D ‘Alembert ’s Principle and Lagrange Equations of Motion

1991 ◽  
pp. 565-590
Author(s):  
Franz Ziegler
Keyword(s):  
Equations Of Motion ◽  
Lagrange Equations

scholarly journals A least action principle for interceptive walking

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Soon Ho Kim ◽  
Jong Won Kim ◽  
Hyun Chae Chung ◽  
MooYoung Choi
Keyword(s):  
Social Systems ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Action Principle ◽  
Lagrange Equations ◽  
Least Action Principle ◽  
Least Action ◽  
The Least Action Principle ◽  
Principle Of Least Action ◽  
Human Participants

AbstractThe principle of least effort has been widely used to explain phenomena related to human behavior ranging from topics in language to those in social systems. It has precedence in the principle of least action from the Lagrangian formulation of classical mechanics. In this study, we present a model for interceptive human walking based on the least action principle. Taking inspiration from Lagrangian mechanics, a Lagrangian is defined as effort minus security, with two different specific mathematical forms. The resulting Euler–Lagrange equations are then solved to obtain the equations of motion. The model is validated using experimental data from a virtual reality crossing simulation with human participants. We thus conclude that the least action principle provides a useful tool in the study of interceptive walking.

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.

Stability and Accuracy Analysis of Baumgarte’s Constraint Violation Stabilization Method

1995 ◽  
Vol 117 (3) ◽  
pp. 446-453 ◽  
Author(s):  
S. Yoon ◽  
R. M. Howe ◽  
D. T. Greenwood
Keyword(s):  
Equations Of Motion ◽  
Integration Step ◽  
State Variables ◽  
Stabilization Method ◽  
Lagrange Equations ◽  
Step Size ◽  
Constraint Violation ◽  
Truncation Errors ◽  
Numerical Stability Analysis ◽  
Stability And Accuracy

When Baumgarte’s Constraint Violation Stabilization Method (CVSM) is used in the simulation of Lagrange equations of motion with holonomic constraints, it is shown that, with suitable assumptions on the integration step size h and the eigenvalues (λ’s) of the linearized system, the constraint variables are effectively integrated by the same algorithm as that used for the state variables. A numerical stability analysis of the constraint violations can be performed using this so-called pseudo-integration equation. A study is also made of truncation errors and their modeling in the continuous time domain. This model can be used to determine the effectiveness of various constraint controls and integration methods in reducing the errors in the solution due to truncation errors. Examples are presented to illustrate the use of a higher-order truncation error model which leads to an accurate quantitative steady-state analysis of the constraint violations.

Weyl–Euler–Lagrange equations on twistor space for tangent structure

2016 ◽  
Vol 13 (07) ◽  
pp. 1650095
Author(s):  
Zeki Kasap
Keyword(s):  
Mathematical Modeling ◽  
Dynamical System ◽  
Classical Mechanic ◽  
Twistor Space ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Equation System ◽  
Lagrange Equations ◽  
Four Manifolds ◽  
Riemannian Geometries

Twistor spaces are certain complex three-manifolds, which are associated with special conformal Riemannian geometries on four-manifolds. Also, classical mechanic is one of the major subfields for mechanics of dynamical system. A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space for classical mechanic. Euler–Lagrange equations are an efficient use of classical mechanics to solve problems using mathematical modeling. On the other hand, Weyl submitted a metric with a conformal transformation for unified theory of classical mechanic. This paper aims to introduce Euler–Lagrage partial differential equations (mathematical modeling, the equations of motion according to the time) for the movement of objects on twistor space and also to offer a general solution of differential equation system using the Maple software. Additionally, the implicit solution of the equation will be obtained as a result of a special selection of graphics to be drawn.

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.

Electrostatically Actuated Coupled MEMS Resonators

2011 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Kyle N. Taylor
Keyword(s):  
Nonlinear Response ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
System Structure ◽  
Lagrange Equations ◽  
Electrostatically Actuated ◽  
Beam System ◽  
Mems Resonators ◽  
The Mathematical Model ◽  
Steady State Solutions

This paper deals with the nonlinear response of an electrostatically actuated cantilever beam system composed of two micro beam resonators near natural frequency. The mathematical model of the system is obtained using Lagrange equations. The equations of motion are nondimensionalized and then the method of multiple scales is used to find steady state solutions. Both AC and DC actuation voltages of the first beam are considered, while the influence on the system of DC on the second beam is explored. Graphical representations of the influence of the detuning parameters are provided for a typical micro beam system structure.

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 A NOTE ON THE (ANTI-)BRST INVARIANT LAGRANGIAN DENSITIES FOR THE FREE ABELIAN 2-FORM GAUGE THEORY

2010 ◽  
Vol 25 (28) ◽  
pp. 2457-2467
Author(s):  
SAURABH GUPTA ◽  
R. P. MALIK
Keyword(s):  
Gauge Theory ◽  
Vector Field ◽  
Lagrange Multiplier ◽  
Field Condition ◽  
Equations Of Motion ◽  
Present Theory ◽  
Lagrange Equations ◽  
Symmetry Transformations ◽  
The Absolute ◽  
Lagrangian Densities

We show that the previously known off-shell nilpotent [Formula: see text] and absolutely anticommuting (sb sab + sab sb = 0) Becchi–Rouet–Stora–Tyutin (BRST) transformations (sb) and anti-BRST transformations (sab) are the symmetry transformations of the appropriate Lagrangian densities of a four (3+1)-dimensional (4D) free Abelian 2-form gauge theory which do not explicitly incorporate a very specific constrained field condition through a Lagrange multiplier 4D vector field. The above condition, which is the analogue of the Curci–Ferrari restriction of the non-Abelian 1-form gauge theory, emerges from the Euler–Lagrange equations of motion of our present theory and ensures the absolute anticommutativity of the transformations s(a)b. Thus, the coupled Lagrangian densities, proposed in our present investigation, are aesthetically more appealing and more economical.

scholarly journals Study on Elastic Dynamic Model for the Clamping Mechanism of High-Speed Precision Injection Molding Machine

2015 ◽  
Vol 2015 ◽  
pp. 1-13
Author(s):  
Xue-feng Chen ◽  
Jian-guo Hu ◽  
Yan-sheng Xu ◽  
Zhong-ming Xu ◽  
Hong-bo Wang
Keyword(s):  
Coordinate System ◽  
Dynamic Model ◽  
High Speed ◽  
Equations Of Motion ◽  
Elastic Vibration ◽  
Lagrange Equations ◽  
Beam Elements ◽  
Model Matrix ◽  
Absolute Coordinate System ◽  
Plastic Injection

This work centered on the double-toggle clamping mechanism with diagonal-five points for the high-speed precise plastic injection machine. Based on Lagrange equations, the differential equations of motion for the beam elements are established, in a rotating coordinate system and an absolute coordinate system, respectively. 43 generalized coordinates and a model matrix for the mechanism are created and some coordinate matrices are derived. By coupling the coordinate transformation and matrix manipulation, a high nonlinear and strong time-variant elastic dynamic model is obtained. Based on the dynamic model, a Kineto-Elasto Dynamics (KED) analysis and a Kineto-Elasto Static (KES) analysis are carried out, respectively. By comparing and analyzing the simulation results of KED and KES, the regularity of elastic vibration of the clamping mechanism in high-speed clamping process has been revealed.

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