This paper reviews how mathematicians and physicists deal with mechanical systems whose usual equations break down because of hidden constraints. The authors explain two geometric frameworks that cover most such cases. Symplectic geometry handles conservative, energy-preserving systems. Contact geometry handles systems with loss or gain, that is, dissipative systems. The review brings together the geometry behind both settings and the step‑by‑step methods — called constraint algorithms — that carve out the part of phase space where well‑defined motion exists.

The authors start from Lagrangian models that are “almost‑regular,” meaning that the standard map from velocities to momenta (the Legendre map) is not invertible but still has a well‑behaved image. That image is the primary constraint manifold M0 in the cotangent bundle, and it inherits a closed but possibly degenerate two‑form ω0 and a Hamiltonian H0. On M0 one tries to solve the Hamiltonian equation ♭0(XH) = dH0, where ♭0 is the map given by contraction with ω0. Because ω0 can be degenerate, this equation need not have a solution everywhere. The paper explains how to detect the points where solutions exist and how to refine M0 into smaller submanifolds M1, M2, … until a final allowed region Mf is found.

At a high level the pre‑symplectic constraint algorithm works by two simple checks repeated in turn. First, one demands that the derivative of the Hamiltonian lie in the image of ♭0; this yields M1. Second, one imposes that any vector field that solves the equation is tangent to the constraint surface; this yields M2. Repeating these two types of tests produces a chain of nested manifolds M0 ⊃ M1 ⊃ M2 ⊃ … that stops when it stabilises at Mf. The authors stress that solutions on Mf may be non‑unique and that the algorithm can terminate in different ways depending on the system. They also compare this geometric viewpoint with the more familiar Dirac–Bergmann algebraic method for constraints.