Noether's Theorem
Introduction
Last time, we formulated an abstract way of looking at the world, through the Principle of Least Action.
But, apart from unifying all of Physics under one equation, it was not clear what the benefit of such
an abstraction was. Tonight we get to see this, by studying an amazing result called Noether's theorem.
Noether's theorem states that whenever there is a symmetry, then you get a conserved quantity.
Thus, this theorem provides a surprising connection between symmetries and conservation laws.
In order to understand the above paragraph, let us describe what we mean by ``conservation '' and
``symmetry'', in simple English. (A more detailed treatment is given in the links above.)
- Symmetry: A symmetry is an invariance in the laws of Physics. For example: Let us assume that
there is a glass of water on a table. If I apply sufficient force, then I can knock the glass off
the table. Now, let us say that I move the glass 10 cm to the right of its present position.
If I apply the same force on the glass, I will again succeed in knocking the glass off the table.
In other words, the laws of Physics will work equally well, no matter where I move the object.
We call this as space translation symmetry.
Similarly, we have time translation symmetry; the same laws of Physics will apply no matter
what time I chose to perform the action. Thus, it does not matter whether I act on the glass
now, or 10 hours from the present moment; if I hit it, it will get knocked down. - Conservation Law: When we say that a quantity is ``conserved'', we mean that it does not
change over time. We have come across many ``conservation laws'' in high-school Physics
such as: ``conservation of linear momentum'', ``conservation of angular momentum'', ``conservation
of energy'' etc.
What they are saying is that, for a closed system, the total amount of momenta or
energy etc., remains the same across time.
So, Noether's theorem tells you that if you have a symmetry, then you have a conservation law that
corresponds to it. In terms of specific examples:
- If you have linear translation symmetry (i.e. linear invariance), then linear momentum is conserved.
(Law of conservation of linear momentum.) - If you have rotational invariance, then angular momentum is conserved.(Law of conservation of
angular momentum.) - If you have time translation invariance, then energy is conserved.(Law of conservation of energy.)
This is a surprising and deep property of nature, which holds good for all physical systems (quantum
mechanics, general relativity etc. etc.)
There is one more thing: Using Noether's theorem we can obtain extremely general definitions of these
conserved quantities. For example: Energy is defined as simply that quantity that is conserved because
of time translation invariance. In other words, energy is a mere consequence of the fact that the laws of
Physics are the same, across time.
The above way of thinking about energy is very abstract, when compared to our usual ideas of energy.
It is easier to understand it mathematically rather than through English. Fortunately,
we do not need any complicated maths (really nothing beyond what we did last time), to derive this
beautiful result. The derivation is given below, for the pleasure of those who are interested.
1 Getting Started
If you don’t know what these series of posts are about, then see this page for why I am doing this,
my sources, how to help etc.
Please note that this section builds heavily from what we did last time. Many terms and techniques
will not make sense without understanding the previous post. However, if you still have doubts,
leave me a comment and I will try and answer them. (Keep in mind though, that I am a layman too,
so there is a chance that my answer may be wrong. However, I will try my best to give a right answer! :))
The roadmap here is as follows: We shall present 2 versions of Noether's theorem.
The first version is for space translation symmetry.
We then shall present the slightly more involved, second version of the theorem for time translation symmetry
after it; this version makes use of the results from the first version of the theorem.
Finally, we will finish off by showing how energy conservation is a consequence of time translation symmetry.
2 Space translation symmetry
Let us define a transformation in the position co-ordinates in the manner given below.
Note, that we are defining an infinitesimal transformation here. (This derivation will not work for
discrete symmetries BTW.)
We shall call this as equation (1).
From (1) we get the following expression for a change in the position
Let us call this as equation (2).
Now, recall the definition of ``action'', from last time
We call this as equation (3).
Now, we need to find out the change in the action on account of changing the position
co-ordinates by an infinitesimal transformation. How do we do this?
Here is the proof sketch:
We shall start by assuming that the original system followed from a Principle of Least Action.
We then use the same integration by parts trick that we used last time, except that this
time, the end-points of the trajectory will move across space. To incorporate this change,
we have to add in an end-point correction term. (this follows from the integration by
parts expression.)
All of this is worked out below:
Notice that in equation (5) above, the integral resembles the expression we got while deriving
the Euler-Lagrange equation last time. Because, we assume that the system follows the
Principle of Least Action, it can safely be set to zero. Thus, leaving only the end point
correction term which is seen in equation (6).
If we assume space translation invariance, then this term can be set to zero. We get
equation (8) simply by substitution from equation (2).
Thus, the quantity that is conserved is:
We shall call this as equation (9). This is what we have been searching for; the quantity
which is conserved because of space translation invariance.
Now, let us see what quantity is conserved because of time translation invariance.
3 Time translation symmetry
Let us change the time of the system by an infinitesimal amount:
We shall call this as equation (10).
The corresponding change in the positions of all the components of the
trajectory is therefore:
Call this as equation (11).
The above equation follows from basic maths which says that, if you change an argument to
a function, then the function shifts in a direction, opposite to that of the shift in the
argument. In this case, adding a positive term to the argument, shifts the function to the left.
From equation (11), we get the corresponding change in the position when we change
the time by an infinitesimal amount. This is shown below:
All right! Now we are ready to derive the corresponding change in the action when we
change the time. The proof sketch is as follows:
The derivation is exactly the same as in the previous case of space translation symmetry,
except that the end-points of the trajectory will move not only across space, but also
across time.
Thus, we simply do what we did in the previous derivation, except now we have an extra
term for the change in the end-points across time.
Let us work this out below:
Notice above that equations (16) and (17) are in the same pattern as equations (5) and (6)
respectively, except for the addition of an extra term. This extra, correction term is for
the movement of the end-points across time.
Let us now leave this expression as it is for the time being while we work out this correction
term. We will then add this correction term back into our expression to get the final answer.
3.1 Motion of the end-points across time
Ok. How do we get an expression for the shift in the end-points of the trajectory, across
time? Here will use a simple trick. Note that we need to make two corrections; at both ends
of the trajectory. This is illustrated below:
Let us call this as equation (18).
Now, if we know the value of a function at a particular point, then the value of that
function, for a point which is located at an infinitesimal increment away from it, is
approximated by:
Value of function at new point = Value of increment X Value of function at original point.
This trick is illustrated by equations (19) and (20) below:
The differences in signs in the above two equations is because a portion of the trajectory
is chopped off at the lower end point and a corresponding portion of it is added at the upper
end point, because of a positive shift (see equation (10)) in the trajectory across time.
3.2 Final expression
Ok. Now we are ready to resume work on our earlier job of getting an expression for the
conserved quantity.
Using equations (18), (19) and (20), we re-write equation (17) as follows:
By our assumption of time translation invariance, the above term reduces to zero.
Thus, the quantity that is conserved is:
Let us call this as equation (24).
This is the answer that we have been searching for.
4 What is Energy?
Ok. Now we have derived an expression (equation (24)), for the quantity that is conserved
because of time translation invariance. It is now very easy to see that it corresponds to the energy
of the system. We do this as follows:
We simply negate the above expression to define a quantity called as the Hamiltonian (or energy).
It is thus defined as:
Let us call this equation (25).
Since we have simply changed the sign of a conserved quantity, it follows that the above expression
is also conserved.
Ok, now it is not clear how this expression corresponds to the energy of a system. To do that we
need to take an example system. Recall, last time we said that we could model any system we
wanted, by simply plugging in the appropriate Lagrangian for it? Just for fun, let us do that now for
Classical Mechanics and see what the expression for energy turns out.
The Lagrangian for Classical Mechanics is defined as:
The total Kinetic Energy minus the total Potential Energy of the system.
This is expressed mathematically below:
Now, we simply plug in equation (27) into equation (25) to get the total energy. This is shown
below:
Thus, we see that we do indeed get the correct quantity for the total energy of the system. (i.e. the
sum of the Kinetic and the Potential energies.) Thus, the conserved quantity, assuming time translation
symmetry really does correspond to the energy of the system!
Final Thoughts
If you went through the technical details of the derivation, then there are a couple of fine points that
must be mentioned:
- The technique relies heavily on the Principle of Least Action. Thus, it is a necessary condition
for Noether's theorem that the system being studied follow this principle. Luckily for us, this
principle is ubiquitous in nature, (see the penultimate point here.) and so this theorem holds
for all physical systems known to us. - The last section regarding energy smells really fishy! It is almost as if we ``cheated'', by knowing what
the answer for energy should be, and then intentionally designing the Lagrangian, so that we
got this answer at the end of the calculation.
If this made you suspicious, then you are some what justified! :) As we proceed to do more advanced
stuff, it is going to look less and less like the Physics you know; It gets increasingly abstract and it will
involve some nifty and very beautiful re-engineering (of equations).
Next time we will focus on constructing a Langrangian of our own. In that post we will exploit this
engineering ``mentality'' to the hilt and derive some beautiful (but also practical) results.