- #1
JVM
- 5
- 1
Hello,
First of all, I'm sorry for the long post. I've tried to make it self consistent, but some prior knowledge will be assumed in both physics and differential geometry.
Also maybe some math will be displayed wrongly, it's my first time here...
In physics people usually describe Noether's theorem by calculating the variation of the Lagrangian, which for a symmetry of the theory will need to be equal to a total derivative:
\begin{equation}
\delta \mathcal{L} = \epsilon^A \partial_{\mu} (\frac{\delta \mathcal{L}}{\delta \partial_{\mu}\phi^i} \Delta_A \phi^i) = \epsilon^A \partial_{\mu} K_A^{\mu}.
\end{equation}
Then a Noether Current can be defined as
\begin{equation}
J^{\mu}_A = -\frac{\delta \mathcal{L}}{\delta \partial_{\mu}\phi^i} \Delta_A \phi^i + K_A^{\mu}.
\end{equation}
For fields which abide the equations of motion this current is then conserved: $\partial_{\mu} J^{\mu}_A = 0$. This is just because the variation of the action will be set to zero using Stokes and assuming the fluctuations on the boundaries will go to zero fast enough.
In mathematics it is done a little different. So far I've tried to describe everything as good as possible such that a reader with some physics background is familiar with everything. Now comes a little differential geometry and I will assume that some basic stuff is known, so if you're not too familiar with it I guess you shouldn't let this question bore you.
A Liouville vectorfield
\begin{equation}
D \in \mathcal{X}(TQ),
\end{equation}
where Q is some smooth manifold, is defined such that her flow has the following form:
\begin{align*}
\Phi_t : TQ &\rightarrow TQ \\
(q^i,v^i) &\mapsto (\bar{q}^i = q^i, \bar{v}^i = e^t v^i)
\end{align*}.
The energy function associated to a function $L$ is then defined as
\begin{equation}
E_L = D(L) -L.
\end{equation}
The Lagrangian vector field associated to a regular function L on TQ is the the vector field $\Gamma$ such that
\begin{equation}
i_{\Gamma}\omega_L = -dE_L
\end{equation}
where
\begin{equation}
\omega_L = d\theta_L
\end{equation}
is the Poincare-Cartan 2-form associated to L, and d is the exterior derivative. Finally we can now introduce Noether's theorem;
Let Y be a vector field on TQ, such that:
\begin{equation}
\mathcal{L}_Y \theta_L = df
\end{equation}
for some function f, and:
\begin{equation}
Y(E_L) = 0.
\end{equation}
Then Y is a symmetry of $\Gamma$ and associated to Y there is a first integral F, determined by
\begin{equation}
F = f - <Y,\theta_L>.
\end{equation}
And with every first integral F of $\Gamma$ there is a symmetry Y which abides the above mentioned properties.
Now comes the question...
Although in physics we are usually interested in invariance of the action, especially in field theory (for example for space-time symmetries), the Noether theorem I just described gives a one to one correspondence between the symmetry of the Lagrangian and the conserved quantities. Now I know that what I just described is valid for mechanics and not field theory, and when you want to describe field theory in this geometric formalism you need to look at jet bundles. Is it then so that when you use jet bundles you will eventually get a correspondence between the action and conserved quantities?
And am I wrong to think that actually in mechanics the symmetries will be described with the action as well?
First of all, I'm sorry for the long post. I've tried to make it self consistent, but some prior knowledge will be assumed in both physics and differential geometry.
Also maybe some math will be displayed wrongly, it's my first time here...
In physics people usually describe Noether's theorem by calculating the variation of the Lagrangian, which for a symmetry of the theory will need to be equal to a total derivative:
\begin{equation}
\delta \mathcal{L} = \epsilon^A \partial_{\mu} (\frac{\delta \mathcal{L}}{\delta \partial_{\mu}\phi^i} \Delta_A \phi^i) = \epsilon^A \partial_{\mu} K_A^{\mu}.
\end{equation}
Then a Noether Current can be defined as
\begin{equation}
J^{\mu}_A = -\frac{\delta \mathcal{L}}{\delta \partial_{\mu}\phi^i} \Delta_A \phi^i + K_A^{\mu}.
\end{equation}
For fields which abide the equations of motion this current is then conserved: $\partial_{\mu} J^{\mu}_A = 0$. This is just because the variation of the action will be set to zero using Stokes and assuming the fluctuations on the boundaries will go to zero fast enough.
In mathematics it is done a little different. So far I've tried to describe everything as good as possible such that a reader with some physics background is familiar with everything. Now comes a little differential geometry and I will assume that some basic stuff is known, so if you're not too familiar with it I guess you shouldn't let this question bore you.
A Liouville vectorfield
\begin{equation}
D \in \mathcal{X}(TQ),
\end{equation}
where Q is some smooth manifold, is defined such that her flow has the following form:
\begin{align*}
\Phi_t : TQ &\rightarrow TQ \\
(q^i,v^i) &\mapsto (\bar{q}^i = q^i, \bar{v}^i = e^t v^i)
\end{align*}.
The energy function associated to a function $L$ is then defined as
\begin{equation}
E_L = D(L) -L.
\end{equation}
The Lagrangian vector field associated to a regular function L on TQ is the the vector field $\Gamma$ such that
\begin{equation}
i_{\Gamma}\omega_L = -dE_L
\end{equation}
where
\begin{equation}
\omega_L = d\theta_L
\end{equation}
is the Poincare-Cartan 2-form associated to L, and d is the exterior derivative. Finally we can now introduce Noether's theorem;
Let Y be a vector field on TQ, such that:
\begin{equation}
\mathcal{L}_Y \theta_L = df
\end{equation}
for some function f, and:
\begin{equation}
Y(E_L) = 0.
\end{equation}
Then Y is a symmetry of $\Gamma$ and associated to Y there is a first integral F, determined by
\begin{equation}
F = f - <Y,\theta_L>.
\end{equation}
And with every first integral F of $\Gamma$ there is a symmetry Y which abides the above mentioned properties.
Now comes the question...
Although in physics we are usually interested in invariance of the action, especially in field theory (for example for space-time symmetries), the Noether theorem I just described gives a one to one correspondence between the symmetry of the Lagrangian and the conserved quantities. Now I know that what I just described is valid for mechanics and not field theory, and when you want to describe field theory in this geometric formalism you need to look at jet bundles. Is it then so that when you use jet bundles you will eventually get a correspondence between the action and conserved quantities?
And am I wrong to think that actually in mechanics the symmetries will be described with the action as well?