Generalized Hamiltonian Dynamics

Generalized Coordinates
The name generalized coordinates is given to any set of quantities which completely specifies as the state of system. The generalized coordinates are customarily written as $q_{1},q_{2},q_{3},\ldots$ or simply as the $q_{j}$. A set of independent generalized coordinates whose number equals the number $s$ degrees of freedom of the system and which are not restricted by the constraints will be called a proper set of generalized coordinates. In certain instances it may be advantageous to use generalized coordinates whose number exceeds the number of degrees through the use of the Lagrange undetermined multipliers. We shall consider a general mechanical system which consists of a collection of $n$ discrete, point particles. In order to specify the state of such a system at a given time, it is necessary to use $n$ radius vectors. If there exist equation of constraint which relate some of these coordinates to others, then not all of the $3n$ coordintes are independent. In fact, if there are $m$ equations of constraint, then $3n-m$ coordinats are independent, and the system is said to possess $3n-m$ degrees of freedom. In addition to the generalized coordinates, we may define a set of quantities which consists of the time derivatives of the $q_{j}$ such as $\dot{q}_{1},\dot{q}_{2},\dot{q}_{3},\ldots$ or simply $\dot{q}_{j}$. In analogy with the nomenclature for rectangular coordinates we call $\dot{q}_{j}$'s the generalized velocities. We may represent the state of such a system by a point in an $s$-dimensional space called configuration space, each point specifying the configuration of the system at a particular instant. A dynamical path in a configuration space consisting of proper generalized coordinates is automatically consistent with the constraints on the system.
Principle of least action
In physics, the principle of least action is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. The principle of least action is defined by the statement that for each mechanical system there exists a certain integral $S$, called the action. It has a minimum value for the actual motion, so that its variation $\delta S$ is zero. To determine the action integral for a free material particle, the integrand must be a differential of the first order . But only scalar of this kind that one can construct for a free particle is the intercal $ds$, or $\alpha ds$ , where $\alpha$ is some constant. So for a free particle the action must have the form- \begin{equation} S=\alpha\int_{a}^{b}ds\label{eq:2-1} \end{equation} The $\int_{a}^{b}$ is an integral along the world line of the particle at the initial position and at the final position at definite times $t_{1}$and $t_{2}$ and $\alpha$ is some constant characterizing the particle. So the dynamics of a system is governed by the stationarity of the action integral can be represent as an integral with respect to the time- \begin{equation} S=\int_{a}^{b}L(q_{i}(t),\dot{q}_{i}(t))dt\label{eq:2-2} \end{equation} where the Lagrangian function $L=L(q,\dot{q})$ is the Lagrange function of the mechanical system in the generalized coordinates $q_{i}(i=1,2,\ldots N)$ and the velocities $\dot{q_{i}}=\frac{dq_{i}}{dt}$. We assume that the system has $N$ degrees of freedom. Now if $q_{i}(t)=q_{i}^{classical}(t)+\varepsilon\delta q(t)$ then as $S=S(\varepsilon)$ has minimum at $\varepsilon=0$ we have- \begin{eqnarray} \frac{dS(0)}{d\varepsilon} & = & 0\nonumber \\ & = & \int_{t_{i}}^{t_{f}}\left(\frac{\partial L}{\partial q_{i}}\delta q_{i}+\frac{\partial L}{\partial\dot{q_{i}}}\delta\dot{q_{i}}\right)dt\nonumber \\ & = & \int_{t_{i}}^{t_{f}}\left(\frac{\partial L}{\partial q_{i}}+\frac{d}{dt}\frac{\partial L}{\partial\dot{q_{i}}}\right)dt\;\text{ (Integration by parts)}\label{eq:2-3} \end{eqnarray} As $\delta q_{i}$ is arbitrary the variation of the action integral leads to the Lagrange equation of motion- \begin{equation} \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0\label{eq:2-4} \end{equation}
The Hamiltonian
We consider a dynamical system of $N$ degrees of freedom, described in terms of generalized coordinates $q_{n}(n=1,2,\ldots,N)$ and velocities $\dot{q}_{n}$ or $\dfrac{dq_{n}}{dt}$. We assume a Lagrangian $L$, which for the present can be any function of the coordinates and velocities. \begin{equation} L\equiv L(q,\dot{q})\label{eq:2-5} \end{equation} If the Lagrangian is expressed in generalized coordinates, we define generalized momenta according to- \begin{equation} p_{i}=\frac{\partial L}{\partial\dot{q}_{i}}\label{eq:2-6} \end{equation} For the development of the theory we introduce a variation procedure,varying each of the quantities $q_{n}$,$\dot{q}_{n}$and $p_{n}$ independently by small quantity $\delta q_{n}$,$\delta\dot{q}_{n}$ and $\delta p_{n}$ of order $\epsilon$ and working to the accuracy of $\epsilon$. As a result of this variation procedure equation \eqref{eq:2-6} will get violated, as its left-hand side will be made to differ from its right-hand side by a quantity of order $\epsilon$. The Hamiltonian $H$ is defined by the equation- \begin{equation} H\equiv p_{i}\dot{q}_{i}-L\label{eq:2-7} \end{equation} where a summation is understood over all values for a repeated suffix in a term. From the equation..... it is clear that the hamitonian $H$ is the function of both position and velocities. The $p_{i}(q,\dot{q})$ is defined by the equation \eqref{eq:2-6}. Since $H$ doesnot explicitly depend on $\dot{q}_{i}$, we have- \begin{align} \frac{\partial H}{\partial\dot{q}_{i}} & =p_{i}-\frac{\partial L}{\partial\dot{q}_{i}}\nonumber \\ & =p_{i}-p_{i}\quad[\text{using \eqref{eq:2-6}}]\nonumber \\ & =0\label{eq:2-8} \end{align} For a slight change of hamiltonian $\delta H$, we have \begin{eqnarray} \delta H & = & \delta(p_{i},\dot{q_{i}})-\delta L\nonumber \\ & = & \dot{q_{i}}\delta p_{i}+p_{i}\delta\dot{q_{i}}-\frac{\partial L}{\partial q_{i}}\delta q_{i}-\frac{\partial L}{\partial\dot{q_{i}}}\delta\dot{q_{i}}\nonumber \\ & = & \dot{q_{i}}\delta p_{i}-\frac{\partial L}{\partial q_{i}}\delta q_{i}+\left(p_{i}-\frac{\partial L}{\partial\dot{q_{i}}}\right)\delta\dot{q_{i}}\nonumber \\ & = & \dot{q_{i}}\delta p_{i}-\frac{\partial L}{\partial q_{i}}\delta q_{i}\label{eq:2-9} \end{eqnarray} From \eqref{eq:2-9} we can see that $\delta H$ does not depend on the $\delta\dot{q}$'s.
Equations of motion
Equations of motion are equations that describe the behaviour of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables. Normally spatial coordinates and time are used, but others are also possible, such as momentum components and time. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system.If the dynamics of a system is known, the equations are the solutions to the differential equations describing the motion of the dynamics. If the potential energy of a system is velocity-independent, then the linear momentum components in rectangualr coordinates are gieven by \begin{equation} p_{i}=\frac{\partial L}{\partial\dot{q}_{i}}\label{eq:2-10} \end{equation} By analogy we extend this result to the case in which the Lagrangian is expressed in generalized coordinates and define the generalized momenta according to \begin{equation} p_{i}=\frac{\partial L}{\partial\dot{q}_{i}}\label{eq:2-11} \end{equation} Now the Lagrange equation of motion are then expressed by- \begin{equation} \dot{p}_{i}=\frac{\partial L}{\partial q_{i}}\label{eq:2-12} \end{equation} And from \eqref{eq:2-7} we can define the Hamiltonian as- \begin{equation} H=\sum_{i}p_{i}\dot{q}_{i}-L\label{eq:2-13} \end{equation} Now the Lagrangian is considered to be a function of the generalized coordinates, the generalized velocities, and possibly the time. The dependence of $L$ on the time may arise either if the constraints are time dependent or if the transformation equations connecting the rectangualar and generalized coordinates explicitly contain the time. We may solve \eqref{eq:2-11} for the generalized velocities and express them as- \begin{equation} \dot{q}_{i}=\dot{q}_{i}(p_{j},q_{j},t)\label{eq:2-14} \end{equation} Thus in \eqref{eq:2-13} we may express the Hamiltonian as- \begin{equation} H(p_{i},q_{i},t)=\sum_{j}p_{j}\dot{q}_{j}-L(\dot{q}_{i},q_{i},t)\label{eq:2-15} \end{equation} This equation is written in a manner which stresses the fact that the Hamiltonian is always considered as a function of the $(p_{i},q_{i},t)$. Therefore the total differential of $H$ may be calculated by \begin{equation} dH=\sum_{k}\left(\frac{\partial H}{\partial q_{i}}dq_{i}+\frac{\partial H}{\partial p_{i}}dp_{i}\right)\label{eq:2-15-a} \end{equation} whereas the Lagrangian is a function of $(p_{i},q_{i},t)$ set.
Hamilton's equations of motion
From \eqref{eq:2-9} and \eqref{eq:2-15-a} if we identify the coefficients of $\delta p_{i}$ and $\delta q_{i}$ \begin{align} \dot{q}_{i} & =\frac{\partial H}{\partial p_{i}}\label{eq:2-16}\\ \dot{p}_{i} & =-\frac{\partial H}{\partial q_{i}}\label{eq:2-17} \end{align} where the dot denotes the ordinary derivative with respect to time of generalized momenta $p_{i}=p_{i}(t)$ and the generalized coordinates $q_{i}=q_{i}(t)$, where $i=1,2,...n$. Equation \eqref{eq:2-16} and \eqref{eq:2-17} are Hamiton's equations of motion. Because of their symmetrical appearance, they are also known as the canonical equations of motion.
Poisson Bracket
In canonical coordinates on the phase space the poisson bracket of two function $A(p_{i},q_{i},t)$ and $B(p_{i},q_{i},t)$ is defined by \begin{equation} \{A,B\}_{PB}=\sum_{i=1}^{N}\left(\frac{\partial A}{\partial q_{i}}\frac{\partial B}{\partial p_{i}}-\frac{\partial A}{\partial p_{i}}\frac{\partial B}{\partial q_{i}}\right)\label{eq:2-18} \end{equation} Poisson brackets are antisymmetric and it also satisfies the Jacobi identity. Possion brackets deform to quantum commutator in Hilbert space. The Hamilton's equations of motion have an equavalent expression in terms of the Poisson bracket. This may be most directly demostrated in an explicit coordinate frame. Suppose $A(p_{i},q_{i},t)$ is a function on the mainfold. Then we have- \begin{align} \frac{d}{dt}A(p_{i},q_{i},t) & =\left(\frac{\partial A}{\partial q_{i}}\frac{dq_{i}}{dt}+\frac{\partial A}{\partial p_{i}}\frac{dp_{i}}{dt}\right)\nonumber \\ & =\left(\frac{\partial A}{\partial q_{i}}\dot{q}_{i}+\frac{\partial A}{\partial p_{i}}\dot{p}_{i}\right)\nonumber \\ & =\left(\frac{\partial A}{\partial q_{i}}\frac{\partial H}{\partial p_{i}}+\frac{\partial A}{\partial p_{i}}\frac{\partial H}{\partial q_{i}}\right)\nonumber \\ & =\{A,H\}\label{eq:2-19} \end{align} So in general \eqref{eq:2-19} implies \begin{equation} \frac{dA}{dt}=\dot{A}=\{A,H\}\label{eq:2-20} \end{equation} Further, by taking $p=p(t)$ and $q=q(t)$ to be solutions to Hamilton's equations \begin{align} \dot{q}_{i} & =\frac{\partial H}{\partial p_{i}}=\{q_{i},H\}\label{eq:2-21}\\ \dot{p}_{i} & =-\frac{\partial H}{\partial q_{i}}=\{p_{i},H\}\label{eq:2-22} \end{align}
Strong and Weak equation
We shall now have to distinguish between two kinds of equations. When we apply the variation, equation \eqref{eq:2-5} remain valild to the accuracy $\epsilon$. On the other hand equation \eqref{eq:2-6} gets violated by a quantity of order $\epsilon$ under the variation. The former kind of equation is called the strong equation. The latter kind of equation is called weak equation. At this stage let us introduce the weak equality sign '$\approx$' for constraints equations. And for the strong equation we introduce the sign '$\equiv$'. We have the following rules governing algebric work with weak and strong equations. \begin{align} \text{if }A & \equiv0\text{ then }\delta A=0;\label{eq:2-23}\\ \text{if }X & \approx0\text{ then }\delta X\neq0;\label{eq:2-24} \end{align} in general. From the relation $X\approx0$ emphasize that $X$ is numerically restricted to be zero but does not identically vanish throughout phase space. This means, in particular, that it has nonzero Poisson brackets with the canonical variables. We can also deduce that- \begin{equation} \delta X^{2}\approx2X\delta X\approx0\label{eq:2-25} \end{equation} On the other hand, the strong equation holds throughout phase space and not just on the submanifold $X\approx0$ and can be written as- \begin{equation} X^{2}\equiv0\label{eq:2-26} \end{equation} Similarly from two weak equation $X_{1}\approx0$ and $X_{2}\approx0$ we can deduce the strong equation \begin{equation} X_{1}X_{2}\equiv0\label{eq:2-27} \end{equation} It may be that the $N$ quantities $\dfrac{\partial L}{\partial\dot{q}_{i}}$ on the right-hand side of \eqref{eq:2-6} are all independent functions of the $N$ velocities $\dot{q}_{i}$. In this case equations \eqref{eq:2-6} determinde each $\dot{q}$ as a function fo the $q$'s and $p$'s. This case will be referred to as the standard case, and is the only one usually considered in dynamical theory. If the $\dfrac{\partial L}{\partial\dot{q}}$ are not independent functions of the velocities, we can eliminate the $\dot{q}$'s from the equations \eqref{eq:2-6} and obtain one or more equations. \begin{equation} \phi(q,p)\approx0\label{eq:2-28} \end{equation} involving only $q$'s and $p$'s . We may suppose equation \eqref{eq:2-28} to be written in such a way that the variation procedure changes $\phi$ by a quantity of order $\epsilon$,since if it changes $\phi$ by a quantity of order $\epsilon^{k}$ ,we have only to replace $\phi$ by $\phi^{\frac{1}{k}}$in \eqref{eq:2-28} and the desired condition will be fulfilled . We now have equation \eqref{eq:2-28} violated by the order $\epsilon$ when we apply the variation, so it is correctly written as weak equation. We shall need to use a complete set of independent equations of the type \eqref{eq:2-28} say \begin{equation} \phi_{i}(q,p)\approx0\label{eq:2-29} \end{equation} where $i=1,2,\ldots,n$. The condition of the independence means that none of the $\phi$'s is expressible linearly in terms of the others, with functions of the $q$'s and $p$'s as coefficients. The condition of completeness means that any function fot he $q$'s and $p$'s which vnishes on account of equation \eqref{eq:2-6} and changes by the order $\epsilon$ with the variation procedure is expressible as a linear function of the $\phi_{i}$ with functions of the $q$'s and $p$'s as coefficients.