# What's the name of the symmetry $L \to L + \frac{d \Lambda}{dt}$?

In the Lagrangian formulation of Classical Mechanics, we have the freedom to add a total time derivative of an arbitrary function $$\Lambda$$ to the Lagrangian:

$$L \to L + \frac{d \Lambda}{dt} .$$

Does this symmetry of the Lagrangian have any particular name?

• The arbitrariness of the function $\Lambda$ is not general. It's restricted. So, if the Lagrangian is $L(q,\dot{q},t)$ then the function $\Lambda$ must not depend on $\dot{q}$ that is $\Lambda(q,t)$. – Frobenius Feb 7 at 3:33

Some authors call that a gauge transformation of the Lagrangian function. Others, don't give any specific name and may object to the previous one.

For a reference for the former denomination, see for instance F. Scheck, Mechanics, Springer, 2010.

• Ah yes, I've also found the term "mechanical gauge transformations" in the book "Analytical Mechanics" by Nolting. I also discovered in Tong's "Classical Dynamics" notes the comment "The Lagrangian L is not unique. We may make the transformation [...] for any function f and the equations of motion remain unchanged. [...] (As an aside: A system no longer remains invariant under these transformations in quantum mechanics. The number α is related to Planck’s constant, while transformations of the second type lead to rather subtle and interesting effects related to the mathematics of topology)." – JakobH Feb 19 at 9:07
• Here are some further references which explicitly call this kind of transformation a gauge transformation: Structure and Interpretation of Classical Mechanics by Gerald Jay Sussman et. al, Theoretical Physics 2 by Nolting, Variational Principles in Classical Mechanics by Cline, A Primer of Analytical Mechanics by Strocchi, Classical Dynamics by Saletan. In particular, the last one defines: "a Lagrangian undergoes a gauge transformation whenever a total time derivative is added to it." – JakobH Mar 1 at 9:15
1. I would simply call the operation$$^1$$ $$L(q,\dot{q},\ldots, q^{(N)},t)\quad \longrightarrow \quad L(q,\dot{q},\ldots, q^{(N)},t) \quad+\quad \frac{dF(q,\dot{q},\ldots, q^{(N-1)},t)}{dt}\tag{1}$$ for "adding a total derivative term to the Lagrangian", nothing else.

2. A quasi-symmetry transformation or a gauge transformation are by definition specified at the level of the fundamental variables of the theory (in this case, the $$q$$s and $$t$$). The operation (1) doesn't in general fulfill this.

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$$^1$$ The operation (1) should be amended with a prescription for the possible new boundary conditions (BCs). Note that the operation (1) [with the new BCs] may render the functional/variational derivative of the action $$S$$ ill-defined. However, if the functional derivative $$\frac{\delta S}{\delta q}$$ exists both before and after the operation (1), it is unchanged, cf. e.g. this Phys.SE post.