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Consider a dynamical system with degrees of freedom which is described
by the generalized coordinates , for . Suppose that
neither the kinetic energy, , nor the potential energy, , depend
explicitly on the time, . Now, in conventional dynamical systems, the potential energy is generally independent of the , whereas the kinetic
energy takes the form of a homogeneous quadratic function of
the . In other words,

(744) 
where the depend on the , but not on the .
It is easily demonstrated from the above equation that

(745) 
Recall, from Section 9.8, that generalized momentum conjugate to the th
generalized coordinate is defined

(746) 
where is the Lagrangian of the system, and we have made use of the fact that is independent of the . Consider the
function

(747) 
If all of the conditions discussed above are satisfied then Equations (745)
and (746)
yield

(748) 
In other words, the function is equal to the total energy of the system.
Consider the variation of the function . We have

(749) 
The first and third terms in the bracket cancel, because
. Furthermore, since Lagrange's equation
can be written
(see Section 9.8), we obtain

(750) 
Suppose, now, that we can express the total energy of the system, , solely
as a function of the and the , with no explicit
dependence on the . In other words, suppose that we
can write . When the energy is written
in this fashion it is generally termed the Hamiltonian of the system. The variation of the Hamiltonian function takes the form

(751) 
A comparison of the previous two equations yields
for . These firstorder differential equations are known
as Hamilton's equations. Hamilton's equations are often a
useful alternative to Lagrange's equations, which take the
form of secondorder differential equations.
Consider a onedimensional harmonic oscillator. The kinetic and potential
energies of the system are written
and
, where is the displacement, the mass, and .
The generalized momentum conjugate to is

(754) 
Hence, we can write

(755) 
So, the Hamiltonian of the system takes the form

(756) 
Thus, Hamilton's equations, (752) and (753), yield
Of course, the first equation is just a restatement of Equation (754), whereas the second is Newton's second law of motion for the
system.
Consider a particle of mass moving in the central potential .
In this case,

(759) 
where are polar coordinates. The generalized momenta conjugate to and are
respectively.
Hence, we can write

(762) 
Thus, the Hamiltonian of the system takes the form

(763) 
In this case, Hamilton's equations yield
which are just restatements of Equations (760) and (761), respectively,
as well as
The last equation implies that

(768) 
where is a constant. This can be combined with Equation (766)
to give

(769) 
where . Of course, Equations (768) and (769) are the
conventional equations of motion for a particle moving in a central potentialsee Chapter 5.
Next: Exercises
Up: Hamiltonian Dynamics
Previous: Constrained Lagrangian Dynamics
Richard Fitzpatrick
20110331