? Newton
One of the most famous formulas in physics is Newton’s second law, \[ \boldsymbol{F} = m \boldsymbol{a}. \] It is named after Isaac Newton (1643–1727), who in all likelihood was never actually hit on the head by any apples, and states that ‘force equals mass times acceleration’. It shines in its simplicity, but contemporary mathematical physicists would much rather write it as
\[ \frac{\mathrm{d} z}{\mathrm{d}t} = \{H,z\} . \]
In this formula, $z$ can be any quantity related to the system and $H$ is the Hamilton function, which represents the total energy of the system. But what is this strange bracket? And why would any sane person write a simple idea like Newton’s second law in such an obscure way?
Isaac kilogram metre per square second
There are a few possible answers to this last question. I could start singing praise for some abstract geometric beauty, but it also provides an easy explanation of Emmy Noether’s famous theorem on the relation between symmetries and conserved quantities. A conserved quantity, as the name suggests, is something that does not change. The most common example is probably the conservation of energy in physics, but there can be many other conserved quantities.
Noether’s (first) theorem states that a system has a conserved quantity if and only if it possesses a related symmetry. Translation symmetry corresponds to conservation of (linear) momentum, for example, and rotational symmetry to angular momentum.
1️⃣??? Hamilton
William Rowan Hamilton (the musical)
Let’s consider a physical system consisting of one point particle of mass $m$ moving through space. The state of the system is given by its position $\boldsymbol{x} = (x_1,x_2,x_3)$ and momentum $\boldsymbol{p} = (p_1,p_2,p_3)$. In mechanics, momentum is simply the product of mass and velocity, $\boldsymbol{p} = m \boldsymbol{v}$, so you could also say the state is given by position and velocity. But it turns out that using momentum leads to more elegant mathematics. If we know the state of the system at some point in time, then the states at all future (and past) times are determined by Newton’s second law.
Suppose we can write the potential energy of the system as a function $V(\boldsymbol{x})$ of the position. Then the force on the particle is given by minus the gradient of this function, $F(\boldsymbol{x}) = -\boldsymbol{\nabla} V(\boldsymbol{x})$, so Newton’s second law can be written as
\[ \frac{\mathrm{d}^2 \boldsymbol{x}}{\mathrm{d}t^2} = – \frac{1}{m} \boldsymbol{\mathsf{\nabla}} V(\boldsymbol{x}), \]
or, written in components,
\[ \frac{\mathrm{d}^2 x_i}{\mathrm{d}t^2} = – \frac{1}{m} \frac{\partial V(\boldsymbol{x})}{\partial x_i}. \]
Since the kinetic energy of the particle is $|\boldsymbol{p}|^2/(2m)$, the total energy will be given by
\[ H(\boldsymbol{x},\boldsymbol{p}) = \frac{1}{2m} |\boldsymbol{p}|^2 + V(\boldsymbol{x}) . \]
This is called the Hamilton function, after William Rowan Hamilton (1805–1865), who is also famous as the inventor/discoverer of the quaternions. One reason why this function deserves its special name, is that it gives the equations of motion in a very satisfying way. The time derivatives of position and momentum are, up to sign, equal to the partial derivatives of $H$: \[ \frac{\mathrm{d} \boldsymbol{x}}{\mathrm{d}t} = \frac{\partial H}{\partial \boldsymbol{p}} \qquad\text{and}\qquad \frac{\mathrm{d} \boldsymbol{p}}{\mathrm{d}t} = -\frac{\partial H}{\partial \boldsymbol{x}}. \] We call these the ‘equations of motion’. By combining the two equations of motion, you can find an expression for the acceleration $\mathrm{d}^2 \boldsymbol{x}/\mathrm{d}t^2$. You can check that this again gives Newton’s second law, so these equations really do describe the motion of the system.
For our choice of Hamilton function, the first equation of motion will simply be $\mathrm{d} \boldsymbol{x}/\mathrm{d}t = \boldsymbol{p}/m$, but we prefer to write it in the form above. It is not just pleasing to have both equations of motion look so similar, but it also helps us find the time derivative of $H$ itself. By the chain rule we have \[ \frac{\mathrm{d} H}{\mathrm{d}t} = \frac{\partial H}{\partial \boldsymbol{x}} \cdot \frac{\mathrm{d} \boldsymbol{x}}{\mathrm{d}t} + \frac{\partial H}{\partial \boldsymbol{p}} \cdot \frac{\mathrm{d} \boldsymbol{p}}{\mathrm{d}t}, \] where the dot denotes the scalar product between vectors. If you prefer to write out the components, you can expand the first term as \[ \frac{\partial H}{\partial \boldsymbol{x}} \cdot \frac{\mathrm{d} \boldsymbol{x}}{\mathrm{d}t} = \frac{\partial H}{\partial x_1} \frac{\mathrm{d} x_1}{\mathrm{d}t} + \frac{\partial H}{\partial x_2} \frac{\mathrm{d} x_2}{\mathrm{d}t} + \frac{\partial H}{\partial x_3} \frac{\mathrm{d} x_3}{\mathrm{d}t} \] and the second term in a similar way. Using the equations of motion we can see the two terms in the expression for $\mathrm{d} H/\mathrm{d}t$ cancel: \[ \frac{\mathrm{d} H}{\mathrm{d}t} = \frac{\partial H}{\partial \boldsymbol{x}} \cdot \frac{\partial H}{\partial \boldsymbol{p}} – \frac{\partial H}{\partial \boldsymbol{p}}\cdot \frac{\partial H}{\partial \boldsymbol{x}} = 0, \] so we can call $H$ a conserved quantity, because it doesn’t change in time. Hence any system that fits into this Hamiltonian framework automatically satisfies conservation of energy.
? Kepler
Johannes Kepler
When thinking about physics, it is always useful to have a particular system in mind. Consider for example a planet orbiting the sun (where we approximate both by point masses and assume that the sun is fixed at the origin). This is known as the Kepler system, after Johannes Kepler (1571–1630), who figured out the laws of planetary motion long before Newton came up with a theory of gravity.
The Kepler system is governed by the Hamilton function, \[ H(\boldsymbol{x},\boldsymbol{p}) = \frac{1}{2m} |\boldsymbol{p}|^2 – \frac{1}{|\boldsymbol{x}|}. \] Its equations of motion are \[ \frac{\mathrm{d} \boldsymbol{x}}{\mathrm{d}t} = \boldsymbol{p} \qquad\text{and}\qquad \frac{\mathrm{d} \boldsymbol{p}}{\mathrm{d}t} = \frac{-\widehat{\boldsymbol{x}}}{|\boldsymbol{x}|^2}, \] where $\widehat{\boldsymbol{x}}$ denotes the unit vector in the direction of $\boldsymbol{x}$. This is Newton’s second law with the inverse square central force \[ F(\boldsymbol{x}) = -\frac{\widehat{\boldsymbol{x}}}{|\boldsymbol{x}|^2}. \]
If the planet is moving relatively slowly, its motion will consist of repeated orbits, tracing out an ellipse. If it is speeding, then its orbit will either be a parabola or a hyperbola, so it will approach the sun only once before rushing off into interstellar space. Let’s ignore the possibility of such a speedy galactic nomad and assume that we are dealing with an elliptic orbit.
The Kepler system has some notable properties. One of these is that it is rotationally symmetric (see below). This can be seen from the formula for the Hamilton function: it only depends on the lengths of $\boldsymbol{x}$ and $\boldsymbol{p}$, not their directions, so it will not change if we rotate these vectors.
The diagram below shows a planet (blue) orbiting the sun (yellow), tracing an ellipse. It is shown near its closest point to the sun, together with its velocity vector.
Rotational symmetry means that if we instantly rotate both the planet and its velocity vector through some angle around the sun, then the new orbit will be the same ellipse as before, rotated by the same angle.
The conserved quantity corresponding to rotational symmetry is the angular momentum $\boldsymbol{L} = \boldsymbol{x} \times \boldsymbol{p}$ (see below). We can check this by verifying that its time derivative is zero. Using the product rule we find \begin{align*} \frac{\mathrm{d} \boldsymbol{L}}{\mathrm{d}t} &= \frac{\mathrm{d} \boldsymbol{x}}{\mathrm{d}t} \times \boldsymbol{p} + \boldsymbol{x} \times \frac{\mathrm{d} \boldsymbol{p}}{\mathrm{d}t} \\ &= \boldsymbol{p} \times \boldsymbol{p} + \boldsymbol{x} \times \frac{-\widehat{\boldsymbol{x}}}{|\boldsymbol{x}|^2}, \end{align*} which is zero because the cross product of parallel vectors is always zero. Hence $\boldsymbol{L}$ is indeed a conserved quantity.
Angular momentum ($\boldsymbol{L} = \boldsymbol{x} \times \boldsymbol{p}$) is a measure of how much an object is spinning (around a reference point). It increases both when the speed of rotation increases and when the distance to the reference point grows.
Conservation of angular momentum implies that the planet moves slower when it is further from the the sun. A terrestrial example of conservation of angular momentum can be observed when a figure skater performs a pirouette. They will spin more quickly when they move some of their mass closer to the rotation axis by bringing in their arms.
Our observations on the Kepler system illustrate Noether’s theorem, which states that symmetries and conserved quantities are in one-to-one correspondence. In particular, if a system is rotationally symmetric, then it conserves angular momentum. Other systems may have different symmetries. If a system has translation symmetry (eg billiards on an unbounded table) it conserves the total linear momentum. Even conservation of energy fits into this picture. It corresponds to the time translation symmetry: the laws of physics don’t change over time.
To prove Noether’s theorem, we need one more item in our toolbox.
{?,?} Poisson
Earlier we checked that $H$ is always a conserved by calculating its time derivative. What if we want to know the time derivative of some other function $z$ of position and momentum? We can calculate \begin{align*} \frac{\mathrm{d} z}{\mathrm{d}t} &= \frac{\partial z}{\partial \boldsymbol{x}} \cdot \frac{\mathrm{d} \boldsymbol{x}}{\mathrm{d}t} + \frac{\partial z}{\partial \boldsymbol{p}} \cdot \frac{\mathrm{d} \boldsymbol{p}}{\mathrm{d}t} \\ &= \frac{\partial z}{\partial \boldsymbol{x}} \cdot \frac{\partial H}{\partial \boldsymbol{p}} – \frac{\partial z}{\partial \boldsymbol{p}} \cdot \frac{\partial H}{\partial \boldsymbol{x}}. \end{align*}
The expression in the last line is usually written as: \[ \{H,z\} := \frac{\partial H}{\partial \boldsymbol{p}} \cdot \frac{\partial z}{\partial \boldsymbol{x}} – \frac{\partial H}{\partial \boldsymbol{x}} \cdot \frac{\partial z}{\partial \boldsymbol{p}} . \] This curly bracket is called the Poisson bracket, after Siméon Denis Poisson (1781–1840), whose name sounds a lot less posh once you remember that ‘poisson’ is French for ‘fish’.
Siméon Denis Poisson
With this definition, we obtain our brackety formula, \[ \frac{\mathrm{d} z}{\mathrm{d}t} = \{H,z\} , \] as a reformulation of the Hamiltonian equations of motion.
We can also go the other way and recover the Hamiltonian equations of motion from the brackety formula. Since $z$ can be any function of position and momentum, we can choose to set it equal to a component of either position or momentum and find \begin{align*} \frac{\mathrm{d} x_i}{\mathrm{d}t} &= \{H,x_i\} = \frac{\partial H}{\partial p_i} &&\text{and}& \frac{\mathrm{d} p_i}{\mathrm{d}t} &= \{H,p_i\} = -\frac{\partial H}{\partial x_i}. \end{align*}
We have now established that $\mathrm{d} z/\mathrm{d}t = \{H,z\}$ is equivalent to the Hamiltonian equations of motion, but why do we care about Poisson brackets? Well, they have some interesting properties:
- A function $z$ of position and momentum is a conserved quantity of the system with Hamilton function $H$ if and only if $\{H,z\} = 0$. This follows immediately from the brackety formula: the vanishing of the Poisson bracket is equivalent to $\mathrm{d} z/\mathrm{d}t = 0$.
- The Poisson bracket is skew-symmetric, meaning that if we swap around its entries, we get the same result but with a minus sign: \[\{z,u\} = -\{u,z\}\] for any two functions $u$ and $z$ of position and momentum.
There are additional properties of the Poisson bracket which make sure that the identification of a time derivative with a bracket makes sense, but we won’t go into those technicalities here. Instead, let’s spin our attention towards Noether.
? $\Leftrightarrow$?? Noether
Emmy Noether
Emmy Noether (1882–1935) was a mathematician of many talents. Much of her work was in abstract algebra, but she is most famous for her theorem stating that conserved quantities of a system are in one-to-one correspondence with its symmetries. She counts as one of the top mathematicians of the interwar period, a status she managed to achieve in the face of cruel discrimination because of her gender and descent. At the University of Göttingen, Germany, where she spent most of her career, she was refused a paid position, despite strong support from her colleagues David Hilbert and Felix Klein. In 1933, she emigrated to the US to escape the Nazi regime.
The University of Göttingen. Image: Daniel Schwen, CC BY-SA 2.5
Let’s put aside the grim history and step into Noether’s mathematical footsteps to find out what symmetries have to do with conserved quantities. Consider two Hamilton functions $H$ and $I$ and the corresponding dynamical systems, \begin{align} \frac{\mathrm{d} z}{\mathrm{d}t} &= \{H,z\} , \label{H}\tag{$H$}\\ \frac{\mathrm{d} z}{\mathrm{d}t} &= \{I,z\} . \label{I}\tag{$I$} \end{align}
In the Kepler problem, for example, the system labelled \eqref{H} would describe the physical motion of a planet and the one labelled \eqref{I} a rotation around the sun. The Hamilton function for a rotation is a component of the angular momentum vector, so in this example we would take $I$ equal to a component of $\boldsymbol{L} = \boldsymbol{x} \times \boldsymbol{p}$.
Now what does it mean for \eqref{I} to be a symmetry of the system \eqref{H}? It means that the motion of \eqref{I} does not change the equation \eqref{H}. Since the dynamics of a system is fully encoded by its Hamilton function, this is equivalent to saying that the system \eqref{I} does not change the Hamilton function $H$. Hence \[ \text{\eqref{I} is a symmetry of \eqref{H}} \quad \iff \quad \text{$H$ is a conserved quantity of \eqref{I}} . \]
We can use property (A) to express this in terms of a Poisson bracket: \[ \text{\eqref{I} is a symmetry of \eqref{H}} \quad \iff \quad \{I,H\}= 0 . \] Next we use property (B): the Poisson bracket is skew-symmetric, hence \[ \text{\eqref{I} is a symmetry of \eqref{H}} \quad \iff \quad \{H,I\}= 0 . \] Or, using property (A) once again: \[ \text{\eqref{I} is a symmetry of \eqref{H}} \quad \iff \quad \text{$I$ is a conserved quantity of \eqref{H}} . \] This is essentially the statement of Noether’s first theorem: the symmetries of a system are related to its conserved quantities.
There is an important thing that we have swept under the rug in this derivation. Not every possible symmetry is generated by a Hamilton function. Hence the correct formulation of Noether’s theorem is that conserved quantities are in one-to-one correspondence with Hamiltonian symmetries. This issue disappears when, instead of Hamiltonian mechanics, we consider Lagrangian mechanics, which is based on the calculus of variations. Within that framework a natural notion of symmetry leads to a one-to-one correspondence between symmetries and conserved quantities.
In fact, Noether’s original paper dealt with the Lagrangian perspective. It included not just the case of mechanics, but also field theory, which deals with partial differential equations. Her main motivation was to understand conservation of energy in Einstein’s theory of gravity. This is a surprisingly subtle problem, because general relativity has an infinite number of symmetries.
When a system has an infinite number of symmetries, the conserved quantities produced by Noether’s first theorem are trivial in some sense. For example, a function which maps position and momentum to a constant would be a trivial conserved quantity: it does not change in time, but that fact does not tell us anything about the dynamical system. Noether’s second theorem is relevant to these systems with infinitely many symmetries. Roughly speaking, it says that if a system has an infinite number of symmetries, then the equations of motion must have a certain redundancy to it: some of the information contained in one of the equations of motion will also be contained in the others.
? Noether’s legacy
It was known before Noether’s time that conserved quantities are related to symmetries. And while her paper was the first one to make this connection precise, her main breakthrough was the lesser known second theorem. Noether’s insights were warmly welcomed by mathematical physicists of the time, including Albert Einstein himself, and are still a key part of modern physics. The power of Noether’s theorems lies in their generality: they apply to any system with the relevant kind of symmetries, and to prove them you don’t need to know the particulars of the system at hand.
Similarly, Poisson brackets allow us to capture essential features of physics with an equation that takes the same form no matter what system it describes. Instead of having to work out all the forces between interacting objects, all you need to put into this framework is the total energy in the form of the Hamilton function. It’s no wonder that mathematical physicists often prefer $\mathrm{d} z/\mathrm{d}t = \{H,z\}$ over $\boldsymbol{F} = m\boldsymbol{a}$.
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