- #1
etotheipi
I came across a derivation of the Noether theorem with a step I don't understand; the transformation of the time and configuration space are written as$$\tau(t,\varepsilon) = t + \varepsilon \delta t, \quad Q^a(q,\varepsilon) = q^a + \varepsilon \delta q^a$$here ##\varepsilon## is an infinitesimal parameter, whilst it looks like ##\delta t## and ##\delta q^a## are in this notation the generators of the time evolution and generalised coordinates respectively. Then they write$$\varepsilon \delta \mathcal{L} = \mathcal{L}(Q(q,\varepsilon), \dot{Q}(q,\varepsilon), \tau(t,\varepsilon)) - \mathcal{L}(q,\dot{q}, t)$$ $$\delta \mathcal{L} = \frac{\partial \mathcal{L}}{\partial q^a} \delta q^a + \frac{\partial \mathcal{L}}{\partial \dot{q}^a} \delta \dot{q}^a + \frac{\partial \mathcal{L}}{\partial t} \delta t$$and from this we can go on to show that if ##\delta \mathcal{L} = \dot{F}## i.e. for some transformation that is a quasi-symmetry, then ##J = \mathcal{H} \delta t - p_a \delta q^a + F## is a constant of the motion.
I don't understand the third line, where does it come from? It's not the chain rule of multivariable calculus, because the generators ##\delta q^a## and ##\delta t## for instance are finite. Maybe I misunderstand how the author has defined the variation of a function in this case. I wondered if someone could help out? Thanks!
I don't understand the third line, where does it come from? It's not the chain rule of multivariable calculus, because the generators ##\delta q^a## and ##\delta t## for instance are finite. Maybe I misunderstand how the author has defined the variation of a function in this case. I wondered if someone could help out? Thanks!