Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/224

or, the components of momentum are the differential coefficients of $T_{\dot {q}}$ with respect to the corresponding velocities.

Again, by equating to zero the coefficients of $\delta q_{1}$, &c.,

${\frac {dT_{p}}{dq_{1}}}+{\frac {dT_{\dot {q}}}{dq_{1}}}=0;$

(18)

or, the differential coefficient of the kinetic energy with respect to any variable $q_{1}$ is equal in magnitude but opposite in sign when T is expressed as a function of the velocities instead of as a function of the momenta.

In virtue of equation (18) we may write the equation of motion (9),

$F_{1}={\frac {dp_{1}}{dt}}-{\frac {dT_{\dot {q}}}{dq_{1}}},$

(19)

or

$F_{1}={\frac {d}{dt}}{\frac {dT_{\dot {q}}}{d{\dot {q_{1}}}}}-{\frac {dT_{\dot {q}}}{dq_{1}}},$

(20)

which is the form in which the equations of motion were given by Lagrange.

565.] In the preceding investigation we have avoided the consideration of the form of the function which expresses the kinetic energy in terms either of the velocities or of the momenta. The only explicit form which we have assigned to it is

$T_{p{\dot {q}}}={\frac {1}{2}}(p_{1}{\dot {q}}_{1}+p_{2}{\dot {q}}_{2}\,+\And c.),$

(21)

in which it is expressed as half the sum of the products of the momenta each into its corresponding velocity.

We may express the velocities in terms of the differential coefficients of T_p with respect to the momenta, as in equation (3),

$T_{p}={\frac {1}{2}}(p_{1}{\frac {dT_{p}}{dp_{1}}}+p_{2}{\frac {dT_{p}}{dp_{1}}}\,+\And c.),$

(22)

This shews that T_p is a homogeneous function of the second degree of the momenta p₁, p₂, &c.

We may also express the momenta in terms of $T_{\dot {q}}$, and we find

$T_{\dot {q}}={\frac {1}{2}}({\dot {q}}_{1}{\frac {dT_{\dot {q}}}{d{\dot {q}}_{1}}}+{\dot {q}}_{2}{\frac {dT_{p}}{d{\dot {q}}_{1}}}\,+\And c.),$

(23)

which shews that $T_{\dot {q}}$ is a homogeneous function of the second degree with respect to the velocities ${\dot {q}}_{1}$, ${\dot {q}}_{2}$, &c.

If we write

$P_{11}{\text{ for }}{\frac {d^{2}T_{\dot {q}}}{d{\dot {q}}_{1}^{2}}},\quad P_{12}{\text{ for }}{\frac {d^{2}T_{\dot {q}}}{d{\dot {q}}_{1}d{\dot {q}}_{2}}},\And c.$

and

$Q_{11}{\text{ for }}{\frac {d^{2}T_{p}}{dp_{1}^{2}}},\quad Q_{12}{\text{ for }}{\frac {d^{2}T_{p}}{dp_{1}dp_{2}}},\And c;$