Hamilton’s Form of the Equations of Motion

Abstract

After the publication of the first edition of Mécanique analytique, the most important step forward in the transformation of the differential equation of motion was made by Poisson in a paper which deals with the method of variation of constants and which appears in Volume 15 of the Polytechnique Journal. Here Poisson introduces the quantity $p = \frac{{\partial T}} {{\partial q'}}$, in place of the quantity q′; now since, as already remarked, T is a homogeneous function of the second degree in the quantities q′ whose coefficients depend on q, p is a linear function of the quantities q′; for the definition of p one has the k equations of the form ${p_i} = {\tilde \omega _i}$, where ${\tilde \omega _i}$ is linear with respect to q′₁, …, q’ _k . If one solves these linear equations for the quantities q′, one then obtains equations of the form q′ _i = K _i where the K _i ’s are linear expressions in p whose coefficients depend on the q. We shall insert these expressions for q′ _i in the equation (8.10) of Lecture 8, i.e., in the equation

$$\frac{{d{p_i}}} {{dt}} = \frac{{\partial \left( {T + U} \right)}} {{\partial {q_i}}} = \frac{{\partial T}} {{\partial {q_i}}} + \frac{{\partial U}} {{\partial {q_i}}}$$

, where $\frac{{\partial U}} {{\partial {q_i}}}$ contains only q, while $\frac{{\partial T}} {{\partial {q_i}}}$ is, besides, a function of the quantities q′, indeed, a homogeneous function of the second degree of these quantities. If we set q′₁ = K _i , then $\frac{{\partial T}} {{\partial {q_i}}}$ is a homogeneous function of second degree in the quantities p _i . Hence the above equations will be of the form

$$\frac{{d{p_i}}} {{dt}} = {P_i}$$

, where P _i is an expression in p and q and in fact of the second degree with respect to p.