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# A. Appendix

## Conservation laws in the presymplectic framework

Cartan's formula $L_X = di_X + i_X d$ implies that the Lie derivative of the Souriau 2-form $\sigma$ vanishes when $i_X \sigma$ is exact,

$$i_X \sigma = -df \qquad (A.1)$$

for some function $f$, which is a constant of the motion [77].

As an example, we consider a NR (Galilean) particle in (1+1) spacetime dimension, with

$$\sigma^G = dp \wedge dx - (p/m)dp \wedge dt, \qquad (A.2)$$

defined on 1+2 dimensional Galilean evolution space $\mathcal{E} = \{t, x, p\}$. A Galilei boost by $b$ is generated by the vector field on $\mathcal{E}$:

$$X^G = (bt)\partial_x + mb\partial_p. \qquad (A.3)$$

Thus (A.1) yields the conserved G-boost momentum:

$$f^G = -mx + pt. \qquad (A.4)$$

Assume that $\sigma$ is exact, $\sigma = d\omega$. Then Cartan's formula implies that $\omega$ is invariant up to a surface term (physically a gauge transformation),

$$L_X \omega = dK \quad \text{with} \quad K = \omega(X) - f, \qquad (A.5)$$

providing us with the conserved quantity, calculated also as

$$f = \omega(X) - K. \qquad (A.6)$$

Eqn (A.5) can be checked also independently, by calculating Lie derivative of the 1-form $\omega = \omega_k dx^k$ using

$$L_X \omega = \left( X^j \frac{\partial \omega_i}{\partial x^j} + \omega_j \frac{\partial X^j}{\partial x^i} \right) dx^i. \qquad (A.7)$$

In the Galilean example above:

$$\omega^G = pdx - (p^2/2m)dt \Rightarrow L_X \omega^G = d(mbx) \equiv dK \qquad (A.8)$$

yielding again (A.4).
Coming to Carroll in (1 + 1) d, the Cartan form and C boost are,

$$ \omega^C = pdx - mds, \quad X^C = mb\partial_p - bx\partial_s, \quad \text{and} \quad L_X\omega^C = 2mbdx = dK, \qquad (A.9) $$

where we used (A.7). Thus we get the conserved quantity

$$ f^C = \omega(X) - K = mbx. \qquad (A.10) $$

The same result can be recovered also from (A.1) with $\sigma^C = dp \wedge dx$,

$$ i_X \sigma^C = -d(mx)b. \qquad (A.11) $$