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If a number of equations of con- Such conditions, which do not involve the velocity components, are called holonomous. THE MECHANICS OF THE ATOM 20 straint be given, an equal number of co-ordinates can be eliminated with their help. There remains then a certain number of independent co-ordinates qf q^ 2 . . of degrees of freedom. , y2 ^ 2 , ... -- These are also valid if the co-ordinates q k refer to arbitrarily moving, or even deformed, systems of reference. 5. The Canonical Equations Each of Lagrange's equations is a differential equation of the second order.

For, if V be regarded as a function of the qk V V or of q k p k> t, or finally of p k canonical transformations. , difference of the integrands is of a function of 2f of the old at ; value. at the limits of integration will be fixed. d q k , the values of According now as we |d=stationary , , p k, t, , , we obtain four principal forms for THE THEORY OF HAMILTON AND JACOBI We choose therefore an arbitrary function V(y 1 q^ . , . 29 t). The condition is fulfilled, if the coefficients of q k and q k and of terms independent of , these quantities, are the same on both sides, that is if o P* (1) = # H=H~ Since in general the y 's can be calculated from the equations of the second line, and then the p k s can be calculated from the equations of fc as functions of the q k and p k> the system (1) replaces the of transformation.

If the zero the interval points are (1, different, it means that the point of intersection of the axis of the top with a sphere, described about the centre of the top, oscillates backwards and forIt wards between two parallel circles. describes a curve shown in fig. 1. In the case of the double root our equations (13) and (14) fail, but the motion can be easily calculated in an elementary manner the case of a regular precession. : FIG. is 1. then a constant, and we have THE MECHANICS OF THE ATOM 44 A general rule for the rigorous solution of the Hamilton-Jacobi cannot be given.

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