Hamilton’s Integral and Lagrange’s Second Form of Dynamical Equations

Abstract

One can substitute another principle in place of the principle of least action where also the first variation of an integral vanishes, and from which one can derive the differential equations of motion in a still simpler way than from the principle of least action. It appears that this principle had not been noticed earlier, because here in general one does not obtain a minimum with the vanishing of the variation, as it happens in the case of the principle of least action. Hamilton is the first to have started out from this principle. We shall use it to formulate the equations of motion in the form Lagrange has given them in Mecanique Analytique. Let, first, the forces X _i , Y _i , Z _i be the partial derivatives of a function U; further let T be half the ‘vis viva’, i.e.,

$$T = \frac{1} {2}\sum {{m_i}v_i^2} = \frac{1} {2}\sum {{m_i}\left\{ {{{\left( {\frac{{d{x_i}}} {{dt}}} \right)}^2} + {{\left( {\frac{{d{y_i}}} {{dt}}} \right)}^2} + {{\left( {\frac{{d{z_i}}} {{dt}}} \right)}^2}} \right\}}$$

; then the new principle is contained in the equation

$$\delta \int {\left( {T + U} \right)dt = 0}$$

((8.1))

. This principle is more general in comparision with that of least action in so far as here U can depend on t explicitly, which was excluded in the earlier principle. There the time had to be eliminated through the principle of vis viva, which holds only when U does not contain t explicitly.