[NOTES/CM-04001] Hamiltonian Formulation of Classical Mechanics

Introduction

We have seen that for a system with Lagrangian $L(\qbf,\dot \qbf,t)$ the equations of motion are given by Euler Lagrange equations
\begin{equation}\Label{EQ01} \frac{d}{dt}\Big(\frac{\partial{L}}{\partial{\dot q_k}}\Big)-\frac{\partial{L}}{\partial{q_k}}=0, \qquad k=1,2,\ldots \end{equation}
It should be noted that these are second order differential equations and describe the motion completely if the initial values of generalized coordinates and velocities are specified. In the Lagrangian approach the generalised coordinates $q_k$ and generalized velocity $\dot q_k$ are the basic variables choice of a Lagrangian is is equivalent to specifying the forces on the system in Newtonian mechanics and Euler Lagrange equation \eqref{EQ01} represent the Laws of motion. Equivalently, the the action principle in Hamilton's form or Weiss statement, gives the laws of dynamics.

We shall now describe another formalism called the Hamiltonian formalism where the basic variables are $q_k,p_k$, The interaction is described by Hamiltonian instead of Lagrangian.The EOM are now first order differential equations in $t$ (see Hamilton's equation below) and the action principle can be now formulated in phase space.

Canonical momenta and Hamiltonian

We define the canonical momentum $p_k$ conjugate to coordinate $q_k$ by
\begin{equation}\Label{EQ02} p_k\stackrel{\rm def}{\equiv}\frac{\partial{L}}{\partial{\dot q_k}}= p_k (q,\dot q,t) \end{equation}
These equations give momenta $p_k$ as a function of $q_k,\dot q,t$ and are inverted to solve for the generalized velocities. Thus the generalized velocities are expressed as functions of $q_k,q_k,t$
\begin{equation}\Label{EQ03} \dot q_k =\dot q_k(\qbf,\pbf,t). \end{equation}
The expression $$\sum_k p_k\dot q_k-L(q,\dot q,t),$$ termed as Hamiltonian, must be expressed in terms of the coordinates $q_k$ and momenta $p_k$ by eliminating velocities using \eqref{EQ03}:
\begin{equation} \sum p_k \dot q_k- L(\qbf,\qbf_k,t)\stackrel{\rm use \eqref{EQ03}}{\longrightarrow}H(\qbf,\pbf,t) \end{equation}
It may be remarked that condition under which the velocities can be expressed in terms of momenta is that the determinant \(\det \Big(\dfrac{\partial^2L}{\partial \dot x_j\partial \dot x_k}\Big)\), of the matrix of the second order mixed partial derivatives with respect to the velocities, be non zero.

Hamilton's Equations

\begin{equation}\dot q_k= \frac{\partial{H}}{\partial{p_k}}~;~~~~~~~~\dot p_k= -\frac{\partial{H}}{\partial{q_k}}.
\end{equation}
Before proceeding further a word of caution is necessary. In Lagrangian formalism $q_k$and $\dot q_k$ are treated as independent for purposes of partial differentiation where as in the Hamiltonian formalism $q_k$and $p_k$ are to be treated as independent variables.
\begin{equation} \frac{\partial}{\partial{q_k}} F(\qbf,\dot{\qbf},t)\rightarrow \frac{\partial}{\partial{q_k}}F(\qbf,\dot{\qbf},t)\Big|_{\dot{q},q_j}, \quad j\ne k \end{equation}

\begin{equation} \frac{\partial}{\partial{\dot q_k}}F(\qbf,\dot {\qbf},t)\rightarrow\frac{\partial}{\partial{\dot{q}_k}} F(\qbf,\dot{\qbf},t)\Big|_{\qbf,\dot{\qbf}_k} ~~\text{here}~~ j\neq k, \end{equation}
where as in Hamiltonian formalism
\begin{equation} \frac{\partial{F}}{\partial{q_k}}\rightarrow\frac{\partial{F}}{\partial{q_k}} \Big|_{q_j,\pbf};~~~~\frac{\partial}{\partial{p}_k}F(\qbf,\pbf,t)\rightarrow\frac{\partial{F}}{ \partial{p}_k}\Big|_{\qbf, p_j}, \qquad\qquad j\ne k. \end{equation}

We begin with the expression
\begin{equation} H = \sum_k p_k\dot q_k-L(q_k,\dot q_k,t) \end{equation} for the Hamiltonian and compute change as $\qbf,\pbf,\dot \qbf$ vary by small amounts
\begin{equation} dH= \sum p_kd\dot q_k +\sum dp_k\dot q_k -\frac{\partial{L}}{\partial{q_k}} dq_k-\frac{\partial{L}}{\partial\dot q_k}\, d\dot q_k-\frac{\partial{L}}{\partial{t}}\delta t. \end{equation} Not all the differentials $dq_k, d\dot q_k, d p_k$ are independent because $q,p,\dot q$ are related. Making use of \eqref{EQ02}, we get
\begin{equation} dH=\sum_k \dot q_k dp_k-\sum_k\frac{\partial{L}}{\partial{q_k}}d q_k-\frac{\partial{L}}{\partial{t}}dt \end{equation} which gives
\begin{equation}\Label{EQ13} \frac{\partial{H}}{\partial{t}}=-\frac{\partial{L}}{\partial{t}} \end{equation} and
\begin{equation}\Label{EQ12} \frac{\partial{H}}{\partial{p_k}}= \dot q_k; ~~\frac{\partial{H}}{\partial{q_k}} = -\frac{\partial{L}}{\partial{q_k}} = -\dot p_k. \end{equation} Using EOM \eqref{EQ01} and the definition \eqref{EQ02} of \(p_k\), in the last term, we arrive at \begin{equation} \boxed{\dot q_k =\frac{\partial{H}}{\partial{p_k}},\qquad \dot p_k.=-\frac{\partial{H}}{\partial{q_k}}.} \end{equation} These equations are the required Hamilton's equations of motion.