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A Treatise on the Application of Generalised Coordinates to by Henry William Watson

By Henry William Watson

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Extra resources for A Treatise on the Application of Generalised Coordinates to the Kinetics o

Example text

H and _T 7 -P F is the expression for 2 ^ with -j- written for p. Secondly let T be given, and let q, p be a set of velocities and momenta in a motion such that T has the given value, 7 / and that p= A. ~ > and let q +q , jQ +p motion having the same kinetic energy. 2/^ + 22^ = or be those in any other Then ................... (1) Therefore (2) Now let q" denote a set of velocities proportional to the q set where r is some numerical quantity, and con so that q = r q\ sequently p = rp" .

G. of the second then the kinetic energy T = \ { T is such that body first axis and parallel to the round an two axes, 25 EXAMPLES OF LAGRANGE s EQUATIONS. ] whence Lagrange equations become s dt a^fy + m ab^r (mf + m or m ah dtc where </> [<cos(i/f ^ and <I> and to [\js <)] cos(\/r m db sin(x// </>)] = 2 cos\/f +m 2 +& (6 ma&\//- </> 2 ) \j/- +m sini//- 6 sin </> (^ are the generalised force components corresponding \|/.

Thomson 330, p. ) Here there are two degrees of freedom, and the coordinates may be conveniently taken to be (i) $ the inclination of the plane containing the axes to a fixed plane through the first axis, and (2) \|/ the inclina tion of the^plane nf lliu UXCB to a plane through the second axis and the centre of gravity of the second body. With this notation it is easily seen that if a be the distance between the axes, and 6 the distance of the C. G. of the second body from the second axis, then the (velocity) 2 of the C.

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