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 on that space, the “directions” at each point, and the corresponding functional gradient 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)

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).↵