Saturday, 20 September 2008

Lagrangian Mechanics: From the Euler-Lagrange Equation to Simple Harmonic Motion

I already wrote about obtaining Newton's Laws from the Principle of Least Action. Now I'm going to analyse a simple mass-spring system; effectively just a case of substituting in a suitable potential for the spring.



Let us go through all of the steps, once more. We'll choose our coordinate x to be the displacement from the equilibrium length of the spring, as shown on the diagram. The kinetic energy of the moving mass is then just
T=\frac{m\dot{x}^2}{2}
. The potential of a spring stretched (or compressed) x metres from its equilibrium length is given by
V=\frac{kx^2}{2}
, where k is the spring constant (N m^-1).

So, for our mass-spring system, the Lagrangian is

L = \frac{1}{2}\left(m\dot{x}^2 - kx^2\right)}


Applying the Euler-Lagrange equation, we obtain

\begin{multiline*}
-kx - \frac{d}{dt}\left(m\dot{x}\right) = 0 \\
\Rightarrow \ddot{x} - \frac{k}{m}x = 0
\end{multiline*}


Comparing the last line above with the general form of Simple Harmonic Motion (SHM),
\ddot{q} - \omega^2 q = 0
we can see that the equation of motion rendered by applying the Euler-Lagrange equation to the Lagrangian of a mass-spring system provides Simple Harmonic Motion with a frequency

\omega = \sqrt{\frac{k}{m}}


This is as expected for the case of a mass-spring system!

3 comments:

  1. It is interesting to notice how different approaches imply different assumptions: when treating the mass-spring system in a Newtonian framework, you start by Hooke's Law of elasticity, while here you have to `make up' a potential energy term.
    Keep up the good work!
    (Oh and you forgot the right bracket when writing down the Lagrangian)

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  2. You don't make up the potential. You could treat it as also starting from Hooke's law: The force F = -kx is represented by the gradient of a potential, i.e. F = -dV/dx

    Therefore the potential V = 1/2 kx^2 as used in the post, by direct integration of Hooke's law!

    Bracket added, thanks for noticing.

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  3. From the point of view of the analysis, be it Newtonian or Lagrangian, Hooke's Law is `made up' too. Our knowledge of Hooke's Law is a result of forces being somehow more easily measurable than potentials.
    So imagine, for a moment, that you knew nothing about Newton's Second Law and forces, but had only ever used the Euler-Lagrange equation to solve problems in mechanics. Then, at least until you `discovered' how to derive Newton's Second Law from the Euler-Lagrange equation, you wouldn't be able to solve the simple spring-mass system unless you `made up' a potential energy term. But if you know nothing about forces, then clearly you don't know about the relationship between a force and a potential, so even if you knew that x'' = -kx for a spring (notice I am not mentioning forces) there would be no way to obtain a potential term, and you would have to look somewhere else!

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