I am trying to find canonical references and the history of trivial symmetries. The earliest text book reference I can find is on page 69 of Quantization of Gauge Systems by Henneaux and Teitelboim.

A trivial symmetry is a symmetry transformation of a classical mechanical or field theory system that reduces to the trivial transformation on-shell* (OS), i.e. in the classical mechanical system with action $S(q,\dot q, \dots)$ $$ q \to q'=q'(q,\dot q,\dots) \quad\mathrm{st}\quad S(q',\dot q'\dots)=S(q,\dot q, \dots) \quad\mathrm{and}\quad q' \xrightarrow{OS} q $$ For infinitesimal symmetries $q \to q'=q+\delta q$ the above is written as (introducing indices $i$ for the coordinates $q$ and the summation convention) $$ q^i \to q^i+\delta q^i \quad\mathrm{st}\quad \delta q^i \frac{\delta S}{\delta q^i}=0 \quad\mathrm{and}\quad \delta q^i \xrightarrow{OS} 0 $$ Theorem (3.1) of Henneaux and Teitelboim says that such a transformation must be proportional* to the equation of motion $$ \delta q^i = \varepsilon^{ij}\frac{\delta S}{\delta q^j} $$ where $\varepsilon^{ij}=-\varepsilon^{ji}$. This all generalises to field theories with both commuting and anticommuting fields. It is also proved in the article Symmetries and physical functions in general gauge theory by Gitman and Tyutin.

The above result means that infinitesimal trivial symmetries form an ideal in the algebra of gauge symmetries and can be basically ignored. In fact, they apparently weren't even really noticed in modern field theory until they turned up as the commutator of some non-trivial symmetries in some supergravity calculations. (This is stated without reference in Remarks on Gauge Invariance and First-Class Constraints.)

A more mathsy discussion can be found in (eg) Rigid Symmetries and Conservation Laws in Non-Lagrangian Field Theory.

So, to summarise, my questions are:

- What is a canonical reference to be given when introducing trivial symmetries?
- Where is theorem (3.1) of Henneaux and Teitelboimm first given?
- What are the historical references for trivial gauge symmetries?*

EDIT - Footnotes:

On-shell (OS) means that $q$ satisfies its equation of motion $\delta S/\delta q = 0$.

Actually, it is normally written using a DeWitt-like condensed notation, so the index contraction actually includes an integration over time (or spacetime in field theories) - this is because there can also be terms with time derivatives of the equations of motion.

Where were trivial symmetries first discussed in classical mechanics? Where were they first discussed in field theories? etc...