Lagrangian and Hamiltonian formulation for continuous system and fields.

Many of the systems can be described as continuous system in the sense that various properties are continuous functions of the coordinates. For example, the mass of the system may be distributed continuously throughout space rather than be concentrated in the discrete particles. A continuous string has thus a linear mass density, each of the points along it takes part in the oscillations and the complete motion can be described by specifying the position coordinates of all points. A system composed of discrete particles that approximates the continuous rod is an infinite chain of equal mass points spaced a distance a apart and connected by uniform mass less springs having force constants k. It will be assumed that the mass points can more only along the length of the chain. Denoting the displacement of the ith particle from its equation position by ηi, the K.E is

The particular form of L in 4 and of the corresponding equations of motion has been chosen for convenience in going to the limit of a continuous rod as a approaches zero. It is clear that m/a reduces to μ, the mass per unit length of the continuous system, but the limiting value ka may not be so obvious. For an elastic rod obeying Hooke's law,

F= Yξ

where ξ is the elongation per unit length and Y is Young's modulus. Now the extension of a length a of a discrete system per unit length, will be ξ = (ηi+1 - ηi)/a. The force necessary to sketch the spring by this amount is

F = k(ηi+1 - ηi) = ka (ηi+1 - ηi) /a

So that ka must correspond to the Young's modulus of the continuous rod. in going from the discrete to the continuous case, the integer i identifying the particular mass point becomes the continuous position coordinate x; instead of the variable ηi we have η(x).

Further, the quantity

(ηi+1 - ηi)/a = η(x+a) - η(x) / a

occurring in Li obviously approaches the limit dη/ dx as a , playing the role of dx approaches zero. Finally the summation over a discrete number of particles becomes an integral over x, the length of the rod and the Lagrangian 4 appear as

which clearly defines a second derivatives of η. Hence the equation of motion for the continuous elastic rod is

The familiar wave equation in one dimension with the propagation velocity

v = √Y/ μ .......8

Equation 8 is the well known formula for the velocity if longitudinal elastic waves.

In 3 dimensional case,

L = ∫∫∫L dxdydz.....................9

where L is known as the Lagrangian density. For the longitudinal vibrations of the continuous rod, the Lagrangian density is

corresponding to the continuous limit of the quantity Li appearing in 4. It is the Lagrangian density, rather than the Lagrangian itself, that will be used to describe the motion of the system.

This note is taken from Classical Mechanics, MSC physics, Nepal.