Mathematics of Physics and Engineering

Hamilton's Equations 93

where Q is the generalized force from (2.3.16), that is,

Q = F • dr/dq = -mg k • dr/dq. (2.3.41)

We therefore accomplished our objective and verified the Lagrange method
for our example by explicitly deriving equation (2.3.40). Once (2.3.39) and
(2.3.41) are taken into account, equation (2.3.40) becomes a second-order
ODE for the function q = q(t). A reader interested in solving this equation
should try Problem 2.9, page 421.
Notice that the unknown, constraint-induced, reaction force N appears
in Newton's equation (2.3.33), but not in the Lagrange equation (2.3.40).
Of course, the Newton-Euler method, requiring elimination of the unknown
force N, results in the same equation of motion; see Problem 2.9 for details.

2.3.3 Hamilton's Equations

Lagrange's equations (2.3.27) are a system of k second-order ODEs. A

change of variables reduces this system to Ik first-order ODEs. While there

are many changes of variables to achieve this reduction, one special change

transforms (2.3.27) into a particularly elegant form, known as Hamilton's

equations.

Given the Lagrange function (2.3.26), introduce new variables p =

iPu • • • ,Pk) by

*=!£•

(2

3

and assume that equations (2.3.42) are solvable in the form

q = f(p,q) (2.3.43)

for some vector function / = (/i,. •-,/&)• For example, if L =

\J2j=i(mj(lj ~ hjQj)i tnen (2.3.42) becomes pj = rrijqj or qj = Pj/rrij.

In applications to mechanics, <jj usually has the dimension of distance and

rrij is the mass, so that pj has the dimension of momentum. This is why

the variable pj defined by (2.3.42) is called the generalized momentum.

Together with (2.3.42), equations (2.3.27) become a system of k first-

order ODEs