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Continuum theory of granular materials
Published in M. Oda, K. Iwashita, Mechanics of Granular Materials, 2020
(The choice of the expression of Equation (2.2.34) for the specific power is based on Lagrange’s principle of analytical mechanics. A homogenized dynamical system is obtained by choosing Equations (2.2.30) and (2.2.31) as virtual displacements or, in the present case, as virtual velocities.) In the Cosserat theory, the gradients of displacement and/or velocity gradients are neglected. In this case, Equation (2.2.32) becomes g¯i21=l[γijkj+l2εijlκlk(kj+hj)kk]
Extrema and Variational Calculus
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
Newton formulated the laws of motion in his 1687 volumes, collectively called the Philosophiae Naturalis Principia Mathematica, or simply the Principia. However, Newton’s development was geometrical and is not how we see classical dynamics presented when we first learn mechanics. The laws of mechanics are what are now considered analytical mechanics, in which classical dynamics is presented in a more elegant way. It is based upon variational principles, whose foundations began with the work of Euler and Lagrange and have been refined by other now-famous figures in the eighteenth and nineteenth centuries.
Analytic mechanics
Published in Louis Komzsik, Applied Calculus of Variations for Engineers, 2018
Analytic mechanics is a mathematical science, but it is of high importance for engineers as it provides analytic solutions to fundamental problems of engineering mechanics. At the same time it establishes generally applicable procedures. Mathematical physics texts, such as [5] and [6], laid the foundation for these analytic approaches addressing physical problems.
The use of the conservation of living force before Helmholtz
Published in Annals of Science, 2023
The examples given here shows a continuity between the way the conservation of living force was employed within the tradition of analytical mechanics and the way Helmholtz later used it. The comparison also highlights the main difference between the two concepts, namely the limited validity of the principle to particular mechanical systems within the analytical tradition, versus the general validity of Helmholtz’s ‘principle of conservation of force’. Helmholtz was correct in viewing his principle as an extension of the old mechanical one through his assumption that all forces are conservative mechanical ones (and his oblivious denial of hard collisions),50 to which one should add his focus on the generally conserved quantity – the sum of the living force and his ‘tensional force’. Caneva has shown that many of Helmholtz’s contemporaries failed to see the novelty of his work due to its close connection to the mechanical tradition.51 The similarity between his employment of the generalized principle and earlier employments of the mechanical principle of conservation makes their reaction even more understandable. It also suggests the debt of Helmholtz and his contemporaries to the tradition of analytical mechanics.
Motion analysis of a multi-joint system with holonomic constraints using Riemannian distance
Published in Advanced Robotics, 2022
Masahiro Sekimoto, Suguru Arimoto
The equations of motion of a multi-joint system in the presence of constraints are governed by the Lagrange-d'Alembert principle, D'Alembert principle, or Gauss's principle of least constraint, as known in analytical mechanics [21, 22]. The equations of motion represent the necessary conditions for the minimum of the Riemannian metric (i.e. square root of energy) in Riemannian geometry, whereas they represent the necessary conditions for the minimum of a Lagrangian (i.e. energy) in analytical mechanics. The mathematical relations between these minimizations have been discussed [21, 22]. However, the present study focuses on the evaluated values (i.e. the curve lengths) as well as the necessary conditions. The primary objective is not to obtain the optimal walking motion but to establish the curve length-based motion analysis.
Flexible multibody system modelling of an aerial rescue ladder using Lagrange’s equations
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Simon Densborn, Oliver Sawodny
Variational methods in analytical mechanics, especially the application of Lagrange’s equation of the second kind, allow the derivation of equations of motion from expressions of the kinetic and potential energy of the modelled system. The Lagrange’s equation of the second kind is given by