First-class constraint: Difference between revisions

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{{distinguish|Primary constraint}}

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In a constrained Hamiltonian system, a dynamical quantity is called a '''first class constraint''' if its Poisson bracket with all the other constraints vanishes on the '''constraint surface''' (the surface implicitly defined by the simultaneous vanishing of all the constraints). A '''second class constraint''' is one that is not first class.

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===An example: a particle confined to a sphere===

Before going on to the general theory, let's look atconsider a specific example step by step to motivate the general analysis.

Let's startStart with the [[action (physics)|action]] describing a [[Newtonian dynamics|Newtonian]] particle of [[mass]] m constrained to a surface of radius R within a uniform [[gravitational field]] ''g''. When one works in Lagrangian mechanics, there are several ways to implement a constraint: one can switch to generalized coordinates that manifestly solve the constraint, or one can use a Lagrange multiplier while retaining the redundant coordinates so constrained.

In this case, the particle is constrained to a sphere, therefore the natural solution would be to use angular coordinates to describe the position of the particle instead of Cartesian and solve (automatically eliminate) the constraint in that way (the first choice). For didacticpedagogical reasons, instead, consider the problem in Cartesian coordinates, with a Lagrange multiplier term enforcing the constraint.

The action is given by

:<math>S=\int dt L=\int dt \left[\frac{m}{2}(\dot{x}^2+\dot{y}^2+\dot{z}^2)-mgz+\frac{\lambda}{2}(x^2+y^2+z^2-R^2)\right]</math>

<math>S=\int dt L=\int dt \left[\frac{m}{2}(\dot{x}^2+\dot{y}^2+\dot{z}^2)-mgz+\frac{\lambda}{2}(x^2+y^2+z^2-R^2)\right]</math>

where the last term is the [[Lagrange multiplier]] term enforcing the constraint.

Of course, as indicated, we could have just used different [[coordinates]] and written it as

<math>S=\int dt \left[\frac{mR^2}{2}(\dot{\theta}^2+\sin^2(\theta)\dot{\phi}^2)+mgR\cos(\theta)\right]</math>

instead, without extra constraints, but let'swe look at the former coordinatization to illustrate constraints.

instead, but let's look at the former coordinatization.

The [[conjugate momentum|conjugate momenta]] are given by

:<math>p_x=m\dot{x}</math>, <math>p_y=m\dot{y}</math>, <math>p_z=m\dot{z}</math>, <math>p_\lambda=0</math> .

<math>p_x=m\dot{x}</math>, <math>p_y=m\dot{y}</math>, <math>p_z=m\dot{z}</math>, <math>p_\lambda=0</math>.

Note that we can't determine <math>\dot{\lambda}</math> from the momenta.

The [[Hamiltonian mechanics|Hamiltonian]] is given by

:<math>H=\vec{p}\cdot\dot{\vec{r}}+p_\lambda \dot{\lambda}-L=\frac{p^2}{2m}+p_\lambda \dot{\lambda}+mgz-\frac{\lambda}{2}(r^2-R^2)</math>.

We can'tcannot eliminate <math>\dot{\lambda}</math> at this stage yet. We are here treating <math>\dot{\lambda}</math> as a shorthand for a function of the [[symplectic space]] which we have yet to determine and ''not'' an independent variable. For notational consistency, define <math>u_1=\dot{\lambda}</math> from now on. The above Hamiltonian with the <{{math|''p''<sub>p_\lambda''λ''</mathsub>}} term is the "naive Hamiltonian". Note that since, on-shell, the constraint must be satisfied, one cannot distinguish, on-shell, between the naive Hamiltonian and the above Hamiltonian with the undetermined coefficient, <math>\dot{\lambda}=u_1</math>, on-shell.

<math>H=\vec{p}\cdot\dot{\vec{r}}+p_\lambda \dot{\lambda}-L=\frac{p^2}{2m}+p_\lambda \dot{\lambda}+mgz-\frac{\lambda}{2}(r^2-R^2)</math>.

We can't eliminate <math>\dot{\lambda}</math> at this stage yet. We are here treating <math>\dot{\lambda}</math> as a shorthand for a function of the [[symplectic space]] which we have yet to determine and ''not'' an independent variable. For notational consistency, define <math>u_1=\dot{\lambda}</math> from now on. The above Hamiltonian with the <math>p_\lambda</math> term is the "naive Hamiltonian". Note that since, on-shell, the constraint must be satisfied, one cannot distinguish between the naive Hamiltonian and the above Hamiltonian with the undetermined coefficient, <math>\dot{\lambda}=u_1</math>, on-shell.

We have the [[primary constraint]]
:{{math|''p_λ''{{=}}0}}.

We require, on the grounds of consistency, that the [[Poisson bracket]] of all the constraints with the Hamiltonian vanish at the constrained subspace. In other words, the constraints must not evolve in time if they are going to be identically zero along the equations of motion.

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From this consistency condition, we immediately get the [[First_class_constraints#Constrained_Hamiltonian_dynamics_from_a_Lagrangian_gauge_theory|secondary constraint]]

:{{math|''r''²-−''R''²{{=}}0}} .

By the same reasoning, this constraint should be added into the Hamiltonian with an undetermined (not necessarily constant) coefficient {{mvar|u}}<mathsub>u_22</mathsub>. At this point, the Hamiltonian is

:<math>

H = \frac{p^2}{2m} + mgz - \frac{\lambda}{2}(r^2-R^2) + u_1 p_\lambda + u_2 (r^2-R^2)