
130 Classical Mechanics
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or, simply,
H = E = T + V = p
2
/2m+ kx
2
/2.
Hamilton’s equations give
Hp pm xHxkxp Ht//
nd
.
The first equation is the definition of momentum; the second equation is the equation of motion
,
and the last equation is the statement of conservation of energy.
Example 5.2
A particle of mass m is attracted to a fixed point O by an inverse square force FGMr kr
r
=− =−
(Figure 5.1). Solve this problem using Hamilton’s equations.
Solution:
As shown in Figure 5.1, we use plane polar coordinates (r, θ). The velocity of the particle is given
by
vrere
r
=+
θ
θ
.
Hence,
Tmrr
1