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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.
For non‑conservative systems the paper develops the parallel story in contact geometry. Contact manifolds are odd‑dimensional and do not have a conserved phase‑space volume. Because of this they are natural for describing dissipation, where volumes may focus or defocus. The authors present the geometry of pre‑contact manifolds and then give a constraint algorithm that closely mirrors the pre‑symplectic case. They note a related idea called contact reduction, which can arise from a particular symmetry, but say that this review does not treat that reduction in detail and points readers to other references.
Why does this matter? Many physically interesting models are singular or gauge invariant. Without a clear geometric way to impose constraints, their equations of motion are ill‑posed or ambiguous. The review lays out a systematic, geometric procedure to find the admissible phase‑space region where Hamiltonian evolution makes sense, both for energy‑preserving and for dissipative systems. Important caveats are emphasised: the methods assume certain topological regularity (for example reflexivity and closedness of some maps), degeneracies can make solutions non‑unique or absent, and contact systems do not conserve phase‑space volume. The paper is a review, so it collects and explains known geometric tools and illustrates them with examples rather than introducing new experimental data.