Application of the preceding investigation to the integration of partial differential equations of the first order, and in particular, to the case of mechanics. The theorem on the third integral derived from two given integrals of differential equations of dynamics

Abstract

In order to apply the results obtained in the investigation of the previous lecture on the simultaneous solutions of linear partial differential equations to the case which led us to this investigation and from whcih we proceed to the integration of the partial differential equation H = h (p.290), we shall first replace the n + 1 independent variables x₀, x₁, …, x _n by an even number 2n of variables x₁, x₂, …, x_2n, where indices we shall begin with 1 instead of 0, so that the expression A(f) and B(f) are now defined through the equations

$$\begin{gathered} A\left( f \right) = {A_1}\frac{{\partial f}} {{\partial {x_1}}} + {A_2}\frac{{\partial f}} {{\partial {x_2}}} + \cdots + {A_{2n}}\frac{{\partial f}} {{\partial {x_{2n}}}}, \hfill \\ B\left( f \right) = {B_1}\frac{{\partial f}} {{\partial {x_1}}} + {B_2}\frac{{\partial f}} {{\partial {x_2}}} + \cdots + {B_{2n}}\frac{{\partial f}} {{\partial {x_{2n}}}}, \hfill \\ \end{gathered}$$

and the 2n equations of constraint

$$C = B\left( {{A_i}} \right) - A\left( {{B_i}} \right) = 0$$

hold for i = 1, 2, …, 2n. Further, we may put p and q in place of the 2n independent variables so that

$${x_1} = {q_1},{x_2} = {q_2}, \ldots ,{x_n} = {q_n},{x_{n + 1}} = {p_1},{x_{n + 2}} = {p_2}, \ldots ,{x_{2n}} = {p_n},$$

and finally, let the coefficients A _i , B _i be determined through the equations

$$\begin{gathered} {A_1} = \frac{{\partial \varphi }} {{\partial {p_1}}},{A_2} = \frac{{\partial \varphi }} {{\partial {p_2}}}, \cdots ,{A_n} = \frac{{\partial \varphi }} {{\partial {p_n}}}, \hfill \\ {A_{n + 1}} = - \frac{{\partial \varphi }} {{\partial {q_1}}},{A_{n + 2}} = - \frac{{\partial \varphi }} {{\partial {q_2}}}, \cdots ,{A_{2n}} = - \frac{{\partial \varphi }} {{\partial {q_n}}}, \hfill \\ {B_1} = \frac{{\partial \psi }} {{\partial {p_1}}},{B_2} = \frac{{\partial \psi }} {{\partial {p_2}}}, \cdots ,{B_n} = \frac{{\partial \psi }} {{\partial {p_n}}}, \hfill \\ {B_{n + 1}} = - \frac{{\partial \psi }} {{\partial {q_1}}},{B_{n + 2}} = - \frac{{\partial \psi }} {{\partial {q_2}}}, \cdots ,{B_{2n}} = - \frac{{\partial \psi }} {{\partial {q_n}}} \hfill \\ \end{gathered}$$

.