# Analysis of flows

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In [theoretical physics](/source/theoretical_physics), 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,{{Citation needed|date=January 2014}} 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](/source/Configuration_space_(physics)). The [on-shell solution](/source/on-shell_solution)s are given by the [variational problem](/source/Calculus_of_variations) of extremizing the action subject to [boundary condition](/source/boundary_condition)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 [generator](/source/generating_set_of_a_group) 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](/source/translational_symmetry), timelike translations are ''not'' flows because in general they change the boundary conditions{{why|date=February 2014}}. However, now take the case of a [simple harmonic oscillator](/source/simple_harmonic_oscillator), 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](/source/gauge_symmetry) because it is not local.

Flows can be given as [derivations](/source/Differential_algebra) 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](/source/Euler%E2%80%93Lagrange_equations), or in other words, consider the equivalence class of functionals/flows which agree on shell.

=== Flows in the Hamiltonian formalism ===
* [First class constraints](/source/First_class_constraints)<ref name=FysikSuSePDF>{{cite web |author1=Ingemar Bengtsson |title=Constrained Hamiltonian Systems |url=http://3dhouse.se/ingemar/Nr13.pdf |publisher=Stockholm University |access-date=29 May 2018}}</ref>
* [Second class constraints](/source/Second_class_constraints)
* [BRST formalism](/source/BRST_formalism)
* [Batalin–Vilkovisky formalism](/source/Batalin%E2%80%93Vilkovisky_formalism)

==References==
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Category:Theoretical physics

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