Physics - Lagrangian methods

Prerequisites

If you are not familiar with this subject you may like to look at the following pages first:

Lagrangian methods

This is a method for analysing mechanical systems based on energy conservation. For many types of mechanical system this method will allow its behavior to be defined with fewer equations than would otherwise be required.

The Lagrangian for classical mechanics is taken to be the difference between the kinetic energy and the potential energy.

This is known as the 'Action' (lookup: principle of least action).

Constraints

A mechanical system may be made up of particles or solid bodies. If unconstraind then each particle has 3 degrees of freedom (x,y and z coordinates) and each solid body has 6 degrees of freedom (3 for position and 3 for orientation) .

However in a mechanism the objects are constraind in various ways so that the position/orientation of each object depends on the other objects in the system. These constrains will therefore allow us to greatly simplify the equations describing the system in most cases.

Examples of these contraints are things like:

Generalised Coordinates, Generalised Forces

Each component is therefore likely to have fewer degrees of freedom than its throretical maximum. It therefore makes sense to define coordinates along the paths that objects are free to move.

Virtual Displacements and Virtual Work

Lagranges equations for a single particle

Lagranges equations for a system of particles

Lagranges equations for rigid bodies

Lagranges equations for a moving coordinate system

D'Alemberts Equation

For a single particle:

m (ax dx + ay dy + az dz ) = Fx dx + Fy dy + Fz dz

where:

 


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flag flag flag flag flag flag New Foundations for Classical Mechanics (Fundamental Theories of Physics). This is very good on the geometric interpretation of this algebra. It has lots of insights into the mechanics of solid bodies. I still cant work out if the position, velocity, etc. of solid bodies can be represented by a 3D multivector or if 4 or 5D multivectors are required to represent translation and rotation.

 

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Matlab.

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