[NOTES/CM-03008] Hamilton's Principle

A solution to the Euler Lagrange equations of motion requires specification of generalized coordinates and generalized velocities at initial time.

Alternatively, a solution to the Euler Lagrange equations can be obtained by specifying the generalized coordinates at the initial and final times \(t_1\) and \(t_2\). As we shall see below, the action principle formulates the path followed, between two end points, as being the path which makes the action extremum.

Infinitesimally close paths

Let~$C$ be a given path,\( \{{\mathbf q}(t)| t_1\le t\le t_2\}\)  ,connecting the points \({\mathbf q}_1\) at \(t_1\) to point \({\mathbf q}_2\) times \(t_2\). Let $C\Prime$ be another path\(\{{\mathbf q}^\prime(t)|t_1^\prime\le t\le t_2^\prime\}\), which differs infinitesimally from the path~$C$. The path $C\Prime$ starts from $\qbf\Prime_1$ at time $t_1\Prime$ and ends at $\qbf_2\Prime$ at time $t_2\Prime$. The values of coordinates for the two paths at time \(t\) are $\qbf(t)$ and \(\qbf\Prime(t)\) for the two paths. We will say that $C\Prime$ is infinitesimally different from the path $C$, if the quantities defined by \begin{equation} \Delta t_1 = t_1\Prime-t_1, \qquad\Delta t_2 = t_2\Prime-t_2, \end{equation} \begin{equation} \Delta \qbf_1 =\qbf^\prime(t_1^\prime)_-\qbf(t_1),\qquad\Delta \qbf_2= \qbf^\prime(t_2^\prime)- \qbf(t_2), \end{equation} and \begin{equation} \delta \qbf(t)= \qbf\Prime(t)-\qbf(t), \qquad t_1 \leq t \leq t_2,\label{EQ09} \end{equation} are infinitesimal quantities. For our present purpose, it will be unimportant weather we take $(t_1,t_2)$, or $(t_1\Prime,t_2\Prime)$, as the range of $t$ in equation \eqref{EQ09}.

The difference in velocities for the two paths is computed by using
\begin{eqnarray}
\delta \mathbf{\dot q(t)}
&=& \frac{d}{dt}\qbf\Prime(t) - \frac {d}{dt}\qbf(t)
=\frac{d}{dt}(\delta \qbf(t)).
\end{eqnarray}

Computing variation of action

To formulate Hamilton's principle,  we compute variation of action functional when path is varied from $C$ to $C\Prime$

\begin{eqnarray}
\Phi[C\Prime]-\Phi[C] 
&=& \int_{t_1\Prime}^{t_2\Prime} L(\qbf\Prime(t),\dot
\qbf\Prime(t),t)dt-\int_{t_1}^{t_2}L(\qbf(t),\dot \qbf(t),t)dt\\
&=& \int_{t_1\Prime}^{t_1}L(\qbf\Prime(t),\dot \qbf\Prime(t),t)dt
+ \int_{t_1}^{t_2}L(\qbf\Prime(t),\dot \qbf\Prime(t),t)dt
+\int_{t_2}^{t_2\Prime}L(\qbf\Prime(t),\dot \qbf(t),t)dt\nonumber\\
&& \hspace{3cm} - \int_{t_1}^{t_2}L(\qbf(t),\dot \qbf(t),t)dt
\\
&\approx& (t_1-t_1\Prime)L(\qbf(t_1),\dot \qbf(t_1),t)~+\int_{t_1}^{t_2}\{L(\qbf\Prime(t),\dot
\qbf \Prime(t),t)-L(\qbf(t),\dot \qbf(t),t)\} dt \nonumber\\
&&\hspace{3cm} +(t_2\Prime -t_2)L(\qbf_2,\dot \qbf_2,t_2) \\[3mm]
&& \qquad \qquad\qquad\qquad\mbox{\HighLight{What lies behind the last step?}} \label{BBX1} \\
&\approx& - \Delta t_1 L(\qbf(t_1),\dot \qbf(t_1),t_1)~+\Delta t_2L(\qbf(t_2),\dot\qbf(t_2),t_2)\nonumber\\
&&\hspace{3cm}+\int_{t_1}^{t_2}\{L(\qbf\Prime(t),\dot \qbf\Prime(t),t) -
L(\qbf(t),\dot \qbf(t),t)\} dt
\label{EQ14}
\end{eqnarray}

We substitute for $\qbf\Prime(t)$ in the last term of \eqref{EQ14} \begin{equation} \qbf\Prime(t) = \qbf(t)+ \delta \qbf(t), \end{equation} and use the fact that the paths $C\Prime$ and $C$ differ by infinitesimal amount to get

\begin{eqnarray}
\int_{t_1}^{t_2} [L(\qbf\Prime,\dot \qbf\Prime,t)- L(\qbf,\dot
\qbf,t)]~dt~
&=&\int_{t_1}^{t_2}\left[L\Big(
\qbf+\delta \qbf,\dot{\qbf}(t)+\frac{d}{dt}\delta \qbf(t)\Big)-L(\qbf,\dot
\qbf,t)\right]\, dt\\
&\approx&\int_{t_1}^{t_2}\sum_k\Big(\frac{\partial{L}}{\partial{q_k}}
\delta q_k+
\frac{\partial{L}} {\partial{\dot q_k}}\delta\dot q_k\Big)dt+
\text{second order terms}\\
&=&\int_{t_1}^{t_2}\sum_k \left(\frac{\partial{L}}{\partial{q_k}} \delta q_k -
\frac{d}{dt}\Big(\frac{\partial{L}}{\partial{\dot q_k}}\Big)
\delta q_k\right)\,dt +\sum_k \frac {\partial {L}}{\partial{\dot q_k}}
\delta q_k\Big|_{t_1}^{t_2} \label{EQ19}
&& \text{\HighLight{Integration by parts has been done in the second term }}
\end{eqnarray}

Substituting \eqref{EQ19} in \eqref{EQ14} we write the final expression as \begin{equation}\label{EQ21} \Phi[C\Prime]-\Phi[C]\approx \int_{t_1}^{t_2}\Big(\sum_k\frac{\partial{L}}{\partial{ q_k}}-\frac{d}{dt}\Big(\frac{\partial{L}}{\partial{\dot q_k}}\Big)\Big)\delta q_k +\Big[L\Delta t + \sum_k \frac{\partial{L}}{\partial{\dot q_k}}\delta q_k\Big]_{t_1}^{t_2} \end{equation}

•  It must be remembered that so far we have kept the variation paths to be general one;  there is no restriction that the paths must have the same end points.

Condition for an extremum

We first consider special class of variations of path which keep the end points fixed, {\it i.e.} \begin{equation} \Delta t_1 = \Delta t_2 =0 \end{equation} \begin{equation} \Delta q_k(t_1)=0;~~~~\Delta q_k(t_2)=0. \end{equation} For such variations we get \begin{equation} \Delta \Phi(C)=\int_{t_1}^{t_2}\sum_k\Big(\frac{\partial{L}}{\partial{q_k}}-\frac{d}{dt }\frac{\partial{L}}{\partial{\dot q_k}}\Big )\delta q_k(t) dt.\label{EQ24} \end{equation}
Since the generalized coordinates are independent and the variations \(\delta \qbf\) are arbitrary, right the right hand side of \eqref{EQ24} vanishes if and only if the Euler Lagrange equations, , \begin{equation} \frac{d}{dt}\Big( \frac{\partial{L}}{\partial{\dot q_k}}\Big)- \frac{\partial{L}}{\partial{q_k}}=0 \end{equation} are satisfied {\it i.e.} $ \qbf(t)$ is solution of EOM. This is summarised into the following statement of action principle.

Action principle --- The statement

Given configurations $\qbf_1,\qbf_2$ at times $t_1$ and $t_2$, the actual dynamical path $C$ followed by a system is that for which the action is stationary {\it i.e.} $C$ is that path about which infinitesimal variations, {\it with fixed end points}, do not produce any change in $\Phi$. If \(C\Prime\) is any other path infinitesimally close to \(C\), then $$\delta \Phi=\Phi [C\Prime]-\Phi [C]=0.$$ In other words, the action is stationary for the actual trajectory followed by the the system.