23 February 2023
C. Hinsley
In reading Peter Olver’s Applications of Lie Groups to Differential Equations, I have had to compute the prolongations of vector fields in Lie algebras many times, which is a time-consuming and error-prone process. Instead I would like to collect the results of these computations for reuse.
In general, a differential equation of $p$ many independent variables and $q$ many dependent variables specifies a diffiety which naturally projects onto an open subspace $M$ of $X \times U \cong \R^p \times \R^q$. A vector field $\vec{v} \in TM$ may be defined, to be written as a linear combination of partial differentials
$$ \vec{v} = \xi^{(1)}\frac{\partial}{\partial x^{(1)}} + \dots + \xi^{(p)}\frac{\partial}{\partial x^{(p)}} + \phi^{(1)}\frac{\partial}{\partial u^{(1)}} + \dots + \phi^{(q)}\frac{\partial}{\partial u^{(q)}}, $$
where each coefficient function $\xi^{(i)}, \phi^{(j)}$ is a function of $x^{(1)}, \dots, x^{(p)}, u^{(1)}, \dots, u^{(q)}$ for $1 \leq i \leq p, 1 \leq j \leq q$.
Let $k$ be a positive integer. The $k$-jet space $X \times U^{(k)}$ is, in essence, a product space of $X \times U$ with $\R$ adjoined for each derivative of the dependent variables $u^{(j)}$ with respect to the independent variables $x^{(i)}$ of order (as a repeated partial derivative) at most $k$; the coordinates of extension are the derivatives themselves. The $n$th prolongation of $\vec{v}$, $\textrm{pr}^{(n)}(\vec{v})$, is $\vec{v}$ embedded naturally (preserving the coordinate functions) into $X \times U^{(k)}$ but where for the solutions in $M$ of a differential equation the local deformations specified by $\vec{v}$ are in agreement with the local deformations in the derivatives of those solutions specified by the coefficient functions of the respective derivatives. The $n$th prolongation of $\vec{v}$ may be written (with shorthand for the partial derivative operators)
$$ \textrm{pr}^{(n)}\vec{v} = \vec{v} + \sum_{|J| \leq n} \sum_{\alpha=1}^q \phi_{(\alpha)}^J \partial_{u_J} $$
each coefficient function $\phi_{(\alpha)}^J$ again being defined on $M$, and with $J$ being a multi-index of independent variables, with $|J|$ being the number of variables appearing in $J$ counting multiplicities.
The formula for the coefficient functions, per Olver, is
$$ \phi_{(\alpha)}^J \overset{\text{def}}{=} \mathcal{D}J\left[\phi^{(\alpha)} - \sum{i=1}^p \xi^{(i)}u_{x^{(i)}}^{(\alpha)}\right] + \sum_{i=1}^p \xi^{(i)}u_{x^{(i)}, J}^{(\alpha)}, $$
where $\mathcal{D}_J$ is the total derivative with respect to the variables in the multi-index $J$.
Below I give a table of the prolongations for various coefficient functions of vector fields on $X \times U$ with varied $p \equiv \dim X, q \equiv \dim U$. You can permute the indices in the formulas to obtain other prolongations than are listed.
| Coefficient function | $p$ | $q$ | Expansion in the monomials |
|---|---|---|---|
| $\phi_{(1)}^{x^{(1)}}$ | 1 | 1 | $(\phi_{x^{(1)}}^{(1)}) \cdot 1 + (\phi_{u^{(1)}}^{(1)} - \xi_{x^{(1)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)} + (-\xi_{u^{(1)}}^{(1)}) \cdot (u_{x^{(1)}}^{(1)})^2$ |
| $\phi_{(1)}^{x^{(1)}x^{(1)}}$ | 1 | 1 | $(\phi_{x^{(1)}x^{(1)}}^{(1)}) \cdot 1 + (2\phi_{x^{(1)}u^{(1)}}^{(1)} - \xi_{x^{(1)}x^{(1)}}^{(1)}) \cdot u_{x^{(1)}} + (\phi_{u^{(1)}u^{(1)}}^{(1)} - 2\xi_{x^{(1)}u^{(1)}}^{(1)}) \cdot (u_{x^{(1)}})^2 + (-\xi_{u^{(1)}u^{(1)}}^{(1)}) \cdot (u_{x^{(1)}})^3 + (-2\xi_{x^{(1)}}^{(1)}) \cdot u_{x^{(1)}x^{(1)}} + (-3\xi_{u^{(1)}}^{(1)}) \cdot u_{x^{(1)}}u_{x^{(1)}x^{(1)}}$ |
| $\phi_{(1)}^{x^{(1)}}$ | 2 | 1 | $(\phi_{x^{(1)}}^{(1)}) \cdot 1 + (\phi_{u^{(1)}}^{(1)} - \xi_{x^{(1)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)} + (-\xi_{x^{(1)}}^{(2)}) \cdot u_{x^{(2)}}^{(1)} + (-\xi_{u^{(1)}}^{(1)}) \cdot (u_{x^{(1)}}^{(1)})^2 + (-\xi_{u^{(1)}}^{(2)}) \cdot u_{x^{(1)}}^{(1)}u_{x^{(2)}}^{(1)}$ |
| $\phi_{(1)}^{x^{(1)}x^{(1)}}$ | 2 | 1 | $(\phi_{x^{(1)}x^{(1)}}^{(1)}) \cdot 1 + (2\phi_{x^{(1)}u^{(1)}}^{(1)} - \xi_{x^{(1)}x^{(1)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)} + (-\xi_{x^{(1)}x^{(1)}}^{(2)}) \cdot u_{x^{(2)}}^{(1)} + (\phi_{u^{(1)}u^{(1)}}^{(1)} - 2\xi_{x^{(1)}u^{(1)}}^{(1)}) \cdot (u_{x^{(1)}}^{(1)})^2 + (-2\xi_{x^{(1)}u^{(1)}}^{(2)}) \cdot u_{x^{(1)}}^{(1)}u_{x^{(2)}}^{(1)} + (-\xi_{u^{(1)}u^{(1)}}^{(1)}) \cdot (u_{x^{(1)}}^{(1)})^3 + (-\xi_{u^{(1)}u^{(1)}}^{(2)}) \cdot (u_{x^{(1)}}^{(1)})^2u_{x^{(2)}}^{(1)} + (\phi_{u^{(1)}}^{(1)} - 2\xi_{x^{(1)}}^{(1)}) \cdot u_{x^{(1)}x^{(1)}}^{(1)} + (-2\xi_{x^{(1)}}^{(2)}) \cdot u_{x^{(1)}x^{(2)}}^{(1)} + (-3\xi_{u^{(1)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)}u_{x^{(1)}x^{(1)}}^{(1)} + (-2\xi_{u^{(1)}}^{(2)}) \cdot u_{x^{(1)}}^{(1)}u_{x^{(1)}x^{(2)}}^{(1)} + (-\xi_{u^{(1)}}^{(2)}) \cdot u_{x^{(2)}}^{(1)}u_{x^{(1)}x^{(1)}}^{(1)}$ |
| $\phi_{(1)}^{x^{(1)}x^{(2)}}$ | 2 | 1 | $(\phi_{x^{(1)}x^{(1)}}^{(1)}) \cdot 1 + (\phi_{x^{(2)}u^{(1)}}^{(1)} - \xi_{x^{(1)}x^{(2)}}^{(1)}) \cdot u^{(1)}{x^{(1)}} + (\phi{x^{(1)}u^{(1)}}^{(1)} - \xi_{x^{(1)}x^{(2)}}^{(2)}) \cdot u_{x^{(2)}}^{(1)} + (-\xi_{x^{(2)}u^{(1)}}^{(1)}) \cdot (u_{x^{(1)}}^{(1)})^2 + (\phi_{u^{(1)}u^{(1)}}^{(1)} - \xi_{x^{(1)}u^{(1)}}^{(1)} - \xi_{x^{(2)}u^{(1)}}^{(2)}) \cdot u_{x^(1)}u_{x^{(2)}} + (-\xi_{x^{(1)}u^{(1)}}^{(2)}) \cdot (u_{x^{(2)}}^{(1)})^2 + (-\xi_{u^{(1)}u^{(1)}}^{(1)}) \cdot (u_{x^{(1)}}^{(1)})^2u_{x^{(2)}}^{(1)} + (-\xi_{u^{(1)}u^{(1)}}^{(2)}) \cdot u_{x^{(1)}}^{(1)}(u_{x^{(2)}}^{(1)})^2 + (-\xi_{x^{(2)}}^{(1)}) \cdot u_{x^{(1)}x^{(1)}}^{(1)} + (\phi_{u^{(1)}}^{(1)} - \xi_{x^{(1)}}^{(1)} - \xi_{x^{(2)}}^{(2)}) \cdot u_{x^{(1)}x^{(2)}}^{(1)} + (-\xi_{x^{(1)}}^{(2)}) \cdot u_{x^{(2)}x^{(2)}}^{(1)} + (-2\xi_{u^{(1)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)}u_{x^{(1)}x^{(2)}}^{(1)} + (-\xi_{u^{(1)}}^{(2)}) \cdot u_{x^{(1)}}^{(1)}u_{x^{(2)}x^{(2)}}^{(1)} + (-\xi_{u^{(1)}}^{(1)}) \cdot u_{x^{(2)}}^{(1)}u_{x^{(1)}x^{(1)}}^{(1)} + (-2\xi_{u^{(1)}}^{(2)}) \cdot u_{x^{(2)}}^{(1)}u_{x^{(1)}x^{(2)}}^{(1)}$ |
| $\phi_{(1)}^{x^{(1)}}$ | 1 | 2 | $(\phi_{x^{(1)}}^{(1)}) \cdot 1 + (\phi_{x^{(1)}}^{(1)} - \xi_{x^{(1)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)} + (\phi_{u^{(2)}}^{(1)}) \cdot u_{x^{(1)}}^{(2)} + (-\xi_{u^{(1)}}^{(1)}) \cdot (u_{x^{(1)}}^{(1)})^2 + (-\xi_{u^{(2)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)}u_{x^{(1)}}^{(2)}$ |
| $\phi_{(1)}^{x^{(1)}}$ | 1 | 3 | $(\phi_{x^{(1)}}^{(1)}) \cdot 1 + (\phi_{x^{(1)}}^{(1)} - \xi_{x^{(1)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)} + (\phi_{u^{(2)}}^{(1)}) \cdot u_{x^{(1)}}^{(2)} + (\phi_{u^{(3)}}^{(1)}) \cdot u_{x^{(1)}}^{(3)} + (-\xi_{u^{(1)}}^{(1)}) \cdot (u_{x^{(1)}}^{(1)})^2 + (-\xi_{u^{(2)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)}u_{x^{(1)}}^{(2)} + (-\xi_{u^{(3)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)}u_{x^{(1)}}^{(3)}$ |
| $\phi_{(1)}^{x^{(1)}}$ | 1 | $q > 1$ | $\displaystyle (\phi_{x^{(1)}}^{(1)}) \cdot 1 + (\phi_{x^{(1)}}^{(1)} - \xi_{x^{(1)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)} + \sum_{\alpha=2}^q (\phi_{u^{(\alpha)}}^{(1)}) \cdot u_{x^{(1)}}^{(\alpha)} + \sum_{\alpha=1}^q (-\xi_{u^{(\alpha)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)}u_{x^{(1)}}^{(\alpha)}$ |
| $\phi_{(1)}^{x^{(1)}}$ | 2 | 2 | $(\phi_{x^{(1)}}^{(1)}) \cdot 1 + (\phi_{u^{(1)}} - \xi_{x^{(1)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)} + (-\xi_{x^{(1)}}^{(2)}) \cdot u_{x^{(2)}}^{(1)} + (\phi_{u^{(2)}}) \cdot u_{x^{(1)}}^{(2)} + (-\xi_{u^{(1)}}^{(1)}) \cdot (u_{x^{(1)}}^{(1)})^2 + (-\xi_{u^{(1)}}^{(2)}) \cdot u_{x^{(1)}}^{(1)}u_{x^{(2)}}^{(1)} + (-\xi_{u^{(2)}}^{(1)}) \cdot u_{x^{(1)}}^{(1)}u_{x^{(1)}}^{(2)} + (-\xi_{u^{(2)}}^{(2)}) \cdot u_{x^{(2)}}^{(1)}u_{x^{(1)}}^{(2)}$ |