[NOTES/CM-04003] Variational Principles in Phase Space

Time development as motion in phase space

In canonical form of dynamics, we have $2N$ variables $q_k,p_k$ which satisfy first order differential equations. As time passes, $q_k,p_k$ values change and at any given time the state of the system is described completely by values $(\qbf,\pbf)$. The values of $(\qbf,\pbf)$ can represented by a point in $2N$ dimensional space called {\bf phase space}. As system evolves in time, the point representing the state of the system traces out a path in phase space.
\begin{equation}
\Phi[C]=\int_{t_1}^{t_2}\Big (\sum p_k\dot q_k-H\Big )dt.
\end{equation}
This motion of phase space points is described by the Hamilton's equations.

Variational principles in phase space

The Hamilton's equations of motion can be derived from a principle of least action in phase space. Without loss of generality, we will restrict our discussion to system with one degree of freedom.
A path in phase space \(\Gamma\) from time \(t_1\) to \(t_2\) is specified by giving giving \(\{q(t),p(t)|t_1\le t\le t_2 \}\). Another path \(\Gamma\Prime = \{q\Prime(t), p\Prime(t)|t_1\le t\le t_2\}\) will be called infinitesimally close \(\Gamma\Prime\) if the quantities \(\delta q, \delta p\) defined by
\begin{equation}
\delta q(t)= q\Prime(t)- q(t), \qquad \delta p(t)=p\Prime(t)-p(t)
\end{equation}
are infinitesimal quantities. We begin with \(L=p\dot q -H(q,p,t)\) and write the action functional as \begin{equation}
\Phi[\Gamma] = \int_{t_1}^{t_2} \big[p \dot q - H\big] dt.
\end{equation}
With no restrictions on \(\delta q, \delta p\) the variation of the action functional is
\begin{eqnarray}
\phi[\Gamma\Prime] - \Phi[\Gamma]
&=& \int \left[ p(\delta \dot q) + (\delta p) \dot q- \pp[H]{q}\,\delta q - \pp[H]{p}\,\delta p\right] dt.\Label{EQ03}
\end{eqnarray}
Variation in \(\dot q\) is computed by making use of \(\delta \dot q = \dd[\delta q]{t}\). Substituting for \(\delta \dot q\) in \eqRef{EQ03} and doing an integration by parts gives
\begin{eqnarray}
\phi[\Gamma\Prime] - \Phi[\Gamma]
&=& \int \left[ \dd{t}(p\delta q) + (\delta p) \dot q - \pp[H]{q}\,\delta q - \pp[H]{p}\,\delta p\right] dt.\nonumber\\
&=& p\delta q\Big|_{t_1}^{t_2} + \int \left[- (\delta p) \dd[q]{t} + \dot p(\delta q) - \pp[H]{q}\,\delta q - \pp[H]{p}\,\delta p\right] dt.\Label{EQ05}\\
&=& p\delta q\Big|_{t_1}^{t_2} + \int_{t_1}^{t_2} \Big[-\delta q\Big(\dot p + \pp[H]{q}\Big) + \delta p\Big(\dot q - \pp[H]{p}\Big)\Big]\label{PhiVar}
\end{eqnarray}
\(\delta q=0\) at both end points\\}
So far we have not restricted the path \(\Gamma\Prime\) in any way. We now consider variations with \(\delta q=0\) at the end points. Then \(\delta \Phi\) vanishes if and only if
\begin{eqnarray}
\delta q\Big(\dot p + \pp[H]{q}\Big) - \delta p\Big(\dot q - \pp[H]{p}\Big) =0
\end{eqnarray}
Since the variations \(\delta q(t)\) and \(\delta p(t)\) are independent and arbitrary, the above relation implies Hamiltonian equations
\begin{equation}
\dot q =\pp[H]{p}, \qquad \dot p = -\pp[H]{q}.
\end{equation}
It may be noted that there is no condition on variation of momentum \(p\) at the end points.

Variations about a classical path without fixed end points
We now consider the case when the path \(\Gamma\) is classical trajectory, but the variations are not restricted to fixed end points. The second term in \eqref{PhiVar} vanishes, we have for several degrees of freedom,
\begin{equation}
\delta \Phi = \sum_k p_k \delta q_k \Big|_{t_1}^{t_2}
\end{equation}
This result shows that the variation in the action about a classical path depend on the end points only.