viernes, 4 de febrero de 2011

ON LAGRANGE'S UBIQUITI IN MODERN PHYSICS

This post is just to point out a quite dazzling phenomenon I have been encountering during this year of research: Joseph Louis Lagrange.

It happens to be that this Italian gentleman revolutionized the world of Physics some two hundred years ago (see here a beautiful explanation on how), in such a manner that now is nearly impossible not to encounter his surname nearly everywhere when trying to understand them.
The following is a short outline of the mathematical/physical concepts including Lagrange (a larger version can be found in http://en.wikipedia.org/wiki/List_of_topics_named_after_Joseph_Louis_Lagrange):

 Lagrange multipliers

• Lagrange multipliers: These are mathematical artifacts for the solution of optimization problems (http://en.wikipedia.org/wiki/Lagrange_multiplier)
• Euler-Lagrange equation: The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. The two further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Euler-Lagrange equation
Their correspondence ultimately led to the calculus of variations, a term coined by Euler himself in 1766. In classical mechanics, it is equivalent to Newton's laws of motion, but it has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations. Is also related to optimization according to variational principles (http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation).
• Lagrangian function: The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as Lagrangian mechanics.
$L = T - V.\quad$
Lagrangian function
In classical mechanics, the Lagrangian is defined as the kinetic energy, T, of the system minus its potential energy.
• Green-Lagrangian tensor: In continuum mechanics, the finite strain theory also called large strain theory, or large deformation theory, deals with deformations in which both rotations and strains are arbitrarily large. This means to invalidate the assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different and a clear distinction has to be made between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue. The concept of strain is used to evaluate how much a given displacement differs locally from a rigid body displacement . One of such strains for large deformations is the Lagrangian finite strain tensor, also called the Green-Lagrangian strain tensor or Green - St-Venant strain tensor.
• Lagrange description of motion. In continuum mechanics the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an individual parcel through time gives the pathline of the parcel. This can be visualized as sitting in a boat and drifting down a river. The Eulerian specification of the flow field is a way of looking at fluid motion that focuses on specific locations in the space through which the fluid flows as time passes. This can be visualized by sitting on the bank of a river and watching the water pass the fixed location. The Lagrangian approach is also associated to particle based formulation, whereas the Eulerian is referred to as grid based formulations.
 Eulerian description
 Lagrangian description

And this is just a small sample. However, it fully justifies the name of the thesis and this blog's, as our research departs from Lagrange's work to try to find engineering solutions.
It must be said that Lagrange's prolificacy results somehow dazing and stunning, as there are so many fields he got involved in, and none of them of trivial nature.
I hope this quick outline serves others to find a way through all this tangled knowledge.