8. Geometric phase 1 Spherical pendulum A spherical pendulum of unit length swings from a fixed point of support under the constant acceleration of gravity g (Figure 2.7). This motion is equivalent to a particle of unit mass moving on the surface of the unit sphere S2 under the influence of the gravitational (linear) potential V (z) with z = ^e3 · x. The only forces acting on the mass are the reaction from the sphere and gravity. This system is often treated by using spherical polar coordinates and the traditional methods of Newton, Lagrange and Hamilton. The present treatment of this problem is more geometrical and avoids polar coordinates.
    [Show full text]
  • Physics 6010, Fall 2016 Constraints and Lagrange Multipliers. Relevant Sections in Text: §1.3–1.6
    Constraints and Lagrange Multipliers. Physics 6010, Fall 2016 Constraints and Lagrange Multipliers. Relevant Sections in Text: x1.3{1.6 Constraints Often times we consider dynamical systems which are defined using some kind of restrictions on the motion. For example, the spherical pendulum can be defined as a particle moving in 3-d such that its distance from a given point is fixed. Thus the true configuration space is defined by giving a simpler (usually bigger) configuration space along with some constraints which restrict the motion to some subspace. Constraints provide a phenomenological way to account for a variety of interactions between systems. We now give a systematic treatment of this idea and show how to handle it using the Lagrangian formalism. For simplicity we will only consider holonomic constraints, which are restrictions which can be expressed in the form of the vanishing of some set of functions { the constraints { on the configuration space and time: Cα(q; t) = 0; α = 1; 2; : : : m: We assume these functions are smooth and independent so that if there are n coordinates qi, then at each time t the constraints restrict the motion to a nice n − m dimensional space. For example, the spherical pendulum has a single constraint on the three Cartesian configruation variables (x; y; z): C(x; y; z) = x2 + y2 + z2 − l2 = 0: This constraint restricts the configuration to a two dimensional sphere of radius l centered at the origin. To see another example of such constraints, see our previous discussion of the double pendulum and pendulum with moving point of support.
    [Show full text]
  • Hamiltonian Dynamics
    Chapter 5. Hamiltonian Dynamics (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7) 5.1 The Canonical Equations of Motion As we saw in section 4.7.4, the generalized momentum is defined by !L p j = . (5.1) !q! j We can rewrite the Lagrange equations of motion !L d !L " = 0 (5.2) !q dt !q! j j as follow !L p! j = . (5.3) !q j We can also express our earlier equation for the Hamiltonian (i.e., equation (4.89)) using equation (5.1) as H = p q! ! L. (5.4) j j Equation (5.4) is expressed as a function of the generalized momenta and velocities, and the Lagrangian, which is, in turn, a function of the generalized coordinates and velocities, and possibly time. There is a certain amount of redundancy in this since the generalized momenta are obviously related to the other two variables through equation (5.1). We can, in fact, invert this equation to express the generalized velocities as q! = q! q , p ,t . (5.5) j j ( k k ) This amounts to making a change of variables from q ,q! ,t to q , p ,t ; we, therefore, ( j j ) ( j j ) express the Hamiltonian as H (qk , pk ,t) = p jq! j ! L(qk ,q!k ,t) , (5.6) 91 where it is understood that the generalized velocities are to be expressed as a function of the generalized coordinates and momenta through equation (5.5). The Hamiltonian is, therefore, always considered a function of the (qk , pk ,t) set, whereas the Lagrangian is a function of the (qk ,q!k ,t) set: H = H (qk , pk ,t) (5.7) L = L(qk ,q!k ,t) Let’s now consider the following total differential " !H !H % !H (5.8) dH = $ dqk + dpk ' + dt, # !qk !pk & !t but from equation (5.6) we can also write # "L "L & "L dH = q! dp + p dq! ! dq ! dq! ! dt.
    [Show full text]
  • Online Stereo 3D Simulation in Studying the Spherical Pendulum in Conservative Force Field
    European J of Physics Education Vol.4 Issue 3 2013 Zabunov Online Stereo 3D Simulation in Studying the Spherical Pendulum in Conservative Force Field Svetoslav S. Zabunov Sofia University, Bulgaria [email protected] Abstract The current paper aims at presenting a modern e-learning method and tool that is utilized in teaching physics in the universities. An online stereo 3D simulation is used for e-learning mechanics and specifically the teaching of spherical pendulum as part of the General Physics course for students in the universities. This approach was realized on bases of interactive simulations on a personal computer, a part of the free online e-learning system at http://ialms.net/sim/. This system was practically applied with students at Sofia University, Bulgaria, among others. The shown simulation demonstrates the capabilities of the Web for online representations and visualizations of simulated physics processes that are hard to observe in laboratory conditions with all the accompanying parameters, vectors, quantities and trajectories. The discussed simulation allows the study of spherical pendulum both in conservative and non-conservative force fields. The conservative force field is created by the earth gravity force, whose magnitude may be varied in the simulation from positive to negative values, while its direction is always vertical. The simulation also supports various non-conservative forces that may be applied to the pendulum. The current article concentrates on the case of conservative forces acting. Keywords: Simulation of spherical pendulum, 3D-stereo simulation, E-learning physics through stereo 3D visualization. Introduction This article describes the capabilities of the modern web technologies through online stereo 3D simulations of physics phenomena.
    [Show full text]
  • Research Article 3 DOF Spherical Pendulum Oscillations with a Uniform Slewing Pivot Center and a Small Angle Assumption
    Hindawi Publishing Corporation Shock and Vibration Volume 2014, Article ID 203709, 32 pages http://dx.doi.org/10.1155/2014/203709 Research Article 3 DOF Spherical Pendulum Oscillations with a Uniform Slewing Pivot Center and a Small Angle Assumption Alexander V. Perig,1 Alexander N. Stadnik,2 Alexander I. Deriglazov,2 and Sergey V. Podlesny2 1 Manufacturing Processes and Automation Engineering Department, Engineering Automation Faculty, Donbass State Engineering Academy, Shkadinova 72, Donetsk Region, Kramatorsk 84313, Ukraine 2 Department of Technical Mechanics, Engineering Automation Faculty, Donbass State Engineering Academy, Shkadinova72,DonetskRegion,Kramatorsk84313,Ukraine Correspondence should be addressed to Alexander V. Perig; [email protected] Received 11 February 2014; Revised 7 May 2014; Accepted 21 May 2014; Published 4 August 2014 Academic Editor: Mickael¨ Lallart Copyright © 2014 Alexander V. Perig et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The present paper addresses the derivation of a 3 DOF mathematical model of a spherical pendulum attached to a crane boom tip for uniform slewing motion of the crane. The governing nonlinear DAE-based system for crane boom uniform slewing has been proposed, numerically solved, and experimentally verified. The proposed nonlinear and linearized models have been derived with an introduction of Cartesian coordinates. The linearizedodel m with small angle assumption has an analytical solution. The relative and absolute payload trajectories have been derived. The amplitudes of load oscillations, which depend on computed initial conditions, have been estimated. The dependence of natural frequencies on the transport inertia forces and gravity forces has been computed.
    [Show full text]
  • Phys 7221 Homework #2
    Phys 7221 Homework #2 Gabriela Gonz´alez September 18, 2006 1. Derivation 1-9: Gauge transformations for electromagnetic potential If two Lagrangians differ by a total derivative of a function of coordinates and time, they lead to the same equation of motion: dF (qi, t) ∂F X ∂F L0(q , q˙ , t) = L(q , q˙ , t) + = L(q , q˙ , t) + + q˙ i i i i dt i i ∂t ∂q j j j as proven in the following lines: ∂L0 ∂L0 ∂F = + ∂q˙i ∂q˙i ∂qi ∂L0 ∂L ∂ dF = + ∂qi ∂qi ∂qi dt d ∂L0 ∂L0 d ∂L ∂L d ∂F ∂ dF − = − + − dt ∂q˙i ∂qi dt ∂q˙i ∂qi dt ∂qi ∂qi dt d ∂F ∂ X ∂ ∂F = + q˙ dt ∂q ∂t j ∂q ∂q i j j i ∂ ∂F X ∂F = + q˙ ∂q ∂t j ∂q i j j ∂ dF = ∂qi dt and thus the equations of motion are the same, derived from either Lagrangian: d ∂L0 ∂L0 d ∂L ∂L − = − dt ∂q˙i ∂qi dt ∂q˙i ∂qi 1 (What goes wrong if F = F (q, q,˙ t)?) A particle moving in an electromagnetic field, with scalar potential φ and vector potential A, has a generalized potential function U = φ − A · v and a Lagrangian 1 L = T − U = mv2 − q (φ − A · v) 2 Consider a gauge transformation for the scalar and vector potentials of the form ∂ψ(r, t) φ0 = φ − ∂t A0 = A + ∇ψ(r, t) The change in potential energy under this transformation is U 0 = φ0 − A0 · v ∂ψ = U − q + ∇ψ · v ∂t ∂ψ dr = U − q + ∇ψ · ∂t dt ∂ψ X ∂ψ dxi = U − q + ∂t ∂xi dt dψ = U − q dt and thus, the new Lagrangian differs from the original one by a total derivative: dψ d(qψ) L0 = T − U 0 = T − U + q = L + dt dt and thus we know the equations of motion are the same.
    [Show full text]
  • Animating Spherical Pendulum Using Jacobi's Elliptic Functions
    Animating Spherical Pendulum Using Jacobi's Elliptic Functions A.C. Baroudy [email protected] Abstract Some years ago I have written a Maple document ( already on Maple's online home page) on the subject of animating a simple pendulum for large angles of oscillation. This gave me the chance to test Maple command JacobiSN(time, k). I was very much pleased to see Maple do a wonderful job in getting these Jacobi's elliptic functions without a glitch. Today I am back to these same functions for a similar purpose though much more sophisticated than the previous one. The idea is: 1- to get the differential equations of motion for the Spherical Pendulum (SP), 2- to solve them, 3- to use Maple for finding the inverse of these Elliptic Integrals i.e. finding the displacement z as function of time, 4- to get a set of coordinates [x, y, z] for the positions of the bob at different times for plotting, 5- finally to work out the necessary steps for the purpose of animation. It turns out that even with only 3 oscillations where each is defined with only 20 positions of the bob for a total of 60 points on the graph, the animation is so overwhelming that Maple reports: " the length of the output exceeds 1 million". Not withstanding this warning, Maple did a perfect job by getting the animation to my satisfaction. Note that with only 60 positions of the bob, the present article length is equal to 11000 KB! To be able to upload it, I have to save it without running the last command related to the animation.
    [Show full text]
  • Chapter 1 HW Solution
    ME 516 Spring 2005 Chapter 1 HW Solution Problem 4. Using fixed Cartesian coordinates, the acceleration of the particle is given by a =x ¨i +y ¨j (1) The approach is to integratex ¨ to getx ˙ then x; then gety ˙,y ¨. Use (1.3.15) to get radius of curvature ρ. Sincex ¨(t) = 1 + 0.2t, integration plus initial conditions yields x˙(t) = t + 0.1t2 (2) t2 0.1t3 x(t) = + − 0.6 (3) 2 3 Then since the constraint equation is y = −x2 + x, we have (using the chain rule) dy dx y˙(t) = = (1 − 2x)x ˙ (4) dx dt d d dx d dx˙ y¨(t) = [(1 − 2x)x ˙] = [·] + [·] = −2x ˙ 2 + (1 − 2x)¨x (5) dt dx dt dx˙ dt Evaluate bothx ¨(t) andy ¨(t) at t = 0.5, and you get a =x ¨i +y ¨j = 1.1i + 1.5845j (6) Radius of curvature is given by (note y0 = dy/dx and y00 = d2y/dx2) 3/2 1 + y02 ρ = = 5.209 (7) |y00| Problem 8. (a) Here we have hθ z = (8) 10π which gives position vector hθ r = Re + k (9) R 10π Note that R is constant, and one can either differentiate r with respect to time or use text equations [1.3.47] and [1.3.48] to find velocity and acceleration. The results are hθ˙ v = Rθ˙e + k (10) θ 10π hθ¨ a = −Rθ˙2e + Rθ¨e + k (11) r θ 10π (b) The normal unit vector is opposite the radial unit vector,while the tangent unit vector is upward along the vehicle path, so en = −er (12) et = cos φeθ + sin φk (13) 1 ME 516 Spring 2005 where h φ = tan−1 (14) 10πR Radius of curvature ρ is found using the fact that along a curved path the normal acceleration is related to radius of curvature and velocity.
    [Show full text]
  • Eindhoven University of Technology BACHELOR the Spherical
    Eindhoven University of Technology BACHELOR The spherical pendulum different solutions on a 2-sphere Dohmen, Jesse L.P. Award date: 2017 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain The Spherical Pendulum: Different Solutions on a 2-Sphere Bachelor Final Project Applied Mathematics Eindhoven University of Technology Supervisor: J.C. van der Meer by Jesse Dohmen 0899975 August 13, 2017 1 1 Introduction A well-known problem from mathematical physics is the simple (mathematical) pendulum, in which the movement of a pendulum through a plane is described. The trajectory of the pendulum is along a circle. By generalising this movement to a 3-dimensional space, we obtain the spherical pendulum.
    [Show full text]
  • Plane and Spherical Pendulums
    Plane and Spherical Pendulums Spherical Pendulum All spherical pendulums are similar to each other. Upon appropriate non-dimensionalization of time and action, they are described by one and the same dimensionless Lagrangian θ_2 '_ 2 L = + sin2 θ + cos θ ; (1) 2 2 where θ and ' are the polar and azimuthal angles, respectively. Consistent with the axial rotational symmetry of the problem, the variable ' is cyclic and corresponding equation of motion yields the law of conservation of the z-component of the (non-dimesionalized) angular momentum, lz: 2 '_ sin θ = lz : (2) This provides us with a simple relation between '(t) and θ(t): l Z dt '_ = z , '(t) = l : (3) sin2 θ z sin2 θ(t) This relation (and many other relations as well, as we will see in what follows) reveals a deep qualitative analogy between the problem of spherical pendulum and the problem of a two-dimensional particle performing a bound motion in a central potential. In both cases, we have a cyclic (azimuthal) angle associated with the rotational symmetry; the behavior of pendulums's polar angle θ is qualitatively equivalent to that of the polar radius of the 2D particle trapped in a central potential.1 The Lagrange equation for θ is θ¨ =' _ 2 sin θ cos θ − sin θ : (4) Relation (3) allows us to exclude' _ from (4), thus getting a closed equation of motion for θ(t). It is both convenient and instructive to directly arrive at the first integral of the resulting closed equation of motion from the conservation of energy, in a direct analogy with our treatment of a 2D particle in a central potential.
    [Show full text]