##### Constraint Formalism

##### Overview of Constrainted system

The dynamical phase space variables of these theories are not all independent, rather some of these variables have to satisfy constraints following from the structure of the theory. Such systems are known as constrained systems and the naive passage to the Hamiltonian description for such a system starting from the Lagrangian description fails. In this case, there is a systematic procedure due to Dirac which allows us to go from the Lagrangian description of a theory to the Hamiltonian description . Let us consider the dynamics of system is governed by the stationary of the action integral \begin{equation} S=\int Ldt\label{eq:3-1} \end{equation} where the Lagrangian $L=L(q_{i},\dot{q}_{i})$, is the function of the coordinates $q_{i}\;(i=1,2,3\ldots N)$, the velocities $\dot{q}_{i}=\dfrac{dq_{i}}{dt}$. We assume that the system has $N$ degrees of freedom. Now the canonical conjugate momentum variables $p_{i}$ are defined by \begin{equation} p_{i}=\frac{\partial L}{\partial\dot{q}_{i}}\label{eq:3-2} \end{equation} Now to go from the configuration space to the phase space of the system we can uniquely define the Hamitonian of the system through the Legendre tranformation \begin{equation} H=p_{i}\dot{q}_{i}-L\label{eq:3-3} \end{equation} And we know that the variation of $H$ thus involves the variation of the $q$'s and $p$'s only. Hence $H=H(q,p)$ is a function of $q$ and $p$. In this case, we can go from the configuration space of the system to the phase space \begin{equation} (q_{i},\dot{q}_{i})\longrightarrow(q_{i},p_{i})\label{eq:3-3-a} \end{equation} and uniquely define the Hamiltonian of the system through the Legendre transformation . \begin{equation} H(q_{i},p_{i})=p_{i}\dot{q}_{i}-L(q_{i},\dot{q}_{i})\label{eq:3-3-b} \end{equation} Now the conditions for the action to be stationary are the Euler-Lagrange equations \begin{equation} \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0\label{eq:3-4} \end{equation} Equation \eqref{eq:3-4} can be written in more detail as\- \begin{equation} \ddot{q}_{j}\frac{\partial^{2}L}{\partial\dot{q}_{j}\partial\dot{q}_{i}}=\frac{\partial L}{\partial q_{i}}-\dot{q}_{j}\frac{\partial L}{\partial q_{j}\partial\dot{q}_{i}}\label{eq:3-5} \end{equation} From thes equation we immediately see that the accelaration $\ddot{q}$ at a given time are uniquely determined by the positions and the velocities at that time if and only if the matrix $\dfrac{\partial^{2}L}{\partial\dot{q}_{j}\partial\dot{q}_{i}}$ can be inverted. i.e. the determinant of the matrix does not vanish. \begin{equation} det\left(\frac{\partial^{2}L}{\partial\dot{q}_{j}\partial\dot{q}_{i}}\right)\neq0\label{eq:3-6} \end{equation} But the difficulty in passage to a Hamitonian description arises when this is non invertible. For an example let us consider a bosonic theory where the transformation to phase space in \eqref{eq:3-3-a} can be written in as a matrix \begin{equation} \left(\begin{array}{c} q\\ p \end{array}\right)=M\left(\begin{array}{c} q\\ \dot{q} \end{array}\right)=\left(\begin{array}{cc} \mathbb{I} & 0\\ \tilde{m} & m \end{array}\right)\left(\begin{array}{c} q\\ \dot{q} \end{array}\right)\label{eq:3-7} \end{equation} where the the elements $m$,$\tilde{m}$ represents $N\times N$ matrices. The inverse of the matrix $M$ in \eqref{eq:3-7} can be easil seen to have the form \begin{equation} M^{-1}=\left(\begin{array}{cc} \mathbb{I} & 0\\ -m^{-1}\tilde{m} & m^{-1} \end{array}\right)\label{eq:3-8} \end{equation} So the inverse of $M$ exist only if the $m$ is invertable. From \eqref{eq:3-7} we also note that \begin{equation} p_{i}=\tilde{m}_{ij}(q)q_{j}+m_{ij}(q)\dot{q}_{j}\label{eq:3-9} \end{equation} So that \begin{equation} m_{ij}(q)=\frac{\partial p_{i}}{\partial\dot{q}_{j}}=\frac{\partial^{2}L}{\partial\dot{q}_{j}\partial\dot{q}_{i}}\label{eq:3-10} \end{equation} From this equation, it is clear that the if Lagrangian descirbing the theory satifies \begin{equation} det\left(\frac{\partial^{2}L}{\partial\dot{q}_{j}\partial\dot{q}_{i}}\right)=0\label{eq:3-11} \end{equation} then the transformation \eqref{eq:3-3-a} is not invertible. In this case, not all the conjugate momenta can be thought of as independent variables leading to the fact that not all of $N$ independent velocities can be expressed in terms of independent momenta. So there exist constraints between various dynamical variables and that system is by definition is called the constrained system.##### Example of a constrained system

where $\lambda_{1}$and $\lambda_{2}$ are the lagrange's undermined multipliers and $a$ and $b$ defined as- \begin{align} \left(a(t),b(t)\right) & =\left(cos(\theta(t)),sin(\theta(t))\right)\label{eq:3-16}\\ \therefore\left(\dot{a}(t),\dot{b}(t)\right) & =\left(-sin(\theta(t))\dot{\theta},cos(\theta(t))\dot{\theta}(t)\right)\label{eq:3-17} \end{align} Here the dynamical variables are $a,b,\lambda_{1},$ and $\lambda_{2}$. And comparing ......... and .......... we can see that here $\dot{a}=-b$ and $\dot{b}=a$. But the trouble comes when we try to pass to Hamilonian- \begin{equation} p_{i}=\frac{\partial L}{\partial\dot{q}_{i}}\label{eq:3-18} \end{equation} Because $\dot{\lambda}_{i}$ cannot be solved in terms of $p_{\lambda_{i}}$ and from the lagrangian we have- \begin{equation} p_{\lambda_{1}}=\frac{\partial L}{\partial\dot{\lambda}_{1}}=0;\quad p_{\lambda_{2}}=\frac{\partial L}{\partial\dot{\lambda}_{2}}=0\label{eq:3-19} \end{equation} So in this case $\lambda_{1}$ and $\lambda_{2}$ are two constraints.

#### Dirac theory of constraints

##### Primary constraints

The discussion so far assumes that the transformation to phase space is invetible so that all the velocities can be expressed uniquely in terms of independent momenta leading to the unique Hamiltonia of the system. In the Hamiltonian formalism, one treats the moments as independent functions of the velocities. Velocities can be expressed in terms of momenta and coordinates.Now if our action integral is- \begin{equation} I=\int_{t_{i}}^{t_{f}}L(q,\dot{q})dt\label{eq:3-20} \end{equation} 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:3-21} \end{equation} From \eqref{eq:3-2} we can express the canonical conjugate momentum $p_{i}$ as \begin{equation} p_{i}=\frac{\partial L}{\partial\dot{q}_{i}}\label{eq:3-22} \end{equation} In the Hamiltonian formalism , one treats the momenta as independent functions of the velocities. However there may exist several relation connecting the coordinate and momentum varibles- \begin{equation} \phi_{m}(q,p)=0\label{eq:3-23} \end{equation} where $m=1,2,\ldots M$ These $M$ in equation \eqref{eq:3-23} are the primary constraints of the Hamiltonian formalism.##### Hamilton's Equations in constraints system:

We now define the Hamiltonian function- \begin{equation} H=p_{i}\dot{q}_{i}-L\label{eq:3-24} \end{equation} Now for slight change of $\delta H$ we can write- \begin{eqnarray} \delta H & = & \dot{q_{i}}\delta p_{i}-\frac{\partial L}{\partial q_{i}}\delta q_{i}\label{eq:3-25} \end{eqnarray} The variation of $H$ thus involves the variation of the $q$'s and $p$'s only. Hence $H=H(q,p)$ is a function of $q$ and $p$. In presence of the constraints \eqref{eq:3-23} , the Hamiltonian \eqref{eq:3-24} is not unique. One may add a linear combination of the $\phi_{i}$'s to the Hamiltonian $H$ to get another Hamiltonian. \begin{equation} H^{\prime}=H+c_{i}\phi_{i}\label{eq:3-26} \end{equation} The unknown coefficient $c_{i}$ can be any function of q and p. Now , equation \eqref{eq:3-25} holds for any variation of the $q_{i}$and $p_{i}$ provided the constraints \eqref{eq:3-23} are preserved.But , this means that all the q's and p's cannot be varied independently. Now, from \eqref{eq:3-23}- \begin{equation} \delta\phi_{m}=\frac{\partial\phi_{m}}{\partial q_{i}}\delta q_{i}+\frac{\partial\phi_{m}}{\partial p_{i}}\delta p_{i}\label{eq:3-27} \end{equation} multiplying these equations with the unknown Lagrange multipliers $\lambda^{m}$ and adding to \eqref{eq:3-25}, we get \begin{eqnarray} \delta H+\lambda^{m}\delta\phi_{m} & = & \dot{q_{i}}\delta p_{i}-\frac{\partial L}{\partial q_{i}}\delta q_{i}\nonumber \\ \Rightarrow\frac{\partial H}{\partial q_{i}}\delta q_{i}+\frac{\partial H}{\partial p_{i}}\delta p_{i}+\lambda^{m}\left(\frac{\partial\phi_{m}}{\partial q_{i}}\delta q_{i}+\frac{\partial\phi_{m}}{\partial p_{i}}\delta p_{i}\right) & = & \dot{q_{i}}\delta p_{i}-\frac{\partial L}{\partial q_{i}}\delta q_{i}\label{eq:3-28} \end{eqnarray} inclusion of the $\lambda^{m}$ makes q and p independent , so then we find by comparing the coefficient of $\delta q_{i}$ and $\delta p_{i}$ - \begin{eqnarray} \dot{q_{i}} & = & \frac{\partial H}{\partial p_{i}}+\lambda^{m}\frac{\partial\phi_{m}}{\partial p_{i}}\nonumber \\ \Rightarrow-\frac{\partial L}{\partial q_{i}} & = & \frac{\partial H}{\partial p_{i}}+\lambda^{m}\frac{\partial\phi_{m}}{\partial p_{i}}\label{eq:3-29} \end{eqnarray} But $\frac{\partial L}{\partial q_{i}}=\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q_{i}}}\right)=\dot{p_{i}}$. Hence, the Hamilton's equation of motion can be found as \begin{equation} \dot{q_{i}}=\frac{\partial H}{\partial p_{i}}+\lambda^{m}\frac{\partial\phi_{m}}{\partial p_{i}}\label{eq:3-30} \end{equation} \begin{equation} \dot{p_{i}}=-\frac{\partial H}{\partial q_{i}}-\lambda^{m}\frac{\partial\phi_{m}}{\partial q_{i}}\label{eq:3-31} \end{equation} Now the equations \eqref{eq:3-30} and \eqref{eq:3-31} can be written compactly using Poisson bracket defined by \begin{equation} \{f,g\}\equiv\frac{\partial f}{\partial q_{i}}\frac{\partial g^{\prime}}{\partial p_{i}}-\frac{\partial f}{\partial p_{i}}\frac{\partial g^{\prime}}{\partial q_{i}}\label{eq:3-32} \end{equation} for two functions $f(q,p)$ and $g(q,p)$. Now, for any function $g=g(q,p)$ we have- \begin{eqnarray} \dot{g} & \equiv & \frac{dg}{dt}\nonumber \\ & = & \frac{\partial g}{\partial q_{i}}\dot{q_{i}}+\frac{\partial g}{\partial p_{i}}\dot{p_{i}}\nonumber \\ & = & \frac{\partial g}{\partial q_{i}}\left(\frac{\partial H}{\partial p_{i}}+\lambda^{m}\frac{\partial\phi_{m}}{\partial p_{i}}\right)-\frac{\partial g}{\partial p_{i}}\left(\frac{\partial H}{\partial q_{i}}+\lambda^{m}\frac{\partial\phi_{m}}{\partial q_{i}}\right)\text{(Using equation \ref{eq:3-30} and \ref{eq:3-31})}\nonumber \\ & = & \left(\frac{\partial g}{\partial q_{i}}\frac{\partial H}{\partial p_{i}}-\frac{\partial g}{\partial p_{i}}\frac{\partial H}{\partial q_{i}}\right)+\lambda^{m}\left(\frac{\partial g}{\partial q_{i}}\frac{\partial\phi_{m}}{\partial p_{i}}-\frac{\partial g}{\partial p_{i}}\frac{\partial\phi_{m}}{\partial q_{i}}\right)\nonumber \\ & = & \{g,H\}+\lambda^{m}\{g,\phi_{m}\}\label{eq:3-33} \end{eqnarray} The poisson bracket in \eqref{eq:3-32} is defined for quantities which can be expressed in terms of the q's and the p's. Let us now extend the meaning of the poisson bracket and assume that they exist for any two quantities . Now \begin{eqnarray} \{g,H+\lambda^{m}\phi_{m}\} & = & \{g,H\}+\{g,\lambda^{m}\phi_{m}\}\nonumber \\ & = & \{g,H\}+\{g,\lambda^{m}\}\phi_{m}+\lambda^{m}\{g,\phi_{m}\}\nonumber \\ & = & \{g,H\}+\lambda^{m}\{g,\phi_{m}\}\label{eq:3-34} \end{eqnarray} where we have used equation \eqref{eq:3-23}. Hence equation \eqref{eq:3-33} can be rewritten as \begin{equation} \dot{g}=\{g,H+\lambda^{m}\phi_{m}\}\label{eq:3-35} \end{equation} In above formalism, in presence of constraints, one must not use the constraints equation \eqref{eq:3-23} before working out any poisson bracket . To remember this, various equations are written with an equality sign ($\approx$) different from the usual one. Hence, the constraints equations are written as- \begin{equation} \phi_{m}\approx0\label{eq:3-36} \end{equation} These equations are called weak equations, after Dirac. With this in mind, the equations of motion \eqref{eq:3-35} are written as \begin{equation} \dot{g}\approx\{g,H_{T}\}\label{eq:3-37} \end{equation} where \begin{equation} H_{T}\equiv H+\lambda^{m}\phi_{m}\label{eq:3-38} \end{equation} is the total Hamiltonian.##### Secondary Constraints

Let us now examine some of the consequences of the equations of motion \eqref{eq:3-33}. A basics consistency requirement is that the primary constraints be preserved in time. THusm if we take $g$ in \eqref{eq:3-33} to be one of the $\phi_{m}$ we should have $\dot{\phi}=0$. This gives rise to the consistency conditions, \begin{equation} \{\phi_{m},H\}+\lambda^{m}\{\phi_{m},\phi_{n}\}=0\label{eq:3-39} \end{equation} Equation \eqref{eq:3-39} above can either reduce to a relation independent of the $u$'s or it may impose a restriction on the $u$'s. In the former case, if the relation between the $p$'s and the $q$'s is independent of the primary constraints, it is called a secondary constraints.##### Consistency conditions

Now, the constraints $\phi$ have to be zero through all time. Putting $g=\phi_{m}$ and $\dot{g}=0$ in \eqref{eq:3-33} we get \begin{equation} \{\phi_{m},H\}+\lambda^{m}\{\phi_{m},\phi_{n}\}\approx0\label{eq:3-40} \end{equation} Thus, a number of consistency conditions follow, one for each value of m, which are needed to be examined. It is possible that they lead directly to an inconsistency of the type $1=0$. Thenit means tat our original Lagrangian is such that the Lagrange equations of motion are inconsistent. Hence, one cannot take the Lagrangian to be completely arbitrary. We must impose on it the condition that the Lagrange equations of motion do not involve an inconsistency. With this restrictions the equations \eqref{eq:3-40} can be classified into three types- 1) One kind of equation reduces to $0=0$; i.e. it is identically satisfied with the help of the primary constraints 2) Another kind of equation reduces to an equation independent of the primary constraints $\phi$'s , involving only the q's and the p's - \begin{equation} \chi(q,p)=0\label{eq:3-41} \end{equation} which means that we have another constraint on the Hamiltonian varibles. Constraints emerging in this way are called secondery constraints. While primary constraints are consequences of the equation \eqref{eq:3-2} defining the momentum varibles , one has to make use of the Lagrange equations of motion for the secondary constraints.For a secondary constraint present in the theory, we get another consistency condition, requiring $\dot{\chi}\approx0$ and using \eqref{eq:3-33}- \begin{equation} \{\chi,H\}+\lambda^{m}\{\chi,\phi_{m}\}\approx0\label{eq:3-42} \end{equation} If this leads to another secondary constraint, then the process has to be pushed one stage further and we carry on like that until all the consistency conditions are exhusted. Then,we end up with a number of secondary constraints of the type \eqref{eq:3-41}. Together with a number of conditions on the coefficients $\lambda^{m}$ of the type \eqref{eq:3-40}. We can treat the secondary constraints on the same footing as the primary constraints. It is convenient to write them as- \begin{equation} \phi_{k}\approx0;\text{ k=M+1,${\ldots}$,M+K}\label{eq:3-43} \end{equation} where k is the total number of secondary constraints. They are also weak equations which one must not make use of before evaluating any Poisson bracket. So, all the constraints together may be written as- \begin{equation} \phi_{j}\approx0;\text{ j=1,${\ldots}$ ,M+K ${\equiv}J$ } \label{eq:3-44} \end{equation} 3) A third kind of equation in \eqref{eq:3-40} may not reduce to the either of the above mentioned types; it then imposes a condition on the coefficients $\lambda$- \begin{equation} \{\phi_{j},H\}+\lambda^{m}\{\phi_{j},\phi_{m}\}\approx0\label{eq:3-45} \end{equation} where m is summed from 1 to M and j takes on any of the values from 1 to J. We can treat the $\lambda$'s as unknown and then equation \eqref{eq:3-45} can be viewed as a number of non-homogenous linear equations in these unknown $\lambda$, with coefficients which are functions of the q's and the p's. Let a solution of these equations be- \begin{equation} \lambda^{m}=U^{m}(q,p)\label{eq:3-46} \end{equation} There must exist a solution of this type,else the Lagrange equation of motion are in consistent . Let $V^{m}(q,p)$ be the solution of the homogenous equations associated with \eqref{eq:3-45}- \begin{equation} V^{m}\{\phi_{j},\phi_{m}\}=0\label{eq:3-47} \end{equation} To find the most generaal solution of \eqref{eq:3-45}, we must consider al the independent solutions of \eqref{eq:3-46}, which we denote by $V_{a}^{m}(q,p),\; a=1,\ldots A,$. The general solution of \eqref{eq:3-45} is then \begin{equation} \lambda^{m}=U^{m}+\eta_{a}V_{a}^{m}\label{eq:3-48} \end{equation} in terms of the coefficients $\eta_{a}$which cane be arvitrary. The total Hamiltonian in \eqref{eq:3-38} reads- \begin{eqnarray} H_{E} & = & H+\lambda^{m}\phi_{m}+\eta_{a}V_{a}^{m}\phi_{m}\label{eq:3-49}\\ & = & H^{\prime}+\eta_{a}\phi_{a}\label{eq:3-50} \end{eqnarray} where \begin{eqnarray} H^{\prime} & \equiv & H+\lambda^{m}\phi_{m}\label{eq:3-51} \end{eqnarray} and \begin{equation} \phi_{a}=V_{a}^{m}\phi_{m}\label{eq:3-52} \end{equation} $H_{E}$is really the extended Hamiltonian, rather than the total Hamiltonian. In terms of this ectended Hamiltonian \eqref{eq:3-50} we still have the equations of motion \eqref{eq:3-37}. Though, we have now satidied all the consistency requirments, we still have the arvitrary coefficients $\eta_{a}$.Their number is usually less than the number of the coefficients $\lambda^{m}$.The $\lambda$'s are not arbitrary but have to satisfy the consistency conditions while the $\eta$'s are arbitary coefficients.One can take the $\eta$'s to be functions of time t and can still satisfy all the requirements of the dynamical theory. The occurance of arbitrary functions of time in the general solution of the equations of motion means that we have gauge degrees of freedom in the theory. Thus , the dynamical varibles at future times are not completely determined by the initist conditions and the arbitrariness shows up through arbitrary functions appearing in the general solution.