Constrained systems

The most thorough treatment of a gauge system is the Hamiltonian formulation, where a gauge system is represented as a constrained Hamiltonian system. The converse is however not true, not all constrained Hamiltonian systems are gauge systems.

The classical motions of a system are those that make the action stationary under the variations of the variables. These are the Euler-Lagrange equations (5)

$$\frac{\partial L}{\partial q}-\frac{d}{dt}\frac{\partial L} {\partial \dot{q}}=0,$$ (38)

which in more detail read

$$(\dot{q}_{k}\frac{\partial}{\partial q_{k}}+ \ddot{q}_{k}\fr... ... (\frac{dL}{d\dot{q}_{m}})-\frac{\partial L}{\partial q_{m}}=0$$ (39)

implying that

$$\ddot{q}_{k}=(\frac{\partial L}{\partial q_{m}} - \dot{q}_{k... ...{\partial^{2}L}{\partial \dot{q}_k \partial \dot{q}_{m}})^{-1}$$ (40)

This means that if $\partial^{2}L/\partial \dot{q}_{k}\partial \dot{q}_{m}$ is invertible, i.e.

$${\cal{D}} = det(\frac{\partial^{2}L}{\partial \dot{q}_k \partial \dot{q}_{m}}) \neq 0,$$ (41)

whereby the accelerations $\ddot{q}_{m}$ may at a certain moment be given unambiguously as functions of position and momentum. If however ${\cal{D}}$ = 0, the accelerations are not unambiguously determined, and the solutions of the equations of motion may contain arbitrary functions of the time. In this case, furthermore, not all the canonical momenta

$$p_{n}=\frac{\partial L}{\partial \dot{q}_{n}}$$

are independent. This corresponds to the presence of $primary$ $constraints$ $\phi_{m}$ $\approx$ 0 in theory. The wavy sign denotes ``weak equality'', indicating that the right-hand side and the left-hand side are equal, but that this equality is set aside while setting up the formalism. In any solution of the equations of motion, the constraints actually do vanish, but this is ignored while setting up the canonical formalism. $\phi_{m}$ $\approx$ 0 thus indicates that $\phi_{m}$ are not identically zero throughout the phase space, but vanish only on a submanifold in phase space, the primary constraint surface ${\cal{M}}_\phi$. It is essential that ${\dot{\phi}_{m}}$ = $\{\dot{\phi}_{m},H\} \approx$ 0. In the case that this is not satisfied, this relation may nevertheless be defined as satisfied, by introducing yet more, secondary constraints.

A function ${\cal{F}}$ is $first$ $class$ if its Poisson brackets with the constraints on the system satisfy $\{{\cal{F}},\phi_{m}\}$ = 0, where $\phi_{m}$ are the constraints on the system and $m=1,2,..$. First class constraints thus satisfy $\{\chi_{j},\chi_{m}\}$ = 0, while for second class constraints, $\{\chi_{j},\chi_{m}\}$ = $C_{jm}$.

For a system without constraints, there is a 1-1 correspondence between the points in phase space and the physical states of the system. The phase space is equipped with a symplectic structure and a Hamiltonian function, which together define the time evolution of the system. For a system with second class constraints, the 1-1 correspondence instead occurs between the physical states of the system and the points on the constraint manifold.