Mathematics of Physics and Engineering

Hamilton's Equations 95

Substituting (2.3.50) into

1 ™

results in

3fe

£K = Yl aMiQi = QTAq, (2.3.51)

where A = (a,j:e, j,t= 1,... ,3k) is a symmetric matrix and

1 v^ dri dri

(^2) ~i dqj dqe
Clearly, if F(x) — xTBx = Y2i,j bijXjXj is a quadratic form with the
coefficients bij independent of x, then VF = 2Bx and VF • x = 2.F.
Therefore, by (2.3.51),
V f^ = 2£K. (2.3.52)
Since V does not depend on q, it follows that d£x/dqj = dL/dqj = pj,
and (2.3.52) becomes
3fe
Since ff = J2i=i PiQi ~ L and L = £K -V, equality (2.3.48) follows. D
As with most equation related to the Lagrange-Hamilton method, the
main significance of (2.3.46) is its mathematical elegance and high level of
abstraction; it is the level of abstraction that makes (2.3.46) an extremely
interesting object to study in various branches of mathematics. With all
that, equations (2.3.46) simplify the analysis of many concrete problems in
mechanics. Unfortunately, these topics fall outside the scope of this book.

2.4 Elements of the Theory of Relativity

Newtonian mechanics, as discussed in the previous sections, describes the

motion of objects at speeds much smaller than the speed of light; at speeds