Change of variables (PDE)

Suppose that we have a function ${\displaystyle u(x,t)}$ and a change of variables ${\displaystyle x_{1},x_{2}}$ such that there exist functions ${\displaystyle a(x,t),b(x,t)}$ such that

${\displaystyle x_{1}=a(x,t)}$

${\displaystyle x_{2}=b(x,t)}$

and functions ${\displaystyle e(x_{1},x_{2}),f(x_{1},x_{2})}$ such that

${\displaystyle x=e(x_{1},x_{2})}$

${\displaystyle t=f(x_{1},x_{2})}$

and furthermore such that

${\displaystyle x_{1}=a(e(x_{1},x_{2}),f(x_{1},x_{2}))}$

${\displaystyle x_{2}=b(e(x_{1},x_{2}),f(x_{1},x_{2}))}$

and

${\displaystyle x=e(a(x,t),b(x,t))}$

${\displaystyle t=f(a(x,t),b(x,t))}$

In other words, it is helpful for there to be a bijection between the old set of variables and the new one, or else one has to

• Restrict the domain of applicability of the correspondence to a subject of the real plane which is sufficient for a solution of the practical problem at hand (where again it needs to be a bijection), and
• Enumerate the (zero or more finite list) of exceptions (poles) where the otherwise-bijection fails (and say why these exceptions don't restrict the applicability of the solution of the reduced equation to the original equation)

If a bijection does not exist then the solution to the reduced-form equation will not in general be a solution of the original equation.

We are discussing change of variable for PDEs. A PDE can be expressed as a differential operator applied to a function. Suppose ${\displaystyle {\mathcal {L}}}$ is a differential operator such that

${\displaystyle {\mathcal {L}}u(x,t)=0}$

Then it is also the case that

${\displaystyle {\mathcal {L}}v(x_{1},x_{2})=0}$

where

${\displaystyle v(x_{1},x_{2})=u(e(x_{1},x_{2}),f(x_{1},x_{2}))}$

and we operate as follows to go from ${\displaystyle {\mathcal {L}}u(x,t)=0}$ to ${\displaystyle {\mathcal {L}}v(x_{1},x_{2})=0:}$

• Apply the chain rule to ${\displaystyle {\mathcal {L}}v(x_{1}(x,t),x_{2}(x,t))=0}$ and expand out giving equation ${\displaystyle e_{1}}$.
• Substitute ${\displaystyle a(x,t)}$ for ${\displaystyle x_{1}(x,t)}$ and ${\displaystyle b(x,t)}$ for ${\displaystyle x_{2}(x,t)}$ in ${\displaystyle e_{1}}$ and expand out giving equation ${\displaystyle e_{2}}$.
• Replace occurrences of ${\displaystyle x}$ by ${\displaystyle e(x_{1},x_{2})}$ and ${\displaystyle t}$ by ${\displaystyle f(x_{1},x_{2})}$ to yield ${\displaystyle {\mathcal {L}}v(x_{1},x_{2})=0}$, which will be free of ${\displaystyle x}$ and ${\displaystyle t}$.

In the context of PDEs, Weizhang Huang and Robert D. Russell define and explain the different possible time-dependent transformations in details.^[2]

Often, theory can establish the existence of a change of variables, although the formula itself cannot be explicitly stated. For an integrable Hamiltonian system of dimension ${\displaystyle n}$, with ${\displaystyle {\dot {x}}_{i}=\partial H/\partial p_{j}}$ and ${\displaystyle {\dot {p}}_{j}=-\partial H/\partial x_{j}}$, there exist ${\displaystyle n}$ integrals ${\displaystyle I_{i}}$. There exists a change of variables from the coordinates ${\displaystyle \{x_{1},\dots ,x_{n},p_{1},\dots ,p_{n}\}}$ to a set of variables ${\displaystyle \{I_{1},\dots I_{n},\varphi _{1},\dots ,\varphi _{n}\}}$, in which the equations of motion become ${\displaystyle {\dot {I}}_{i}=0}$, ${\displaystyle {\dot {\varphi }}_{i}=\omega _{i}(I_{1},\dots ,I_{n})}$, where the functions ${\displaystyle \omega _{1},\dots ,\omega _{n}}$ are unknown, but depend only on ${\displaystyle I_{1},\dots ,I_{n}}$. The variables ${\displaystyle I_{1},\dots ,I_{n}}$ are the action coordinates, the variables ${\displaystyle \varphi _{1},\dots ,\varphi _{n}}$ are the angle coordinates. The motion of the system can thus be visualized as rotation on torii. As a particular example, consider the simple harmonic oscillator, with ${\displaystyle {\dot {x}}=2p}$ and ${\displaystyle {\dot {p}}=-2x}$, with Hamiltonian ${\displaystyle H(x,p)=x^{2}+p^{2}}$. This system can be rewritten as ${\displaystyle {\dot {I}}=0}$, ${\displaystyle {\dot {\varphi }}=1}$, where ${\displaystyle I}$ and ${\displaystyle \varphi }$ are the canonical polar coordinates: ${\displaystyle I=p^{2}+q^{2}}$ and ${\displaystyle \tan(\varphi )=p/x}$. See V. I. Arnold, `Mathematical Methods of Classical Mechanics', for more details.^[3]