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Hamilton's Equations
Consider a dynamical system with
degrees of freedom that is described
by the generalized coordinates
, for
. Suppose that
neither the kinetic energy,
, nor the potential energy,
, depend
explicitly on the time,
. 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,
![$\displaystyle K = \sum_{i,j = 1,{\cal F}} m_{ij} \dot{q}_i \dot{q}_j,$](img3035.png) |
(B.68) |
where the
depend on the
, but not on the
.
It is easily demonstrated from the previous equation that
![$\displaystyle \sum_{i=1,{\cal F}} \dot{q}_i \frac{\partial K}{\partial \dot{q}_i} = 2 K.$](img3037.png) |
(B.69) |
Recall, from Section B.4, that generalized momentum conjugate to the
th
generalized coordinate is defined
![$\displaystyle p_i = \frac{\partial L}{\partial \dot{q}_i} = \frac{\partial K}{\partial \dot{q}_i},$](img3038.png) |
(B.70) |
where
is the Lagrangian of the system, and we have made use of the fact that
is independent of the
. Consider the
function
![$\displaystyle H = \sum_{i=1,{\cal F}} \dot{q}_i p_i - L = \sum_{i=1,{\cal F}} \dot{q}_i p_i -K + U.$](img3039.png) |
(B.71) |
If all of the conditions discussed previously are satisfied then Equations (B.69)
and (B.70)
yield
![$\displaystyle H = K+ U.$](img3040.png) |
(B.72) |
In other words, the function
is equal to the total energy of the system.
Consider the variation of the function
. We have
![$\displaystyle \delta H = \sum_{i=1,{\cal F}} \left(\delta\dot{q}_i p_i + \dot{...
...t{q}_i} \delta \dot{q}_i - \frac{\partial L}{\partial q_i} \delta q_i\right).$](img3041.png) |
(B.73) |
The first and third terms in the bracket cancel, because
. Furthermore, because Lagrange's equation
can be written
(see Section B.4), we obtain
![$\displaystyle \delta H = \sum_{i=1,{\cal F}} \left(\dot{q}_i \delta p_i - \dot{p}_i \delta q_i\right).$](img3044.png) |
(B.74) |
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
![$\displaystyle \delta H =\sum_{i=1,{\cal F}} \left(\frac{\partial H}{\partial p_i} \delta p_i + \frac{\partial H}{\partial q_i} \delta{q}_i\right).$](img3046.png) |
(B.75) |
A comparison of the previous two equations yields
for
. These
first-order 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
second-order differential equations.
Next: Wave Mechanics
Up: Classical Mechanics
Previous: Hamilton's Principle
Richard Fitzpatrick
2016-01-25