Constraint counting

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In mathematics, constraint counting is a crude but often useful way of counting the number of free functions needed to specify a solution to a partial differential equation.

Einstein strength[edit]

Everyone knows that Albert Einstein said that a physical theory should be as simple as possible, but no simpler. But not everyone knows that he had a quantitative idea in mind.[citation needed]

Consider a second order partial differential equation in three variables, such as the two-dimensional wave equation

 u_{tt} = u_{xx} + u_{yy}.

It is often profitable to think of such an equation as a rewrite rule allowing us to rewrite arbitrary partial derivatives of the function u(t,x,y) using fewer partials than would be needed for an arbitrary function. For example, if u satisfies the wave equation, we can rewrite

 u_{tyt} = u_{tty} = u_{xxy} + u_{yyy}

where in the first equality, we appealed to the fact that partial derivatives commute.

Einstein asked: how much redundancy can we eliminate in this fashion, for a given partial differential equation?

Linear equations[edit]

To answer this in the important special case of a linear partial differential equation, Einstein asked: how many of the partial derivatives of a solution can be linearly independent? It is convenient to record his answer using an ordinary generating function

s(\xi) = \sum_{k=0}^\infty s_k \xi^k

where s_k is a natural number counting the number of linearly independent partial derivatives (of order k) of an arbitrary function in the solution space of the equation in question.

Einstein observed that whenever a function satisfies some partial differential equation, we can use the corresponding rewrite rule to eliminate some of them, because further mixed partials have necessarily become linearly dependent. Specifically, the power series counting the variety of arbitrary functions of three variables (no constraints) is

f(\xi) =  \frac{1}{(1-\xi)^3} = 1 + 3 \xi + 6 \xi^2 + 10 \xi^3 + \dots

but the power series counting those in the solution space of some second order p.d.e. is

g(\xi) = \frac{1-\xi^2}{(1-\xi)^3} = 1 + 2 \xi + 5 \xi^2 + 7 \xi^3 + \dots

which records that we can eliminate one second order partial u_{tt}, three third order partials u_{ttt}, \, u_{ttx}, \, u_{tty} , and so forth.

More generally, the o.g.f. for an arbitrary function of n variables is

s[n](\xi) = 1/(1-\xi)^n = 1 + n \, \xi + \left( \begin{matrix} n \\ 2 \end{matrix} \right) \, \xi^2 + \left( \begin{matrix} n+1 \\ 3 \end{matrix} \right) \, \xi^3 + \dots

where the coefficients of the infinite power series of the generating function are constructed using an appropriate infinite sequence of binomial coefficients, and the power series for a function required to satisfy a linear m-th order equation is

g(\xi) = \frac{1-\xi^m}{(1-\xi)^n}

Next,

 \frac{1-\xi^2}{(1-\xi)^3} = \frac{1 + \xi}{(1-\xi)^2}

which can be interpreted to predict that a solution to a second order linear p.d.e. in three variables is expressible by two freely chosen functions of two variables, one of which is used immediately, and the second, only after taking a first derivative, in order to express the solution.

General solution of initial value problem[edit]

To verify this prediction, recall the solution of the initial value problem

u_{tt} = u_{xx} + u_{yy}, \; u(0,x,y) = p(x,y), \; u_t(0,x,y) = q(x,y)

Applying the Laplace transform u(t,x,y) \mapsto [Lu](\omega,x,y) gives

 -\omega^2 \, [Lu] + \omega \, p(x,y) + q(x,y) + [Lu]_x + [Lu]_y

Applying the Fourier transform [Lu](\omega,x,y) \mapsto [FLU](\omega,m,n) to the two spatial variables gives

 -\omega^2 \, [FLu] + \omega \, [Fp] + [Fq] - (m^2+n^2) \, [FLu]

or

[FLu](\omega,m,n) = \frac{ \omega \, [Fp](m,n) + [Fq](m,n)}{\omega^2 + m^2 + n^2}

Applying the inverse Laplace transform gives

 [Fu](t,m,n) = [Fp](m,n) \, \cos( \sqrt{m^2+n^2} \, t ) + \frac{ [Fq](m,n) \, \sin (\sqrt{m^2+n^2} \, t) }{\sqrt{m^2+n^2}}

Applying the inverse Fourier transform gives

u(t,x,y) = Q(t,x,y) + P_t(t,x,y)

where

P(t,x,y) = \frac{1}{2\pi} \, \int_{(x-x')^2 + (y-y')^2 < t^2} \frac{p(x',y') \, dx' dy'}{ \left[ t^2-(x-x')^2-(y-y')^2 \right]^{1/2}}
Q(t,x,y) = \frac{1}{2\pi} \, \int_{(x-x')^2 + (y-y')^2 < t^2} \frac{q(x',y') \, dx' dy'}{ \left[ t^2-(x-x')^2-(y-y')^2 \right]^{1/2}}

Here, p,q are arbitrary (sufficiently smooth) functions of two variables, so (due their modest time dependence) the integrals P,Q also count as "freely chosen" functions of two variables; as promised, one of them is differentiated once before adding to the other to express the general solution of the initial value problem for the two dimensional wave equation.

Quasilinear equations[edit]

In the case of a nonlinear equation, it will only rarely be possible to obtain the general solution in closed form. However, if the equation is quasilinear (linear in the highest order derivatives), then we can still obtain approximate information similar to the above: specifying a member of the solution space will be "modulo nonlinear quibbles" equivalent to specifying a certain number of functions in a smaller number of variables. The number of these functions is the Einstein strength of the p.d.e. In the simple example above, the strength is two, although in this case we were able to obtain more precise information.

References[edit]

  • Siklos, S. T. C. (1996). "Counting solutions of Einstein's equation". Class. Quant. Grav. 13 (7): 1931–1948. doi:10.1088/0264-9381/13/7/021.  Application of constraint counting to Riemannian geometry and to general relativity.