Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/221

The last term may be written

$\sum {\left({\frac {dT_{p}}{dq}}{\dot {q}}\delta t\right)},$

which diminishes with δt, and ultimately vanishes with it when the impulse becomes instantaneous.

Hence, equating the coefficients of δp in equations (1) and (2), we obtain

${\dot {q}}={\frac {dT_{p}}{dp}},$

(3)

or, the velocity corresponding to the variable q is the differential coefficient of T_p with respect to the corresponding momentum p.

We have arrived at this result by the consideration of impulsive forces. By this method we have avoided the consideration of the change of configuration during the action of the forces. But the instantaneous state of the system is in all respects the same, whether the system was brought from a state of rest to the given state of motion by the transient application of impulsive forces, or whether it arrived at that state in any manner, however gradual.

In other words, the variables, and the corresponding velocities and momenta, depend on the actual state of motion of the system at the given instant, and not on its previous history.

Hence, the equation (3) is equally valid, whether the state of motion of the system is supposed due to impulsive forces, or to forces acting in any manner whatever.

We may now therefore dismiss the consideration of impulsive forces, together with the limitations imposed on their time of action, and on the changes of configuration during their action.

Hamilton's Equations of Motion.

561.] We have already shewn that

${\frac {dT_{p}}{dp}}={\dot {q}}.$

(4)

Let the system move in any arbitrary way, subject to the conditions imposed by its connexions, then the variations of p and q are

$\delta p={\frac {dp}{dt}}\delta t,\quad \delta q={\dot {q}}\delta t.$

(5)

Hence

${\begin{aligned}{\frac {dT_{p}}{dp}}\delta p&={\frac {dp}{dt}}{\dot {q}}\delta t,\\&={\frac {dp}{dt}}\delta q,\end{aligned}}$

(6)