# User:Jftsang/sandbox

${\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s)}$

# Rayleigh–Ritz method

I propose the following cleanup of the article Rayleigh–Ritz method. --jftsang 05:28, 20 April 2014 (UTC)

In mathematics, the Rayleigh–Ritz method is a method for finding an approximation to the smallest eigenvalue of a Sturm-Liouville operator. It is named after Walther Ritz and Lord Rayleigh.

## Description

The Rayleigh-Ritz method is a direct variational method, in which the minimum of a functional defined on a normed linear space is approximated by a linear combination of elements from that space.

Suppose we are given the eigenvalue equation

${\displaystyle Ly=\lambda wy,}$

where ${\displaystyle L}$ is the Sturm-Liouville operator

${\displaystyle Ly=-{\frac {\mathrm {d} }{\mathrm {d} x}}\left(p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right)+q(x)y}$

with appropriate boundary conditions. We seek the smallest eigenvalue of ${\displaystyle L}$. Consider the functionals

${\displaystyle F[y]=\int wyLy\mathrm {d} x}$

and

${\displaystyle G[y]=\int wy^{2}\mathrm {d} x.}$

It can be shown that the problem of finding the smallest eigenvalue of ${\displaystyle L}$ is equivalent to the variational problem of minimising ${\displaystyle F[y]}$, subject to ${\displaystyle G[y]=1}$, which in turn is equivalent to minimising ${\displaystyle F-\lambda G}$. Here ${\displaystyle \lambda }$ is a Lagrange multiplier, the possible values of which being also the eigenvalues of ${\displaystyle L}$. We can minimise ${\displaystyle F-\lambda G}$ by considering a trial function ${\displaystyle {\tilde {y}}}$, which satisfies the boundary conditions and ${\displaystyle G[{\tilde {y}}]=1}$. We make an ansatz about the form of ${\displaystyle {\tilde {y}}}$ and minimise ${\displaystyle F-\lambda G}$ amongst functions of this form. For example, ${\displaystyle {\tilde {y}}}$ might be taken to be a truncated Taylor series; and we conduct the minimisation of ${\displaystyle F-\lambda G}$ by choosing the coefficients of the series properly. The method is effective if the form that we guess for ${\displaystyle {\tilde {y}}}$ can approximate the lowest-eigenvalue eigenfunction well.

## Example

Consider the eigenvalue problem

${\displaystyle Ly=-y''+y=\lambda y}$

subject to ${\displaystyle y\rightarrow 0}$ as ${\displaystyle |x|\rightarrow \infty }$. This is the eigenvalue problem for the quantum harmonic oscillator, in suitable units. We seek approximations to the lowest eigenvalue and eigenfunction. Here ${\displaystyle w=1}$, and

${\displaystyle F-\lambda G=\int (-yy''+y^{2}-\lambda y^{2})\mathrm {d} x.}$

## Applications

### Mechanical engineering

The Rayleigh-Ritz method is widely used in applied mathematics and mechanical engineering for the calculation of the natural vibration frequency of a structure in the second or higher order, as it can give a useful approximation even when the true solution may be intractable.

Typically in mechanical engineering it is used for finding the approximate real resonant frequencies of multi degree of freedom systems, such as spring mass systems or flywheels on a shaft with varying cross section. It is an extension of Rayleigh's method. It can also be used for finding buckling loads and post-buckling behaviour for columns.

### Other applications

The Rayleigh-Ritz method is also widely used in quantum chemistry.