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

 Rh 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., 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), 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 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 Tp with respect to the momenta, as in equation ($$),

This shews that Tp 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 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 and