On the role of constraints and degrees of freedom in the Hamiltonian formalism

Abstract

Unfortunately, the Hamiltonian mechanics of degenerate Lagrangian systems is usually presented as a mere recipe of Dirac, with no explanation as to how it works. Then it comes to discussing conjectures of whether all primary constraints correspond to gauge symmetries, and it goes all the way to absolutely wrong claims such as the statement that electrodynamics or gravity have only two physical components each, with others being spurious. One has to be very careful because non-dynamical, or constrained, does not mean unphysical. I give a pedagogical introduction to the degenerate Hamiltonian systems, showing both very simple mechanical examples and general arguments about how it works. For the familiar field theory models, I explain why the gauge freedom there "hits twice" in the sense of producing twice as many first-class constraints as gauge symmetries, and why primary, and only primary, constraints should be put into the total Hamiltonian.

the final version; 19 pages

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Cited by in corpus (10)

• Geometry and covariance of symmetric teleparallel theories of gravity
• Einstein-Proca theory from the Einstein-Cartan formulation
• Constraint characterization and degree of freedom counting in Lagrangian field theory
• On the degrees of freedom count on singular phase space submanifolds
• Gravitational Waves in New General Relativity
• To Half--Be or Not To Be?
• Geometric multipliers and partial teleparallelism in Poincaré gauge theory
• Hamiltonian Analysis of Pre-geometric Gravity
• Canonical aspects of pregeometric vector-based first order gauge theory
• More on Bianchi I spacetimes and f(T) gravity