In physics, Hamiltonian constraints are a technique used to simplify the solving of certain types of equations of motion that occur in classical mechanics.

In theories with reparametrization invariance with respect to time, if a canonical Hamiltonian formulation using first class constraints of the theory is possible (we might have second class constraints), we'd have "gauge" "symmetries" (which really means nothing more than the time coordinates are completely arbitrary and the description of a state using time coordinates is redundant). If we use the method of first class constraints, it turns out the Hamiltonian H is equal to one of its constraints. To see why this is the case, note that multiplying the "lapse parameter" by a positive number would yield another valid time flow which only differs by a positive constant. Since the difference between two valid time flows (representing different "gauges") is generated by a linear combination of constraints, the Hamiltonian would also have to be a linear combination of constraints.

This means the Hamiltonian time flows maps points on the constrained subspace to points in the same orbit (generated by the constraints). Since physical observables are only defined after quotienting out the orbits, what this means is the Hamiltonian time flows map orbits to the same orbits. So, using Occam's razor we can eliminate time since the time evolution of the orbits is trivial. This might sound bizarre physically, but we don't measure things at a particular (absolute) time. We only measure things relative to a dynamical clock. Even then, it still seems bizarre. Surely the state where an experimenter notices a subsystem is in state A and a dynamical clock is at a state t₁ is physically distinguishable from the state where the experimenter notices the subsystem is in state B and the clock is at state t₂ and he has the memory that when the clock is at state t₁, he noticed that the subsystem is at state A even if the latter is the result of a time evolution of the former. They both lie in the same orbit, but does this call into question the statement that states in the same orbit are physically indistinguishable? In some models, time flows, and hence, the corresponding orbits display "subergodic" properties. One thing to note is we don't know for sure the past exists. All we note is that our current memories of what appears to be the past are consistent and also the appearance of motion is due to a current static neurological configuration of parts of the brain dedicated to "motion" as observed from the fact that there are states of consciousness where time appears to be distorted. The other is to go over to the quantum case and apply the relative state interpretation to a timeless theory. Then, even though we only have one pure state (which doesn't evolve in time because there is no time), because of decoherence, both states mentioned earlier lie in different "worlds". We'd also need to explain why the pure state has "components" which are primarily concentrated on component states with a "consistent correlation of memories/records", but this can be shown to be equivalent to the problem of the second law of thermodynamics (the arrow of time).

Note that for general relativity, though, we actually have infinitely many independent Hamiltonian constraints, one for each spatial point. This is because we have "multi-fingered" time where each spatial point has its own (nondynamical) "clock". The mathematics of this is covered by shift vectors and lapse functions . (The shift vectors are covered by the spatial diffeomorphism constraints ).

Example

The dynamics of a single point particle of mass m with no internal degrees of freedom moving in a pseudo-Riemannian spacetime manifold S with metric g. Let's also assume the parameter τ describing the trajectory of the particle is arbitrary (i.e. we insist upon reparametrization invariance ). Then, its symplectic space is the cotangent bundle T*S with the canonical symplectic form ω. If we coordinatize T*S by its position x in the base manifold S and its position within the cotangent space p, then we have a constraint f=m²-g(x)^-1(p,p)=0. The Hamiltonian H_α=αf/2m for some arbitrary positive number α called the lapse parameter. Note that all the H_{α's agree over the constrained subspace, which means they are all "gauge equivalent". We can even let f be a smooth positive function over the symplectic manifold, intepretated as the lapse parameter being dependent upon the state of the system.}