On the simultaneous solutions of two linear partial differential equations

Abstract

The problem of integrating the given partial differential equation H = h is now reduced to finding n − 1 functions H₁, H₂, …, H_n−1, independent of one another and also of H, of the variables p₁, p₂, …, p _n , q₁, q₂, …, q _n , which satisfy the $\frac{{n\left( {n - 1} \right)}} {2}$ equations of constraint

$$\left( {{H_\alpha },{H_\beta }} \right) = 0$$

(for the values 0, 1, …, n − 1 of the indices α and β), and which one has to set equal to n − 1 mutually independent arbitrary constants h₁, h₂, …, h_n−1. Between any one of these n − 1 functions, e.g. H₁, and the function H known to us, also the equation of constraint (H, H₁) = 0, holds i.e. H₁ satisfies the partial differential equation

$$\begin{gathered} \frac{{\partial H}} {{\partial {p_1}}}\frac{{\partial {H_1}}} {{\partial {q_1}}} + \frac{{\partial H}} {{\partial {p_2}}}\frac{{\partial {H_1}}} {{\partial {q_2}}} + \cdots + \frac{{\partial H}} {{\partial {p_n}}}\frac{{\partial {H_1}}} {{\partial {q_n}}} - \hfill \\ - \frac{{\partial H}} {{\partial {q_1}}}\frac{{\partial {H_1}}} {{\partial {p_1}}} - \frac{{\partial H}} {{\partial {q_2}}}\frac{{\partial {H_1}}} {{\partial {p_2}}} - \cdots - \frac{{\partial H}} {{\partial {q_n}}}\frac{{\partial {H_1}}} {{\partial {p_n}}} = 0 \hfill \\ \end{gathered}$$

or what is the same, H₁ = h₁ is an integral of the system of isoperimetric differential equations¹

$$\begin{gathered} d{q_1}:d{q_2}: \ldots :d{q_n}:d{p_1}:d{p_2}: \ldots :d{p_n} \hfill \\ = \frac{{\partial H}} {{\partial {p_1}}}:\frac{{\partial H}} {{\partial {p_2}}}: \ldots :\frac{{\partial H}} {{\partial {p_n}}}: - \frac{{\partial H}} {{\partial {q_1}}}: - \frac{{\partial H}} {{\partial {q_2}}}: \cdots :\frac{{\partial H}} {{\partial {q_n}}} \hfill \\ \end{gathered}$$

which, for H = T − U, goes over to the system of differential equations of mechanics. A similar relation holds for the functions H₂, …, H_n−1 which satisfy the analogous equations of constraint (H, H₂) = 0, …, (H, H_n−1) = 0. All n − 1 equations

$${H_1} = {h_1},{H_2} = {h_2}, \ldots ,{H_{n - 1}} = {h_{n - 1}}$$

are therefore integrals of the system of isoperimetric differential equations given above.