2
$\begingroup$

In Herbert Goldstein's Classical Mechanics book, p. 75 (2nd ed.) it says:

Equations 3-18 and 3-20 are the two remaining integrations, and formally the problem has been reduced to quadratures...

It is not clear to me what a quadrature is. Is it simply the integration process through some numerical analysis method such as trapezoidal rule? Or is it the process of reducing the order of the initial partial differential equation by using constants of motion?

$\endgroup$
1
  • 3
    $\begingroup$ Quadrature : In mathematics, quadrature is a historical term which means determining area. This term is still used nowadays in the context of differential equations, where "solving an equation by quadrature" means expressing its solution in terms of integrals. $\endgroup$
    – Frobenius
    Commented Sep 19, 2018 at 18:47

1 Answer 1

2
$\begingroup$

As Frobenius mentioned in comment, quadrature is a historic term for integration.

But in modern texts I tend to only see quadrature used in the context of numeric approximation methods, in order to avoid confusing those methods with exact mathematical methods of integration.

In particular quadrature nowadays seems to refer to integration over an explicit range, whereas integration is the inverse of differentiation and does not require a range. See also Antiderivative.

So for example, $$\int_a^by(x) \ \mathrm{d}x \approx \sum_{n=1}^{N}\,y\left(a+\frac n N (b-a)\right)\frac{b-a}N, \tag{a quadrature}$$ $$\int_0^{2\pi}\sin^2(\theta) \ \mathrm{d}\theta = \pi, \tag{an exact integral}$$ and $$\int \sin^2(\theta) \ \mathrm{d}\theta = \frac \theta 2 - \frac 1 4 \sin(2\theta). \tag{an indefinite integral}$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.