The Multiplier for Systems of Differential Equations with Higher Differential Coefficients. Applications to a System of Mass Points Without Constraints

Abstract

All our considerations up to now concerned systems of differential equations with only first order differential coefficients. One can look upon systems of this sort as a special case of those in which differential coefficients of arbitrary order occur. But also, conversely, one can, by increasing the number of variables, reduce a system with higher order differential coefficients to the form of a system containing only first order differential coefficients, so that each becomes a special case of the other. We shall first concern ourselves with this reduction of an arbitrary system into another in which only differential coefficients of the first order occur. Let there be a system of i differential equations of i + 1 variables t, x, y, z, …; of which t is looked upon as the independent and x, y, z, … as the dependent variables. Let the highest differential coefficients which occur in these differential equations be mth in x, nth in y, pth in z, etc. If we further assume that we can solve for these highest differential coefficients, so that the differential equations take the following form:

$$\frac{{{d^m}x}} {{d{t^m}}} = A,\frac{{{d^n}y}} {{d{t^n}}} = B,\frac{{{d^p}z}} {{d{t^p}}} = C, \ldots$$

((15.1))

where the highest differential coefficients of x, y, z etc. which occur in A, B, C, … are of the (m − 1)th, (n − 1)th, (p − 1)th order, then this is the canonical form of the differential equations that are to be studied. Any given system cannot always directly be reduced to this canonical form (15.1); for example, this will not go through if in one of the differential equations the highest differential coefficients $\frac{{{d^m}x}} {{d{t^m}}},\frac{{{d^n}y}} {{d{t^n}}},\frac{{{d^p}z}} {{d{t^p}}}, \ldots$ do not occur.