Hamiltonian formulation 21=
∑^3 N
α=1[
∂H
∂qα∂H
∂pα−
∂H
∂pα∂H
∂qα]
= 0, (1.6.15)
where the second line follows from Hamilton’s equation, eqns. (1.6.11). We will see
shortly that conservation laws, in general, are connected with physical symmetries of
a system and, therefore, play an important role in the analysis of the dynamics.
Hamilton’s equations of motion describe the unique evolution of the coordinates
and momenta subject to a set of initial conditions. In the language of phase space, they
specify a trajectory xt= (q 1 (t),...,q 3 N(t),p 1 (t),...,p 3 N(t)) in the phase space starting
from an initial point x 0. The energy conservation condition
H(q 1 (t),...,q 3 N(t),p 1 (t),...,p 3 N(t)) = constis expressed as a condition on a phase space trajectory. It can also be expressed as a
condition directly on the coordinates and momentaH(q 1 ,...,q 3 N,p 1 ,...,p 3 N) = const,
which defines a 6N−1 dimensional surface in the phase space on which a trajectory
must remain. This surface is known as theconstant-energy hypersurfaceor simply the
constant-energy surface. An important theorem, known as thework–energy theorem,
follows from the law of conservation of energy. Consider the evolution of the system
from a point xAin phase space to a point xB. Since energy is conserved, the energy
HA=HB. But sinceH=K+U, it follows that
KA+UA=KB+UB, (1.6.16)or
KA−KB=UB−UA. (1.6.17)
The right side expresses the difference in potential energy between points A and B and
is, therefore, equal to the work,WAB, done on the system in moving between these
two points. The left side is the difference between the initial and finalkinetic energy.
Thus, we have a relation between the work done on the system and the kinetic energy
difference
WAB=KA−KB. (1.6.18)
Note that ifWAB>0, net work is done on the system, which means that its potential
energy increases, and its kinetic energy must decrease between points A and B. If
WAB<0, work is done by the system, its potential energy decreases, and its kinetic
energy must, therefore, increase between points A and B.
In order to understand the formal structure of a general conservation law, consider
the time evolution of any arbitrary phase space function,a(x). Viewing x as a function
of time xt, the time evolution can be analyzed by differentiatinga(xt) with respect to
time:
da
dt=
∂a
∂xt· ̇xt