The Multiplier of the Equations of Motion of a System Under Constraint in the first Langrange Form

Abstract

We have shown in Lecture 7 (page 59) that the differential equations of a system which is bound through the equations of constraint

$$\varphi = 0,\quad \psi = 0,\quad \tilde \omega = 0, \ldots$$

can be brought to the following form:

$$\begin{gathered} {m_i}\frac{{{d^2}{x_i}}} {{d{t^2}}} = {X_i} + \lambda \frac{{\partial \varphi }} {{\partial {x_i}}} + \mu \frac{{\partial \psi }} {{\partial {x_i}}} + \upsilon \frac{{\partial \tilde \omega }} {{\partial {x_i}}} + \cdots , \hfill \\ {m_i}\frac{{{d^2}{y_i}}} {{d{t^2}}} = {Y_i} + \lambda \frac{{\partial \varphi }} {{\partial {y_i}}} + \mu \frac{{\partial \psi }} {{\partial {y_i}}} + \upsilon \frac{{\partial \tilde \omega }} {{\partial {y_i}}} + \cdots , \hfill \\ {m_i}\frac{{{d^2}{z_i}}} {{d{t^2}}} = {X_i} + \lambda \frac{{\partial \varphi }} {{\partial {z_i}}} + \mu \frac{{\partial \psi }} {{\partial {z_i}}} + \upsilon \frac{{\partial \tilde \omega }} {{\partial {z_i}}} + \cdots , \hfill \\ \end{gathered}$$

where the multipliers λ, µ, v, … are to be determined, as already remarked there, by differentiating the equations $\varphi = 0,\psi = 0,\tilde \omega = 0, \cdots$ twice. When we determine λ, µ, v, …, we find, as we shall show presently, that these quantities are not independent of x’, y’, z’. One cannot therefore set the multiplier M equal to 1 here, one must go back for this determination to equation (15.4) of Lecture 15, p. 132. According to this the multiplier M for the system of differential equations

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

is defined through the equation

$$0 = \frac{{d\;\log M}} {{dt}} + \frac{{\partial A}} {{\partial {x^{\left( {m - 1} \right)}}}} + \frac{{\partial B}} {{\partial {y^{\left( {n - 1} \right)}}}} + \frac{{\partial C}} {{\partial {z^{\left( {p - 1} \right)}}}} + \cdots$$

. In the present case, this gives

$$\begin{gathered} - \frac{{d\;\log M}} {{dt}} = \sum\limits_i {\frac{1} {{{m_i}}}} \left( {\frac{{\partial \varphi }} {{\partial {x_i}}}\frac{{\partial \lambda }} {{\partial {{x'}_i}}} + \frac{{\partial \varphi }} {{\partial {i_i}}}\frac{{\partial \lambda }} {{\partial {{y'}_i}}} + \frac{{\partial \varphi }} {{\partial {z_i}}}\frac{{\partial \lambda }} {{\partial {{z'}_i}}}} \right) \hfill \\ + \sum\limits_i {\frac{1} {{{m_i}}}} \left( {\frac{{\partial \psi }} {{\partial {x_i}}}\frac{{\partial \mu }} {{\partial {{x'}_i}}} + \frac{{\partial \psi }} {{\partial {i_i}}}\frac{{\partial \mu }} {{\partial {{y'}_i}}} + \frac{{\partial \psi }} {{\partial {z_i}}}\frac{{\partial \mu }} {{\partial {{z'}_i}}}} \right) \hfill \\ + \cdots \hfill \\ \end{gathered}$$

, where on the right hand side, to each of the multipliers λ, µ, v, … corresponds a sum. For the application of the theory of multipliers M, it is necessary that the right hand side of this equation be a total differential coefficient.