Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/219

559.]
187
IMPULSE AND MOMENTUM.

that we can no longer assume that the momentum is the time-integral of the force which acts on it.

But the increment δq of any variable cannot be greater than $\dot{q}' \delta t$, where δt is the time during which the increment takes place, and $\dot{q}'$ is the greatest value of the velocity during that time. In the case of a system moving from rest under the action of forces always in the same direction, this is evidently the final velocity.

If the final velocity and configuration of the system are given, we may conceive the velocity to be communicated to the system in a very small time δt, the original configuration differing from the final configuration by quantities δq1, δq2, &c., which are less than $\dot{q}_1 \delta t$, $\dot{q}_2 \delta t$, &c., respectively.

The smaller we suppose the increment of time δt, the greater must be the impressed forces, but the time-integral, or impulse, of each force will remain finite. The limiting value of the impulse, when the time is diminished and ultimately vanishes, is defined as the instantaneous impulse, and the momentum p, corresponding to any variable q, is defined as the impulse corresponding to that variable, when the system is brought instantaneously from a state of rest into the given state of motion.

This conception, that the momenta are capable of being produced by instantaneous impulses on the system at rest, is introduced only as a method of defining the magnitude of the momenta, for the momenta of the system depend only on the instantaneous state of motion of the system, and not on the process by which that state was produced.

In a connected system the momentum corresponding to any variable is in general a linear function of the velocities of all the variables, instead of being, as in the dynamics of a particle, simply proportional to the velocity.

The impulses required to change the velocities of the system suddenly from $\dot{q}_1$, $\dot{q}_2$, &c. to $\dot{q}'_1$, $\dot{q}'_2$, &c, are evidently equal to p1' - p1, 'pp2' - p2, the changes of momentum of the several variables.

Work done by a Small Impulse.

559.] The work done by the force Fl during the impulse is the space-integral of the force, or

 \begin{align} W &= \int {F_1 \, dq_1}, \\ &= \int {F_1 \dot{q}_1 \, dt}. \end{align}