Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/220

If ${\dot {q}}_{1}'$ is the greatest and ${\dot {q}}''_{1}$ the least value of the velocity ${\dot {q}}_{1}$ during the action of the force, W must be less than

${\dot {q}}_{1}'\int {F\,dt}\quad {\text{or}}\quad {\dot {q}}_{1}'(p_{1}'-p_{1}),$

and greater than

${\dot {q}}_{1}''\int {F\,dt}\quad {\text{or}}\quad {\dot {q}}_{1}''(p_{1}'-p_{1}).$

If we now suppose the impulse $\int {F\,dt}$ to be diminished without limit, the values of ${\dot {q}}_{1}'$ and ${\dot {q}}''_{1}$ will approach and ultimately coincide with that of ${\dot {q}}_{1}$ and we may write $p'_{1}-p_{1}=\delta p_{1}$, so that the work done is ultimately

$\delta W_{1}={\dot {q}}_{1}\delta p_{1},$

or, the work done by a very small impulse is ultimately the product of the impulse and the velocity.

Increment of the Kinetic Energy.

560.] When work is done in setting a conservative system in motion, energy is communicated to it, and the system becomes capable of doing an equal amount of work against resistances before it is reduced to rest.

The energy which a system possesses in virtue of its motion is called its Kinetic Energy, and is communicated to it in the form of the work done by the forces which set it in motion.

If T be the kinetic energy of the system, and if it becomes T + δT, on account of the action of an infinitesimal impulse whose components are δp₁, δp₁, &c., the increment δT must be the sum of the quantities of work done by the components of the impulse, or in symbols,

${\begin{aligned}\delta T&={\dot {q}}_{1}\delta p_{1}+{\dot {q}}_{2}\delta p_{2}+\And c.,\\&=\sum {({\dot {q}}\delta p)}.\end{aligned}}$

(1)

The instantaneous state of the system is completely defined if the variables and the momenta are given. Hence the kinetic energy, which depends on the instantaneous state of the system, can be expressed in terms of the variables (q), and the momenta (p). This is the mode of expressing T introduced by Hamilton. When T is expressed in this way we shall distinguish it by the suffix _p, thus, T_p.

The complete variation of T_p is

$\delta T_{p}=\sum \left({\frac {dT_{p}}{dp}}\delta p\right)+\sum \left({\frac {dT_{p}}{dq}}\delta q\right).$

(2)