Lagrange method

Lagrange method . This method reduces the constrained problem in n variables to one without restrictions of n + 1 variables whose equations can be solved.

Summary

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  • 1 General
  • 2 Simple pendulum
  • 3 Newton method
  • 4 Lagrange polynomials
  • 5 Help provided
  • 6 See also
  • 7 Source

General

In optimization problems, Lagrange multipliers, named after Joseph Louis Lagrange , are a method of working with multi-variable functions that we are interested in maximizing or minimizing, and are subject to certain restrictions. This method reduces the constrained problem in n variables to one without restrictions of n + 1 variables whose equations can be solved.

This method introduces a new unknown scalar variable, the Lagrange multiplier, for each constraint and forms a linear combination involving the multipliers as coefficients. Its proof involves partial derivatives, either using total differentials, or its close relatives, the chain rule. The aim is, using some implicit function, to find the conditions so that the derivative with respect to the independent variables of a function is equal to zero.

Simple pendulum

A simple pendulum is called an ideal entity consisting of a point mass suspended by an inextensible and weightless thread , capable of oscillating freely in a vacuum and without friction. When separating the mass from its equilibrium position , it oscillates on both sides of said position, making a simple harmonic movement. The practical realization of a simple pendulum is naturally impossible, but it is accessible to theory.

The simple or mathematical pendulum is so named as opposed to the only real, composite, or physical pendulums that can be built.

Newton method

When considered a simple pendulum, if the particle is moved from the equilibrium position until the wire forms an angle Θ with the vertical, and then it is abandoned starting from rest, the pendulum will swing in a vertical plane under the action of gravity.

The oscillations will take place between the extreme positions Θ and -Θ, symmetrical with respect to the vertical, along an arc of circumference whose radius is the length, \ ell, of the thread. The movement is periodic, but it cannot be guaranteed that it is harmonic.

To determine the nature of the oscillations, the equation of the particle’s motion must be written . The particle moves on an arc of circumference under the action of two forces: its own weight (mg) and the thread tension (N).

Lagrange polynomials

Assuming you know at least

intends:

for

Thus:

for

 

Help provided

  • For Dynamic Optimization Problem Solving: Solving an interpolation problem leads to a linear algebra problem in which a system of equations must be solved. Using a standard monomic basis for the interpolating polynomial , the Vandermonde matrix is ​​arrived at. Choosing a different base, the Lagrange base, we arrive at the simplest form of the identity = δi matrix that can be solved immediately.
  • Langrange Multipliers: Named after Joseph Louis Lagrange , it is a procedure to find the maxima and minima of functions of various variables subject to restrictions

 

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