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Stuart_M
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- TL;DR Summary
- There is a version of Reynolds Transport Theorem for differential forms: what is the conserved quantity?
Consider a fluid flow with density ##\rho=\rho(t,x)## and velocity vector ##v=v(t,x)##. Assume it satisfies the continuity equation
$$
\partial_t \rho + \nabla \cdot (\rho v) = 0.
$$
We now that, by Reynolds Transport Theorem (RTT), this implies that the total mass is conserved
$$
\frac{d}{dt}\int_{\Omega_t} \rho dx = 0,
$$
where ##\Omega_t## is some control volume (which I understand in the following way: I fix a region ##\Omega## at time ##0##; I let ##\Omega## evolve with the flow at time ##t## and I obtain ##\Omega_t=\Phi_t(\Omega)##). I have recently been reading Frankel's "The Geometry of Physics" and I've learnt that using differential forms it is possible to generalise (RTT) in the following way (see Chapter 4, Section 3):
$$
\frac{d}{dt} \int_{S(t)} \alpha = \int_{S(t)} (\partial_t \alpha + L_v \alpha)
$$
where now ##\alpha## are (time-indexed family of) ##p##-forms, ##S## is a fixed ##p##-manifold and ##S(t) = \Phi_t(S)## is the evolution of ##S## under the flow of ##v##, and ##L_v## denotes the Lie derivative. From this we easily deduce that, if a ##p##-form satisfies the "continuity equation"
$$
\partial_t \alpha + L_v \alpha = 0
$$
then the quantity ##\int_{S(t)} \alpha## is constant in time. If ##\alpha## is a (top-dimensional) volume form, then we recover (RTT) and thus the conservation of *mass*. However, in the general case what does the quantity ##\int_{S(t)} \alpha## represent?
I have considered some easy toy examples: if ##\alpha = f## is a ##0##-form (i.e. scalar function), then the "continuity equation" reads ##\partial_t f + v \cdot \nabla f = 0##, whence ##f(t,\Phi_t(x)) = f(0,x)## and indeed this can be expressed as "conservation of the integral of ##f## on 0-dimensional manifolds". However, which quantity is conserved?
Another example [taken from Frankel, ibidem]: in Euler, the vorticity *form* ##\omega=d \nu## (##\nu## being the velocity *co*vector) is invariant under the flow, i.e. solves the continuity equation in the sense of 2-forms. Therefore,
$$
\int_{S(t)} \omega
$$
is constant for any 2-manifold ##S##. Which conservation law am I rediscovering? Which conserved quantity is this?
$$
\partial_t \rho + \nabla \cdot (\rho v) = 0.
$$
We now that, by Reynolds Transport Theorem (RTT), this implies that the total mass is conserved
$$
\frac{d}{dt}\int_{\Omega_t} \rho dx = 0,
$$
where ##\Omega_t## is some control volume (which I understand in the following way: I fix a region ##\Omega## at time ##0##; I let ##\Omega## evolve with the flow at time ##t## and I obtain ##\Omega_t=\Phi_t(\Omega)##). I have recently been reading Frankel's "The Geometry of Physics" and I've learnt that using differential forms it is possible to generalise (RTT) in the following way (see Chapter 4, Section 3):
$$
\frac{d}{dt} \int_{S(t)} \alpha = \int_{S(t)} (\partial_t \alpha + L_v \alpha)
$$
where now ##\alpha## are (time-indexed family of) ##p##-forms, ##S## is a fixed ##p##-manifold and ##S(t) = \Phi_t(S)## is the evolution of ##S## under the flow of ##v##, and ##L_v## denotes the Lie derivative. From this we easily deduce that, if a ##p##-form satisfies the "continuity equation"
$$
\partial_t \alpha + L_v \alpha = 0
$$
then the quantity ##\int_{S(t)} \alpha## is constant in time. If ##\alpha## is a (top-dimensional) volume form, then we recover (RTT) and thus the conservation of *mass*. However, in the general case what does the quantity ##\int_{S(t)} \alpha## represent?
I have considered some easy toy examples: if ##\alpha = f## is a ##0##-form (i.e. scalar function), then the "continuity equation" reads ##\partial_t f + v \cdot \nabla f = 0##, whence ##f(t,\Phi_t(x)) = f(0,x)## and indeed this can be expressed as "conservation of the integral of ##f## on 0-dimensional manifolds". However, which quantity is conserved?
Another example [taken from Frankel, ibidem]: in Euler, the vorticity *form* ##\omega=d \nu## (##\nu## being the velocity *co*vector) is invariant under the flow, i.e. solves the continuity equation in the sense of 2-forms. Therefore,
$$
\int_{S(t)} \omega
$$
is constant for any 2-manifold ##S##. Which conservation law am I rediscovering? Which conserved quantity is this?
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