Wednesday, 24 September 2008

Symmetries, Conserved Quantities and Noether's Theorem

Following on from my posts on Lagrangian and Hamiltonian Mechanics [1, 2, 3, 4], I'd like to discuss one of the most amazing topics in physics. The consideration of the symmetries of physical laws and how those symmetries relate to conserved quantities, the fundamentally beautiful mathematics lying beneath, and the extent to which we can develop theories of the world around us from such simple concepts; these things continually inspire me and provoke my interest.

I'll start by showing that the formulation of Lagrangian and Hamiltonian Mechanics, thus far, allows us to determine several conservation laws. Consider, for example, the homogeneity of space. Space is homogeneous if the motion (or time-evolution) of a particle (or system thereof) is independent of absolute position. That is, the potential does not vary with absolute position (it can still vary with the vector distance between two particles, as an interaction potential, for example!)

If we make a transformation
, then the Lagrangian will also transform as
L \rightarrow L+\delta L
. For a single particle, we can Taylor expand as follows:

L(\mathbf{r}+\delta\mathbf{r},\mathbf{v}) = L(\mathbf{r},\mathbf{v})+\frac{\partial L}{\partial x}\delta x+\frac{\partial L}{\partial y}\delta y+\frac{\partial L}{\partial z}\delta z

Which we can use to write
\delta L = \frac{\partial L}{\partial \mathbf{r}}\cdot\delta\mathbf{r}

\frac{\partial L}{\partial \mathbf{r}}
is a vector quantity; each component is the derivative of L with respect to the corresponding coordinate of r. For a single particle, then,
\frac{\partial L}{\partial \mathbf{r}}=\nabla L

Homogeneity of space requires that
\delta L = 0
. Since
\delta \mathbf{r}
is arbitrary (and therefore not necessarily zero), we have that

\frac{\partial L}{\partial q_i} = 0 ~~~~~~ (\star)

This holds only if L does not depend on absolute position, otherwise there would be a contribution
\delta L
from many of the possible choices of
. Spatial dependence of e.g. V(x) implies spatial variation of L, and momentum would not be conserved.

The Euler-Lagrange Equation applies for each coordinate in the vector r. The sum of these Euler-Lagrange Equations (ELEs) means that
requires that:

\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} = 0 \\
\Rightarrow p_i = \frac{\partial L}{\partial \dot{q}_i} ~~\mathrm{remains~constant}

We have, therefore, demonstrated the conservation of momentum as a result of requiring translational invariance. That is, any canonical momenta whose conjugate coordinates do not appear explicitly in the Lagrangian are conserved.

Turning once again to time symmetries, let us re-derive the conservation of energy. If the Lagrangian is homogeneous in time, i.e.
, then:

\frac{dL}{dt} = \sum_i \frac{\partial L}{\partial q_i}\dot{q}_i + \sum_i\frac{\partial L}{\partial \dot{q}_i}\ddot{q}_i

As L does not depend explicitly on time, there is no term
\frac{\partial L}{\partial t}
on the RHS. Sunstituting
\frac{\partial L}{\partial q_i}
from the ELE,

\frac{dL}{dt} = \sum_i\dot{q}_i\frac{d}{dt}\frac{\partial L}{\partial\dot{q}_i}+\sum_i\frac{\partial L}{\partial\dot{q}_i}\ddot{q}_i = \sum_i\frac{d}{dt}\left(\dot{q}_i\frac{\partial L}{\partial\dot{q}_i}\right)

\Rightarrow \frac{d}{dt}\sum_i\left( \dot{q}_i\frac{\partial L}{\partial\dot{q}_i}-L \right) = 0 \\
\Rightarrow H = \sum_i \dot{q}_i\frac{\partial L}{\partial \dot{q}_i} - L~~~\mathrm{remains~constant}

The conservation of energy holds for any motion in a non-time-varying external field V(x)

We turn now to the isotropy of space, and show that angular momentum is conserved due to rotational invariance of the Lagrangian. Consider rotation by an angle
(with a direction given by
) about a vector. For small rotations,
\mathbf{r}\rightarrow \mathbf{r}+\delta\mathbf{r}
, with
\delta\mathbf{r} = \delta\theta\times r
. Each component of the velocity is also transformed by this rotation,

For a single body, we now impose the requirement that the Lagrangian be unchanged under such a rotation (i.e. we require space to be isotropic).
\delta L = \sum_i\left( \frac{\partial L}{\partial q_i}\cdot\delta r_i + \frac{\partial L}{\partial\dot{q}_i}\cdot\delta v_i \right) = 0

We can replace
\frac{\partial L}{\partial v_i}
by the vector canonical momentum
, and
\frac{\partial L}{\partial q_i}
, leaving:
\left(\dot{\mathbf{p}}\cdot\delta\mathbf{r} + \mathbf{p}\cdot\delta\mathbf{v}\right) = 0 \\
\Rightarrow \dot{\mathbf{p}}\cdot\left(\delta\theta\times \mathbf{r}\right) + \mathbf{p}\cdot\left(\delta\theta\times\mathbf{v}\right) = 0

\mathbf{a}\cdot\left(\mathbf{b}\times\mathbf{c}\right) = \mathbf{b}\cdot\left(\mathbf{c}\times\mathbf{a}\right)


is arbitrary, this requires that
does not change in time, hence angular momentum is a conserved quantity.

Hamilton's Equations
Using the ideas presented above, I'm going to take a moment to derive Hamilton's Equations, which will prove useful later on.

Consider changes in the Lagrangian L, according to
dL = \sum_i\frac{\partial L}{\partial\dot{q}_i}\,d\dot{q}_i + \sum_i\frac{\partial L}{\partial q_i}\,dq_i

This can be written:

\frac{\partial L}{\partial q_i}=\dot{p}_i
\frac{\partial L}{\partial\dot{q}_i}=p_i
\sum_i p_i\,d\dot{q}_i = d\left(\sum_i p_i q_i\right) - \sum_i\dot{q}_i\,dp_i

d\left(\sum_i p_i\dot{q}_i - L\right) = -\sum_i\dot{p}_i\,dq_i+\sum_i\dot{q}_i\,dp_i

The argument of the differential on the left is the Hamiltonian, H,
H(q,p,t)=\sum_i p_i\dot{q}_i - L
, therefore:
dH = -\sum_i\dot{p}_i\,dq_i + \sum_i\dot{q}_i\,dp_i

From here, we can obtain Hamilton's Equations:
\dot{q}_i &= \frac{\partial H}{\partial p_i}\\
\dot{p_i} &= \frac{\partial H}{\partial q_i}

For m coordinates (and m momenta), Hamilton's Equations form a system of 2m first-order differential equations, compared to the m second-order equations in the Lagrangian treatment.

The total time derivative,

\frac{dH}{dt}=\frac{\partial H}{\partial t}+\sum_i\frac{\partial H}{\partial q_i}\dot{q}_i+\sum_i\frac{\partial H}{\partial p_i}\dot{p}_i

Substituting Hamilton's equations for
\dot{q}_i, \dot{p}_i
, the last two terms cancel, so
\frac{dH}{dt} = \frac{\partial H}{\partial t}

and if H does not depend explicitly on time,
and energy is conserved!

Noether's Theorem
The three conserved quantities above were shown to be related to the invariance of the Lagrangian under some symmetry transformation:
  • Translational invariance (homogeneity of space) ==> Conservation of momentum
  • Rotational invariance (isotropy of space) ==> Conservation of angular momentum
  • Time invariance (homogeneity of time) ==> Conservation of energy

Noether's Theorem states that any differentiable symmetry of the Action (integral of the Lagrangian) of a physical system has a corresponding conservation law.

To every differentiable symmetry generated by local actions, there corresponds a conserved current.

`Symmetry' here, refers to the covariance of the form of a physical law with respect to a Lie group of transformations; the conserved quantity is known as a charge and the flow carrying it as a current (c.f. electrodynamics).

Noether's Theorem, which I will discuss in more detail at a later date, is another critical component used to build gauge theories. The key thing to remember right now is that a symmetry (invariance) of the Lagrangian corresponds to a conserved quantity; we can use this result to look for the underlying symmetry behind quantities we know to be conserved (for example, electric charge).

Monday, 22 September 2008

Feedback on `Writing Scientific Documents Using LaTeX'

Since I submitted my article Writing Scientific Documents Using LaTeX to CTAN, I've had several emails offering comments, suggestions and improvements. I plan on taking all of these into account for the next edition, which should be available sometime soon. Meanwhile, I thought I'd just post a little about some of the suggestions here.

The single most common suggestion was that I replace the use of $$ ... $$ to delimit displayed mathematics with the \[ ... \] form. The dollar-variant is apparently deprecated, and the newer square bracket style preferred. The reason for this, as given in, is that the $$ form is a plain TeX command, and should be avoided in LaTeX due to inconsistencies. Check the l2tabu document for more details.

Another frequent comment was that the eqnarray environment is bad, and should always be replaced by the align or similar, from the amsmath package. I originally thought I'd leave a discussion of eqnarray in there for reference, but now it seems better to remove it entirely.

Other comments related to minor corrections and requests for additional explanation in some sections, expanded coverage of BibTeX and more details on table design.

As I said above, I plan to incorporate all of these suggestions into the 5th Edition of the article.

From Lagrangian to Hamiltonian Mechanics

If the Lagrangian L does not depend explicitly on time, and varies with time only through the time-dependence of the coordinate q (and its time derivative), then we can define a constant of motion known as the Hamiltonian, H.

That is, if

L = L(q(t),\dot{q}(t)) \\
\mathrm{not}~~L = L(q(t), \dot{q}(t), t)

then the Hamiltonian H can be defined as
H = \dot{q}\frac{\partial L}{\partial \dot{q}} - L

For the simplest example of a particle of mass m, moving through a potential V(x), with velocity v = dx/dt, we have
L = \frac{m\dot{x}^2}{2} - V(x) = T - V
. Using the above definition of the Hamiltonian,
H = \frac{m\dot{x}^2}{2} + V(x)
which can be identified as the total energy. Since H is a constant of motion, this result corresponds to conservation of energy!

More generally,
\frac{dH}{dt}=\frac{d}{dt}\left(\dot{x}\frac{\partial L}{\partial\dot{x}}\right)-\frac{\partial L}{\partial\dot{x}}\frac{d\dot{x}}{dt}-\frac{\partial L}{\partial x}\frac{dx}{dt}
= v\left( \frac{d}{dt}\frac{\partial L}{\partial \dot{x}} - \frac{\partial L}{\partial x}\right) = 0

since the last term inside the brackets vanishes (it's the Euler-Lagrange Equation!)

Note that if L depends explicitly on time, the above does not hold, and we lose conservation of energy; here, energy can be `used' in thermodynamically irreversible processes. Of course, if we expanded the Lagrangian to take into account a large enough system, we regain conservation of energy for the universe as a whole!

The Hamiltonian H is an integral of the motion, since
\frac{dH}{dt} = 0 \Rightarrow \int_{t_0}^{t_1}\frac{dH}{dt}\,dt = \left[H\right]_{t_0}^{t_1} = 0

It contains only first-order time derivatives of the coordinate q, whereas the Euler-Lagrange equation contains second-order time derivatives,
\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}

As an example of determining the Hamiltonian from a Lagrangian, I'd like to look at the case of Special Relativity (SR). There are two reasons for this... first, the SR Lagrangian looks a bit different from the usual T - V form of classical mechanics. Secondly, Special Relativity will feature prominently in a number of future posts as I start to direct the methods of Lagrangian and Hamiltonian mechanics towards describing gauge theories.

The SR Lagrangian in a potential V(x) is given by
L = \frac{-mc^2}{\gamma} - V(x) = -mc^2\sqrt{1-\frac{v^2}{c^2}} - V(x)

where I have used
v = \dot{x} = \frac{dx}{dt}
and have referred to the rest mass as simply m, while many books use the notation m_0.

The Hamiltonian H can be found as
H = v\frac{\partial L}{\partial v} - L\\
= \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} + V(x)\\
= \gamma mc^2 + V(x)

The last line is readily identified as the total relativistic energy of a particle of rest mass m in a potential V(x), so our interpretation of the Hamiltonian holds!

Canonical Momenta
I'd like to take a moment now to introduce some new nomenclature and notation, and to explain why it is useful here.

First, a reminder that the Euler-Lagrange equation for a coordinate q can be written as
\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q}

We can define a quantity p, known as the canonical momentum conjugate to the coordinate q, as follows
p=\frac{\partial L}{\partial \dot{q}}

and a quantity F, known as the canonical force conjugate to the momentum p, as
F = \frac{\partial L}{\partial q}

These follow the usual definitions such that
T = \frac{p^2}{2m}
F = -\nabla V
, since the Lagrangian L = T - V.

Written in this form, the Euler-Lagrange Equation directly represents Newton's 2nd Law:
F = \dot{\mathbf{p}}

For multiple coordinates
, the Hamiltonian H is given by

H = \sum_i p_i \dot{q}_i - L

is the canonical momentum conjugate to the coordinate

Introducing the concept of canonical momentum conjugate to a coordinate is essential for the future posts I'd like to make on this blog. The reason for this is that when I discuss quantum field theory, the concepts introduced above will play an essential role in the procedure. More on this later!

Meanwhile, although I provided a couple of classical mechanics examples for the use of the Lagrangian, in order to introduce the concept at a level that most people can understand from the world around them, I will refrain from posting similar examples making use of the Hamiltonian, since I wish to progress to more complicated topics.

In order to illuminate the path from here to a discussion of gauge theories, I'd like now to simply list some of the topics I must first cover. As such, the interested reader will notice any such posts! I anticipate that this list will be incomplete, and that I will need to cover certain topics in a lot more detail than others.
  1. Symmetries and Noether's Theorem (some of this)
  2. Special Relativity in four-vector formulation (lots of this)
  3. Relativistic electrodynamics (not much of this)
  4. Relativistic Quantum Mechanics (I've already discussed the Klein-Gordon equation, but I'll go over this and other aspects, again)
  5. Spin & Relativity (Pauli matrices and the gamma matrices)
  6. The Dirac Equation (in some detail!)
  7. U(1) symmetries and the Gauge Principle
  8. Quantum Electrodynamics
  9. SU(2) and Electroweak Unification
  10. Aside on Superconductivity (which may be omitted)
  11. Spontaneous Symmetry Breaking and the Higgs Mechanism

I expect those posts to take some considerable time. I've spent at least two weeks discussing Lagrangian mechanics, and we have a long way to go before I can introduce the Higgs mechanism, which is the ultimate goal of this entire series of posts. The idea is that anyone with a basic grounding in physics and mathematics should be able to learn enough about Lagrangian & Hamiltonian Mechanics, Quantum Mechanics and Special Relativity to be able to appreciate the formulation of the gauge theories of the Standard Model of Particle Physics, and to go a little beyond that and get a look at the Higgs mechanism. Let's hope I make it to the end!

Saturday, 20 September 2008

Lagrangian Mechanics: From the Euler-Lagrange Equation to Simple Harmonic Motion

I already wrote about obtaining Newton's Laws from the Principle of Least Action. Now I'm going to analyse a simple mass-spring system; effectively just a case of substituting in a suitable potential for the spring.

Let us go through all of the steps, once more. We'll choose our coordinate x to be the displacement from the equilibrium length of the spring, as shown on the diagram. The kinetic energy of the moving mass is then just
. The potential of a spring stretched (or compressed) x metres from its equilibrium length is given by
, where k is the spring constant (N m^-1).

So, for our mass-spring system, the Lagrangian is

L = \frac{1}{2}\left(m\dot{x}^2 - kx^2\right)}

Applying the Euler-Lagrange equation, we obtain

-kx - \frac{d}{dt}\left(m\dot{x}\right) = 0 \\
\Rightarrow \ddot{x} - \frac{k}{m}x = 0

Comparing the last line above with the general form of Simple Harmonic Motion (SHM),
\ddot{q} - \omega^2 q = 0
we can see that the equation of motion rendered by applying the Euler-Lagrange equation to the Lagrangian of a mass-spring system provides Simple Harmonic Motion with a frequency

\omega = \sqrt{\frac{k}{m}}

This is as expected for the case of a mass-spring system!

Friday, 19 September 2008

Lagrangian Mechanics: From the Euler-Lagrange Equation to Newton's Laws

Last week I wrote about the Euler-Lagrange Equation, and how it can be obtained from the Principle of Least Action. Today, I'd like to show that this equation is consistent with Newtonian Mechanics.

For reference, the Euler-Lagrange equation for some arbitrary coordinate q is:

\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0

L, the Lagrangian, is the kinetic energy of a system minus the potential.

Let us now consider the case of a free particle of mass m, moving in a potential V(x). The particle's instantaneous speed is given by
v = \frac{dx}{dt}=\dot{x}
. The kinetic energy is the familiar

Applying the Euler-Lagrange Equation, we have:

\frac{\partial L}{\partial x} = -\frac{\partial V(x)}{\partial x} \\
\mathrm{and}~~\frac{\partial L}{\partial \dot{x}} = m\dot{x} \\
\Rightarrow \frac{d}{dt}\frac{\partial L}{\partial \dot{x}} = m\ddot{x}

Putting these together, we have:

-\frac{\partial V(x)}{\partial x} - m\ddot{x} = 0 \\
\Rightarrow m\ddot{x} = -\frac{\partial V}{\partial x}

The right-hand side is the gradient of a potential energy (in 1D). Force can be defined in terms of the gradient of a potential V:

F = -\nabla V

And since
\ddot{x} = \frac{d^2 x}{dt^2}
is acceleration, a, the result of applying the Euler-Lagrange equation to a classical-mechanical Lagrangian is the familiar form of Newton's Second Law:

F = ma

In other words, applying the Euler-Lagrange equation to a suitable Lagrangian provides an equation of motion!

Tuesday, 16 September 2008

Writing Scientific Documents Using LaTeX: Permanent home on CTAN

Following the suggestion of someone who commented on my previous post about my article Writing Scientific Documents Using LaTeX, I submitted the PDF and TeX source to CTAN, the Comprehensive Tex Archive Network. After changing the license to the LPPL, it was accepted into the archive.

As a result, the PDF can now be found permanently at the following location:

The CTAN entry is at

Sunday, 14 September 2008

Lagrangian Mechanics: From the Principle of Least Action to the Euler-Lagrange Equation

A quantity known as the Action, A is defined as

A = \int L \,dt

where L is known as the Lagrangian. The simplest Lagrangian is given as the kinetic energy T of a system minus the potential energy V of the system:

L(x,t) = T(x,t) - V(x,t)

By extremising the Action (minimising it, in this case) we can obtain the Euler-Lagrange Equation, a key equation for Lagrangian and Hamiltonian mechanics. The concept is also used extensively in quantum mechanics and particle physics, particularly when dealing with gauge theories.

Consider an extremised path x(t) from a point x(t0) to a point x(t1), and some excursion from this path, as shown in the figure below. The excursion is given by some small function a(t), and the velocity is changed accordingly:

x(t) \rightarrow x(t) + a(t) \\
v(t) \rightarrow v(t) + \dot{a}(t)

If we take x(t) to be the extremal path from x(t0) to x(t1), with the end-points fixed, and a(t) to be some small but general excursion from that path which must pass through the end-points, we can assert that:

a(t_0) = a(t_1) = 0

The Lagrangian will be changed as a result of these excursions. To first-order in small a(t), the Lagrangian transforms as:

L(x,v) \rightarrow L(x+a, v+\dot{a}) \\
= L(x, v) + a(t)\frac{\partial L}{\partial x} + \dot{a}(t)\frac{\partial L}{\partial v}

The Action therefore transforms according to
A \rightarrow A + \delta A

\delta A = \int_{t_0}^{t_1}\,dt \left( a(t)\frac{\partial L}{\partial x} + \frac{da}{dt}\frac{\partial L}{\partial v} \right)

The second term in the brackets above can be integrated by parts:

\int_{t_0}^{t_1}\,dt \frac{da}{dt}\frac{\partial L}{\partial v} = \left[ a(t)\frac{\partial L}{\partial v} \right]_{t_0}^{t_1} - \int_{t_0}^{t_1}dt\,a(t)\frac{d}{dt}\frac{\partial L}{\partial v}

Since a(t0) = a(t1) = 0, the integrated part (in square brackets) vanishes, leaving the following form for delta A:

\delta A = \int dt\,a(t)\left(\frac{\partial L}{\partial x} - \frac{d}{dt}\frac{\partial L}{\partial v}\right)

For arbitrary a(t), to minimise the action (by having delta A zero), the part in brackets must be zero. This is the Euler-Lagrange Equation, rewritten for a generalised coordinate q (in place of x) and using the time-derivative of q in place of the velocity v, above:

\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0

As I mentioned above, the Euler-Lagrange Equation is critical for the understanding of several aspects of quantum mechanics and gauge theories. Since I plan on making posts about some of these topics in the future, I felt I should begin by explaining Lagrangian mechanics, rather than jumping in at the deep end.

Two-Column Layouts in LaTeX

I was recently asked about using LaTeX to create the two-column layout often seen in scientific journals. I used this layout in some lab reports and project reports.

The first thing to do is to include the multicol package in your preamble:

After that, to place text within a multi-column layout, start the multicols environment (note the s at the end!) and specify the number of columns you'd like:

Finally, end the multi-column layout with:

It's as simple as that! Any sections, subsections, equations etc. between the two will now be automatically formatted into two columns.

There are, however, a few caveats and points to note for formatting considerations.

The first thing to note is that the abstract is often formatted as a single column. To do this, simply place the \begin{multicols}{2} command below your abstract, but above the first \section command.

Secondly, I usually typeset references within the two-column layout. To do this, end the multicols environment after your references section.

Finally, and most importantly, graphics and tables do not always work well with multi-column environments. In order to achieve reasonable results, I used a few workarounds.

If the graphic is large, or has small text, set it normally and it will float outside of the column layout, spanning both columns. For smaller figures, you can create an adjusted figure or table environment which forces LaTeX to set the figure in the column, at the point you typed it in the source text. This is not always what you want, but a reasonable layout can usually be obtained by playing with the exact positioning of the table or graphic commands in the source.

The following, inserted into the preamble, defines two new environments, tablehere and figurehere, which insert tables and figures inline with the column text.
% Figures within a column...
In addition to this, you'll probably need to scale graphics to the column width. This can be achieved with the \resizebox command inside a figure or our new figurehere environment. For example,
\caption{\label{gf-graphs}Graph showing applied RF frequency against magnetic field from the sweep coils, the gradient of the lines reveals information about $g_f$ for each isotope. The left gradient, corresponding to \chem{^{85}{Rb}} is $7.73 \times 10^9 (\pm 1.5 \times 10^8)$. The right gradient, corresponding to \chem{^{87}{Rb}}, is $5.24 \times 10^9 (\pm 6.00 \times 10^7)$ }

\columnwidth is defined by the multicol package to be the width of a text-column!

Saturday, 13 September 2008

Writing Scientific Documents Using LaTeX

Back in 2007, I wrote a short document describing some of the basic LaTeX commands and explaining how to go about creating a document using LaTeX. I did this because I was writing third year lab reports collaboratively with my lab partners (as per the regulations for the third year lab at Warwick) and I wanted us all to be using LaTeX, so the reports would look good and so that editing the sections together would be easier. In order to do this, though, I had to educate my lab partners in the use of LaTeX.

Since then, the document has been read (and requested!) by other physicists, scientists and mathematicians at Warwick and in the wider world. I've updated it several times to add content and to correct minor mistakes. It was not until this week, however, that someone suggested I put it online and make it freely available to anyone who wants it (and can find it).

So, I spent a while tonight polishing it up, adding a few more subsections, and creating what is now the 13-page Fourth Edition. It is available online at