Here is an underemphasized way to frame the relationship between trajectories and symmetries (in the sense of Noether’s theorem)a . Consider the space of all possible trajectories 
for a system, a real-valued Lagrangian functional ![L[q(t)]](https://blog.jessriedel.com/wp-content/ql-cache/quicklatex.com-d3cb37d302796a067c15afcd3af52741_l3.svg "Rendered by QuickLaTeX.com")
on that space, the “directions” 
at each point, and the corresponding functional gradient ![\delta L[q(t)]/\delta q(t)](https://blog.jessriedel.com/wp-content/ql-cache/quicklatex.com-cc7e04061e22c0890a458e9c6fde0570_l3.svg "Rendered by QuickLaTeX.com")
in each direction. Classical solutions are exactly those trajectories such that the Lagrangian is stationary for perturbations in any direction , and continuous symmetries are exactly those directions such that the Lagrangian is stationary for any trajectory . That is,
(1) ![\begin{align*} q(t) \mathrm{\,is\, a\,}\mathbf{solution}\quad \qquad &\Leftrightarrow \qquad \frac{\delta L[q(t)]}{\delta q(t)} = 0 \,\,\,\, \forall \delta q(t)\\ \delta q(t) \mathrm{\,is\, a\,}\mathbf{symmetry} \qquad &\Leftrightarrow \qquad \frac{\delta L[q(t)]}{\delta q(t)} = 0 \,\,\,\, \forall q(t). \end{align*}](https://blog.jessriedel.com/wp-content/ql-cache/quicklatex.com-aa839448b00aee12eec31396a9ef9893_l3.svg "Rendered by QuickLaTeX.com")
There are many subtleties obscured in this cartoon presentation, like the fact that a symmetry , being a tangent direction on the manifold of trajectories, can vary with the tangent point it is attached to (as for rotational symmetries). If you’ve never spent a long afternoon with a good book on the calculus of variations, I recommend it.
Footnotes
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- You can find this presentation in “A short review on Noether’s theorems, gauge symmetries and boundary terms” by Máximo Bañados and Ignacio A. Reyes (H/t Godfrey Miller).↵
