Ordinary differential equations Lagrange equation

Now that we have introduced coordinates and velocities, the next question is how to predict the time evolution of a mechanical system. This is accomplished by solving a set of ordinary differential equations, the equations of motion, which can be derived from the principle of least action. It was discovered by Maupertuis and was further developed by Euler, Lagrange and Hamilton (d Abro (1951)). 

This system of second order ordinary differential equations of motion satisfies the constraints without the need to include the Lagrange multipliers, thereby not increasing the size of an already large problem. This technique will be used to solve for the elastic deformations as well as the rigid body motions of a general flexible mechanism or multibody system. 

The set of equations (3.1.6) is a special case of the Euler equations of the calculus of variations (see, e.g., Arnold (1989)). They are referred to as the Euler-Lagrange equations in the literature. The Euler-Lagrange equations are ordinary second order differential equations for the generalized coordinates qa-... 

In [164] the Variational Iterative Method is reconsidered for initial-value problems in ordinary or partial differential equations. A reconsideration of the Lagrange Multiplier is proposed. The above reconsideration is taken place in order the iteration formula and the convergence analysis to be simplified and facilitated. 

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