• Journal for History of Mathematics
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• v.27 no.3
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• pp.165-182
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• 2014

After algebraic expressions for the roots of 3rd and 4th degree polynomial equations were given in the mid 16th century, seeking such a formula for the 5th and greater degree equations had been one main problem for algebraists for almost 200 years. Lagrange made careful and thorough investigation of various solving methods for equations with the purpose of finding a principle which could be applicable to general equations. In the process of doing this, he found a relation between the roots of the original equation and its auxiliary equation using permutations of the roots. Lagrange's ingenious idea of using permutations of roots of the original equation is regarded as the key factor of the Abel's proof of unsolvability by radicals of general 5th degree equations and of Galois' theory as well. This paper intends to examine Lagrange's contribution in the theory of polynomial equations, providing a detailed analysis of various solving methods of Lagrange and others before him.

• Journal of Ocean Engineering and Technology
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• v.10 no.3
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• pp.96-104
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• 1996

Hamilton's principle is used to derive Euler-Lagrange equations for free surface flow problems of incompressible ideal fluid. The velocity field is chosen to satisfy the continuity equation a priori. This approach results in a hierarchial set of governing equations consist of two evolution equations with respect to two canonical variables and corresponding boundary value problems. The free surface elevation and the Lagrange's multiplier are the canonical variables in Hamilton's sense. This Lagrange's multiplier is a velocity potential defined on the free surface. Energy is conserved as a consequence of the Hamiltonian structure. These equations can be applied to waves in water of finite depth including generalization of Hamilton's equations given by Miles and Salmon.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.33 no.10
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• pp.1038-1044
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• 2009

In this study, vibrations of an electric motor are analyzed when the motor has the interaction between mechanical and electromagnetic behaviors. For this vibration analysis a 3-phase 8-pole brushless DC motor is selected. Vibrations of the motor are influenced by coupled electromechanical characteristics. The variation of air-gap induced by vibration has an influence on the inductance of the motor coil. To analyze dynamic characteristics of the rotor, we studied inductance by the variation of an air-gap. After obtaining the kinetic, potential and magnetic energies for the motor, the equations of motion are derived by using Lagrange's equation. By applying the Newmark time integration method to the equations, the dynamic responses for the displacements and currents are computed.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1992.10a
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• pp.349-353
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• 1992

The purpose of this paper is to develop methods for the dynamic analysis of multibody system that consist of interconnected rigid and deformable component. The equations of motion are derived by using the Lagrange's equation and finite element theory for the elastic mechanism systems. The type of equation of motion is the differential algebraic equation included kinematic nonlinear algebraic equation. The generalized coordinate partitioning method is used for solving this equation. To show the validity of this analysis solver, couple of models were canalized and those results were compared with the commercial package(ADAMS).

• Journal of Astronomy and Space Sciences
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• v.31 no.1
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• pp.91-99
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• 2014

The atmospheric flow in the 3-Cell model of global atmosphere circulation is described by the Lagrange's equation of the non-inertial frame where pressure force, frictional force and fictitious force are mixed in complex form. The Coriolis force is an important factor which requires calculation of fictitious force effects on atmospheric flow viewed from the rotating Earth. We make new Mathematica platform to solve Lagrange's equation by numerical analysis in order to analyze dynamics of atmospheric general circulation in the non-inertial frame. It can simulate atmospheric circulation process anywhere on the earth. It is expected that this pedagogical platform can be utilized to help students studying the atmospheric flow understand the mechanisms of atmospheric global circulation.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.19 no.1
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• pp.183-193
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• 1994

The dynamic equation of a flexible robot manipulator is formulated by the assumed-mode method and the Lagrange equation. The controller is designed for a flexible robot manipulator including a joint actuator. The controller consists of a parmaeter estimator and the adaptive controller. A parameter estimator evaluates ARMA model`s parameter using RLS algorithm. An adaptive controller is designed based on a reference model and a minimum prediction error controller.

• Bulletin of the Society of Naval Architects of Korea
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• v.23 no.1
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• pp.23-32
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• 1986

By virtue of an application of the receptance method, simplified formulae to calculate natural frequencies of stiffened plates having a resiliently mounted or concentrated mass are obtained. Some numerical results are compared with those based on Lagrange's equation of motion and with experimental results. For the problem formulation the stiffened plate is reduced to an equivalent orthotropic plate, a resiliently mounted mass to a spring-mass system, and mode shapes of the plate are assumed with comparison functions consisting of Euler beam functions. The proposed formulae give results in good conformity to both numerical results based on Lagrange's equation of motion and experimental results for in-phase modes of the coupled system. For out-of-phase modes the conformity is assured only in case that the natural frequency of the attached system is less than a half of that the stiffened plate. It is also found that a resiliently mounted mass having its own natural frequency of about two or more times that of the stiffened plate can be reduced to a concentrated mass with assurance of a few percent error in the frequency.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 2007.11a
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• pp.507-508
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• 2007

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.215-228
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• 2007

By using the variational calculus of fractional order, one derives a Hamilton-Jacobi equation and a Lagrangian variational approach to the optimal control of one-dimensional fractional dynamics with fractional cost function. It is shown that these two methods are equivalent, as a result of the Lagrange's characteristics method (a new approach) for solving non linear fractional partial differential equations. The key of this results is the fractional Taylor's series $f(x+h)=E_{\alpha}(h^{\alpha}D^{\alpha})f(x)$ where $E_{\alpha}(.)$ is the Mittag-Leffler function.

• Communications of Mathematical Education
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• v.29 no.4
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• pp.699-722
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• 2015

Based on Lagrange's general theory of algebraic equations, by applying the solution of the equation using the relationship between roots and coefficients to the high school 1st grade class, the purpose of this study is to recognize the significance of symmetry associated with the solution of the equation. Symmetry is the core idea of Lagrange's general theory of algebraic equations, and the relationship between roots and coefficients is an important means in the solution. Through the lesson, students recognized the significance of learning about the relationship between roots and coefficients, and understanded the idea of symmetry and were interested in new solutions. These studies gives not only the local experience of solutions of the equations dealing in school mathematics, but the systematics experience of general theory of algebraic equations by the didactical organization, and should be understood the connections between knowledges related to the solutions of the equation in a viewpoint of the mathematical history.