This article is from Proof Positive, our friendly math newsletter that's delivered to your inbox every Tuesday afternoon. Sign up today and read it first.
Last week, I introduced Emmy Noether, an extraordinary figure in the fields of mathematics and physics. I outlined how Noether’s theorem proves that for every continuous symmetry of a system, there is a conserved quantity.
On supporting science journalism
If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.
But how does that work, exactly? To understand Noether’s reasoning, we need to talk a bit more about the fundamentals of theoretical physics. Today we’re going to dive deep into some concepts coming from calculus and physics.
When solving certain problems in high school physics classes—for example, determining the orbit of a planet around a star or the trajectory of a ball—we use force equations. (In the first case, for instance, the gravitational force between two bodies is set as equal to the mass times the acceleration of the planet.) This approach yields an equation of motion, which tells you when and where the object in question will be.
In college, however, physics students learn a different approach to solving such problems, based on energy rather than force. Of course, the approaches are equivalent, and they lead to the same results. But the energy approach proves more practical in many situations—and it’s also easier to generalize. This is why it’s used to prove Noether’s theorem.
The energy method is somewhat more abstract than the force equilibrium approach. Moreover, prior knowledge in the field of calculus is required to understand the individual calculation steps that ultimately lead to the equations of motion. The fundamental idea, however, is simple: The principle of least action states that nature is lazy. When a system transitions from one state (for example, a ball flies through the air) to another (a ball lands on the ground), it takes the path of least effort. This effort is known in physics as action. This insight stems from Fermat’s principle, according to which light rays choose the shortest path to a destination, and other systems appear to follow this principle as well. By assuming this principle and applying a little calculation, one can derive the equations of motion, such as the orbits of the planets around the sun.
Introducing the Lagrangian: A Fundamental Function in Physics
To fully characterize a dynamic system, such as that of a thrown ball, one must know its velocity and position at every instant. Keeping track of all these quantities simultaneously can be confusing—after all, they’re described by a six-dimensional vector (three spatial coordinates for position and three for velocity) that assumes different values at any given time. Therefore a scalar quantity (meaning a variable number) is used to encode this information: the so-called Lagrangian.
When its value changes, it symbolizes a movement within the system. The action (or the “effort” required to move a system from one state to another within a specific time) is closely related to the Lagrangian: it is given by the sum of the Lagrangians at each individual instant. In other words, the action assigns a numerical value to each possible trajectory of a system. And, as physicists have shown, the correct motion of a physical system corresponds to the principle of least action or the shortest path.
The principle of least action indicates which trajectory is the correct one. In this figure, q represents the generalized coordinate.
Maschen/Wikimedia Commons (CC0 1.0)
In calculus, students learn to find the highest and lowest points of a function within a given interval or across its domain. These highest and lowest points are known collectively as the extrema. You find them through curve sketching: you differentiate and set the result equal to zero. In this case, however, the action isn’t a simple function but a specific type of function called a functional—yes, those two little letters make a difference. The action integrates the Lagrangian over time, and the Lagrangian itself consists of time-dependent functions, such as the velocity and position of the object in question. Therefore, you must proceed more carefully to determine the extrema of the action.
One way to do this is through the calculus of variations. The principle is similar to that used for ordinary functions: you tweak the possible trajectories that the system can follow and find out where the action changes the least. In this way, you obtain equations that correspond to the equations of motion of the system being described—for example, the orbits of planets.
Noether’s Trick: Every Symmetry Brings a Conserved Quantity
After this foray into theoretical physics and calculus, you’re probably wondering what all of this has to do with Noether’s theorem. In fact, the Lagrangian allows us to determine the continuous symmetries of a given system.
If we apply a symmetry transformation (such as a shift in the x coordinates) to the variables of the Lagrangian L without changing anything, then we have found a symmetry. For example, if we want to describe two spheres moving toward each other along the x axis and colliding, the Lagrangian depends solely on their distance: s1 − s2 = q, where q is the generalized coordinate, s1 the position of sphere one and s2 the position of sphere 2. If we shift the positions of both spheres by the same distance α, the Lagrangian remains the same because (s1 + α) − (s2 + α) = q. Therefore the system is symmetric with respect to translation.
Noether investigated how any Lagrangian changes when a variable (such as time or position) is varied by a parameter α. This change in Lis best analyzed by taking the derivative of the Lagrangian with respect to α. If the change as a result of α represents a symmetry transformation, L will not change—consequently, the derivative is zero.
By utilizing some properties of the Lagrangian and performing a few transformations, the derivative of L with respect to α, or (∂L/∂α), becomes the derivative of a new expression Q with respect to time (dQ/dt). And this is also zero—that is, the new expression Q does not change over time and is therefore a conserved quantity! Thus, Noether’s theorem provides a conserved quantity for every symmetry and even gives a formula for calculating this quantity.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.
