Mathematics and physics preliminaries: of hills and plains and other things

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Calculus Of Variations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Space Research ◽  
Lagrange Equations ◽  
Virtual Displacement ◽  
Generalized Coordinates ◽  
Flat Region ◽  
Fixed End ◽  
Mathematics And Physics

The link between mathematics and physics is explained, and how the concepts “coordinates,” “generalized coordinates,” “time,” and “space” have evolved, starting with Galileo. It is also shown that “degrees of freedom” is a slippery but crucial idea. The important developments in “space research”, from Pythagoras to Riemann, are sketched. This is followed by the motivations for finding a flat region of “space”, and for Riemann’s invariant interval. A careful explanation of the three ways of taking an infinitesimal step (actual, virtual, and imperfect) is given. The programme of the Calculus of Variations is described and how this requires a virtual variation of a whole path, a path taken between fixed end-states. This then culminates in the Euler-Lagrange Equations or the Lagrange Equations of Motion. Along the way, the ideas virtual displacement and extremum are explained.

Elements of Classical Field Theory

2021 ◽  
pp. 24-34
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Classical Field ◽  
Lagrange Equations ◽  
Lie Group Of Transformations ◽  
Infinitesimal Transformations ◽  
Conserved Currents

The purpose of this chapter is to recall the principles of Lagrangian and Hamiltonian classical mechanics. Many results are presented without detailed proofs. We obtain the Euler–Lagrange equations of motion, and show the equivalence with Hamilton’s equations. We derive Noether’s theorem and show the connection between symmetries and conservation laws. These principles are extended to a system with an infinite number of degrees of freedom, i.e. a classical field theory. The invariance under a Lie group of transformations implies the existence of conserved currents. The corresponding charges generate, through the Poisson brackets, the infinitesimal transformations of the fields as well as the Lie algebra of the group.

Comparison between Analytical and Vectorial Methods of the Synthesis of Equations of Motion

1983 ◽  
Vol 197 (4) ◽  
pp. 265-274
Author(s):  
Z J Goraj
Keyword(s):  
Angular Momentum ◽  
Analytical Method ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Discrete Systems ◽  
Momentum Equation ◽  
Lagrange Equations ◽  
Weak Points ◽  
Complicated Geometry ◽  
Boltzmann Hamel Equations

In this paper the advantages and weak points of the analytical and vectorial methods of the derivation of equations of motion for discrete systems are considered. The analytical method is discussed especially with respect to Boltzmann-Hamel equations, as generalized Lagrange equations. The vectorial method is analysed with respect to the momentum equation and to the generalized angular momentum equation about an arbitrary reference point, moving in an arbitrary manner. It is concluded that, for the systems with complicated geometry of motion and a large number of degrees of freedom, the vectorial method can be more effective than the analytical method. The combination of the analytical and vectorial methods helps to verify the equations of motion and to avoid errors, especially in the case of systems with rather complicated geometry.

Automatic Generation of Equations of Motion of Mechanical Systems

2014 ◽  
Vol 611 ◽  
pp. 40-45
Author(s):  
Darina Hroncová ◽  
Jozef Filas
Keyword(s):  
Computer Simulation ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Automatic Generation ◽  
Lagrange Equations ◽  
Kinematic Chains ◽  
Transformation Matrices

The paper describes an algorithm for automatic compilation of equations of motion. Lagrange equations of the second kind and the transformation matrices of basic movements are used by this algorithm. This approach is useful for computer simulation of open kinematic chains with any number of degrees of freedom as well as any combination of bonds.

scholarly journals Generalized Lagrange-D’Alembert principle

2012 ◽  
Vol 91 (105) ◽  
pp. 49-58
Author(s):  
Djordje Djukic
Keyword(s):  
Dynamic Equilibrium ◽  
Virtual Work ◽  
Analytical Mechanics ◽  
Principle Of Virtual Work ◽  
Constraint Forces ◽  
Lagrange Equations ◽  
Virtual Displacement ◽  
Generalized Coordinates ◽  
D’Alembert Principle ◽  
Applied Forces

The major issues in the analysis of the motion of a constrained dynamic system are to determine this motion and calculate constraint forces. In the analytical mechanics, only the first of the two problems is analyzed. Here, the problem is solved simultaneously using: 1) Principle of liberation of constraints; 2) Principle of generalized virtual displacement; 3) Idea of ideal constraints; 4) Concept of generalized and ?supplementary" generalized coordinates. The Lagrange-D?Alembert principle of virtual work is generalized introducing virtual displacement as vectorial sum of the classical virtual displacement and virtual displacement in the ?supplementary" directions. From such principle of virtual work we derived Lagrange equations of the second kind and equations of dynamical equilibrium in the ?supplementary" directions. Constrained forces are calculated from the equations of dynamic equilibrium. At the same time, this principle can be used for consideration of equilibrium of system of material particles. This principle simultaneously gives the connection between applied forces at equilibrium state and the constrained forces. Finally, the principle is applied to a few particular problems.

scholarly journals Dynamical analysis of a 2-degrees of freedom spatial pendulum

2018 ◽  
Vol 184 ◽  
pp. 01003 ◽  
Author(s):  
Stelian Alaci ◽  
Florina-Carmen Ciornei ◽  
Sorinel-Toderas Siretean ◽  
Mariana-Catalina Ciornei ◽  
Gabriel Andrei Ţibu
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Dynamical Analysis ◽  
Analysis Software ◽  
Lagrange Equations ◽  
Moments Of Inertia ◽  
Dynamical Simulation ◽  
Principal Moments Of Inertia

A spatial pendulum with the vertical immobile axis and horizontal mobile axis is studied and the differential equations of motion are obtained applying the method of Lagrange equations. The equations of motion were obtained for the general case; the only simplifying hypothesis consists in neglecting the principal moments of inertia about the axes normal to the oscillation axes. The system of nonlinear differential equations was numerically integrated. The correctness of the obtained solutions was corroborated to the dynamical simulation of the motion via dynamical analysis software. The perfect concordance between the two solutions proves the rightness of the equations obtained.

scholarly journals Classical n-body system in geometrical and volume variables: I. Three-body case

2021 ◽  
pp. 2150140
Author(s):  
A. M. Escobar-Ruiz ◽  
R. Linares ◽  
Alexander V. Turbiner ◽  
Willard Miller
Keyword(s):  
Kinetic Energy ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Symmetric Polynomial ◽  
Body System ◽  
Generalized Coordinates ◽  
New Variables ◽  
Three Body ◽  
Mass Dependent

We consider the classical three-body system with [Formula: see text] degrees of freedom [Formula: see text] at zero total angular momentum. The study is restricted to potentials [Formula: see text] that depend solely on relative (mutual) distances [Formula: see text] between bodies. Following the proposal by J. L. Lagrange, in the center-of-mass frame we introduce the relative distances (complemented by angles) as generalized coordinates and show that the kinetic energy does not depend on [Formula: see text], confirming results by Murnaghan (1936) at [Formula: see text] and van Kampen–Wintner (1937) at [Formula: see text], where it corresponds to a 3D solid body. Realizing [Formula: see text]-symmetry [Formula: see text], we introduce new variables [Formula: see text], which allows us to make the tensor of inertia nonsingular for binary collisions. In these variables the kinetic energy is a polynomial function in the [Formula: see text]-phase space. The three-body positions form a triangle (of interaction) and the kinetic energy is [Formula: see text]-permutationally invariant with respect to interchange of body positions and masses (as well as with respect to interchange of edges of the triangle and masses). For equal masses, we use lowest order symmetric polynomial invariants of [Formula: see text] to define new generalized coordinates, they are called the geometrical variables. Two of them of the lowest order (sum of squares of sides of triangle and square of the area) are called volume variables. It is shown that for potentials which depend on geometrical variables only (i) and those which depend on mass-dependent volume variables alone (ii), the Hamilton’s equations of motion can be considered as being relatively simple. We study three examples in some detail: (I) three-body Newton gravity in [Formula: see text], (II) three-body choreography in [Formula: see text] on the algebraic lemniscate by Fujiwara et al., where the problem becomes one-dimensional in the geometrical variables and (III) the (an)harmonic oscillator.

scholarly journals Minimization of dynamic joint reaction forces of the 2-DOF serial manipulators based on interpolating polynomials and counterweights

2015 ◽  
Vol 42 (4) ◽  
pp. 249-260 ◽  
Author(s):  
Slavisa Salinic ◽  
Marina Boskovic ◽  
Radovan Bulatovic
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Lagrange Equations ◽  
Symbolic Form ◽  
Serial Manipulators ◽  
Joint Reaction Forces ◽  
Reaction Forces ◽  
Inertia Forces ◽  
Joint Reaction ◽  
Selection Of

This paper presents two ways for the minimization of joint reaction forces due to inertia forces (dynamic joint reaction forces) in a two degrees of freedom (2-DOF) planar serial manipulator. The first way is based on the optimal selection of the angular rotations laws of the manipulator links and the second one is by attaching counterweights to the manipulator links. The influence of the payload carrying by the manipulator on the dynamic joint reaction forces is also considered. The expressions for the joint reaction forces are obtained in a symbolic form by means of the Lagrange equations of motion. The inertial properties of the manipulator links are represented by dynamical equivalent systems of two point masses. The weighted sum of the root mean squares of the magnitudes of the dynamic joint reactions is used as an objective function. The effectiveness of the two ways mentioned is discussed.

Effects of Temperature and Cross Section Variations on the Vibrational Behavior of Finished Wire Blocks

2014 ◽  
Vol 622-623 ◽  
pp. 95-102
Author(s):  
Mohammadzaman Savari ◽  
Paul Josef Mauk ◽  
Oberrat Bernhardt Weyh
Keyword(s):  
Product Quality ◽  
Torsional Vibration ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Equation System ◽  
Rolling Process ◽  
Essential Elements ◽  
Lagrange Equations ◽  
Cross Sectional ◽  
Coupled Equations

Because of the great speed range in a finishing block of a wire rod mill, the reduction of torsional vibrations makes it possible to achieve closer rolling tolerances. The components transmitting the torque like gear box, coupling etc. can generate non-smooth or alternating torques which affect the product quality. To study the influences of torsion vibrations on the product quality, the dynamic interactions of the block and rolling process are simultaneously analyzed by a simulation model. As an example, a three stand arrangement is considered. The real transmission system is idealized as a structurally discrete torsional vibration model. The generalized rotational coordinates with a large number of rotational degrees of freedom can be reduced among others constants by means of gear ratios. Euler-Lagrange equations are applied to create the coupled equations of motion, which together constitute an ordinary differential equation of order 28. The rolling and main drive torques are defined as excitation for vibration on the right side of the equation system. A DC motor is selected as main drive and the voltage circuit equation of motor is integrated into the system of differential equations. The armature current and its interaction are consequently simulated. The rotational speed of rolls and motor, as well as roll torques, longitudinal stresses in rod and section widths are shown in diagrams as the result and thereby the torsional vibration of the essential elements of the system are studied for different temperatures and cross-sectional variations.

scholarly journals Quadrotor UAV Control for Transportation of Cable Suspended Payload

2019 ◽  
Vol 26 (2) ◽  
pp. 77-84 ◽  
Author(s):  
Tom Kusznir ◽  
Jarosław Smoczek
Keyword(s):  
Degrees Of Freedom ◽  
Sliding Mode ◽  
Equations Of Motion ◽  
Lagrange Equations ◽  
Wide Range ◽  
Time Required ◽  
Interest In Research ◽  
Emergency Equipment ◽  
Pendulum Dynamics ◽  
4 Degrees Of Freedom

Abstract Payload transportation with UAV’s (Unmanned Aerial Vehicles) has become a topic of interest in research with possibilities for a wide range of applications such as transporting emergency equipment to otherwise inaccessible areas. In general, the problem of transporting cable suspended loads lies in the under actuation, which causes oscillations during horizontal transport of the payload. Excessive oscillations increase both the time required to accurately position the payload and may be detrimental to the objects in the workspace or the payload itself. In this article, we present a method to control a quadrotor with a cable suspended payload. While the quadrotor itself is a nonlinear system, the problem of payload transportation with a quadrotor adds additional complexities due to both input coupling and additional under actuation of the system. For simplicity, we fix the quadrotor to a planar motion, giving it a total of 4 degrees of freedom. The quadrotor with the cable suspended payload is modelled using the Euler-Lagrange equations of motion and then partitioned into translation and attitude dynamics. The design methodology is based on simplifying the system by using a variable transformation to decouple the inputs, after which sliding mode control is used for the translational and pendulum dynamics while a feedback linearizing controller is used for the rotational dynamics of the quadrotor. The sliding mode parameters are chosen so stability is guaranteed within a certain region of attraction. Lastly, the results of the numerical simulations created in MATLAB/Simulink are presented to verify the effectiveness of the proposed control strategy.

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