theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

, an analysis of flows is the study of "gauge" or "gaugelike" "symmetries" (i.e. flows the formulation of a theory is invariant under). It is generally agreed that flows indicate nothing more than a redundancy in the description of the dynamics of a system, but often, it is simpler computationally to work with a redundant description.

Flows in classical mechanics

Flows in the action formalism

Classically, the action is a functional on the configuration space. The on-shell solutions are given by the

variational problem The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

of extremizing the action subject to

boundary condition In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...

s. While the boundary is often ignored in textbooks, it is crucial in the study of flows. Suppose we have a "flow", i.e. the

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of a smooth one-dimensional group of transformations of the configuration space, which maps on-shell states to on-shell states while preserving the boundary conditions. Because of the variational principle, the action for all of the configurations on the orbit is the same. This is ''not'' the case for more general transformations which map on shell to on shell states but change the boundary conditions. Here are several examples. In a theory with

translational symmetry In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equatio ...

, timelike translations are ''not'' flows because in general they change the boundary conditions. However, now take the case of a

simple harmonic oscillator In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...

, where the boundary points are at a separation of a multiple of the period from each other, and the initial and final positions are the same at the boundary points. For this particular example, it turns out there ''is'' a flow. Even though this is technically a flow, this would usually not be considered a

gauge symmetry In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...

because it is not local. Flows can be given as

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over the algebra of smooth functionals over the configuration space. If we have a flow distribution (i.e. flow-valued distribution) such that the flow convolved over a local region only affects the field configuration in that region, we call the flow distribution a ''gauge flow''. Given that we are only interested in what happens on shell, we would often take the quotient by the ideal generated by the Euler–Lagrange equations, or in other words, consider the equivalence class of functionals/flows which agree on shell.

Flows in the Hamiltonian formalism

First class constraints A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultaneous vanis ...

Second class constraints A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultaneous vanishi ...

BRST formalism BRST may refer to: * BRST Films, a Serbian video production company * BRST algorithm, an optimization algorithm suitable for finding the global optimum of black box functions * BRST quantization in Yang-Mills theories, a way to quantize a gauge-sym ...

Batalin–Vilkovisky formalism In theoretical physics, the Batalin–Vilkovisky (BV) formalism (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the ghost structure for Lagrangian gauge theories, such as gravity and supergravity, whose ...

Theoretical physics {{theoretical-physics-stub