Top Ten Most Beautiful Equations in Physics

Updated on April 26, 2018
Sam Brind profile image

Sam Brind holds a master's in physics with theoretical physics (MPhys) from the University of Manchester.

Physics can be described simply as the study of our universe and an equation as a piece of maths relating physical quantities e.g. mass, energy, temperature. The rules of our universe, technically speaking physical laws, are almost all written down in the form of equations. The concept of relating the artistic (and subjective) idea of beauty to these mathematical statements may at first seem strange and unnecessary. However, to many physicists the concept is not merely a side effect of their theories but it is intrinsic to a good theory.

Paul Dirac - "A physical law must possess mathematical beauty."

What makes an equation beautiful? This moves away from the empirical fact of whether the equation works, whether it predicts experimental data, to something more personal and subjective. In my opinion there are three criteria to consider: aesthetics, simplicity and significance. The aesthetics is simply whether it looks good when written down. Simplicity is a lack of complicated structure in the equation. The significance of the equation is more of a measure of history, both what it solved and what it lead to in future scientific advancements. Below are my top ten equations (not in any particular order).

Einstein's energy-mass equivalence equation.
Einstein's energy-mass equivalence equation.

1. Einstein's Energy-Mass Equivalence

A consequence of Albert Einstein's theory of special relativity and the most famous equation in physics. This equation states that mass (m) and energy (E) are equivalent. The relation is very simple, only involving multiplication of mass by a very large number (c is the speed of light). Specifically, this equation first showed that even mass not in motion has an intrinsic "rest" energy. It has since been used in nuclear and particle physics.

The biggest impact of this equation and perhaps the event that secured its legacy was the development and subsequent use of atomic bombs at the end of WW2. These bombs horrifically demonstrated the extraction of a huge amount of energy from a tiny amount of mass.

Newton's second law.
Newton's second law.

2. Newton's Second Law

One of the oldest physics equations, formulated by Sir Isaac Newton in his famous book Principia in 1687. It is the cornerstone of classical mechanics, which allows the motion of objects subjected to forces to be calculated. Force (F) is equivalent to mass (m) multiplied by the acceleration of the mass (a). The underline notation indicates a vector, which has both a direction and a magnitude. This equation is now the first to be learnt by every physics student due to it only requiring basic mathematical knowledge but at the same being very versatile. It has been applied to a huge number of problems from the motion of cars all the way up to the orbits of the planets around our sun. It was only usurped by the theory of quantum mechanics in the early 1900s.

The Shrödinger equations.
The Shrödinger equations.

3 .The Schrödinger Equation(s)

Quantum mechanics was the biggest shake up in physics since Newton formulated the foundations of classical mechanics and the Schrödinger equation, formulated by Erwin Schrödinger in 1926, is the quantum analogue of Newton's 2nd law. The equation incorporates two key concepts of quantum mechanics: the wave function (ψ) and operators (anything with a hat over it) which operate on a wave function to extract information. The operator used here is the hamiltonian (H) and extracts the energy. There are two versions of this equation, depending on whether the wavefunction varies in time and space or just in space. Although quantum mechanics is a complicated topic, these equations are elegant enough to be appreciated without any knowledge. They are also a postulate of quantum mechanics, a theory which is one of the pillars of our modern electronic technology.

Maxwell's laws.
Maxwell's laws.

4. Maxwell's Laws

Maxwell's laws are a collection of four equations that were brought together and used to formulate a unified description of electricity and magnetism by scottish physicist James Clerk Maxwell in 1862. They were since refined, using calculus, into the most elegant form shown below or technically speaking in "differential form". The first equation relates the flow of electric field (E) to the charge density (ρ). The second law states that magnetic fields (B) have no monopoles. Whereas electric fields can have a source of positive or negative charge, such as an electron, magnetic fields always come with a north and south pole and hence there is no net "source". The last two equations show that a changing magnetic field creates an electric field and vice versa. Maxwell combined these equations into wave equations for electric and magnetic fields, with their propagation speed being equal to a constant value that was the same as the measured speed of light. This lead him to conclude that light is actually an electromagnetic wave. It would also inspire Einstein's theory of special relativity, which is based upon the speed of light being a constant. These consequences would be huge enough without the obvious fact that these equations led to understanding of electricity which laid the foundations for the digital revolution and the computer you're using to read this article.

Second law of thermodynamics.
Second law of thermodynamics.

5. Second Law of Thermodynamics

Not an equality but an inequality, stating that the entropy (S) of our universe always increases. Entropy can be interpreted as a measure of disorder, hence the law can be stated as the disorder of the universe increasing. An alternative view of the law is heat only flows from hot to cold objects. As well as practical uses during the industrial revolution, when designing heat and steam engines, this law also has profound consequences for our universe. It allows the definition of an arrow of time. Imagine being shown a video clip of a mug being dropped and breaking. The initial state is a mug (ordered) and the final state is a collection of pieces (disordered). You would clearly be able to tell whether the video was being played forward of backward from the flow of entropy. This would also lead on to the big bang theory, with the universe getting hotter as you go into the past but also more ordered, leading towards the most ordered state at zeroth time; a singular point.

The wave equation.
The wave equation.

6. The Wave Equation

The wave equation is a 2nd order partial differentiation equation that describes the propagation of waves. It relates the change of propagation of the wave in time to the change of propagation in space and a factor of the wave speed (v) squared. This equation isn't as groundbreaking as others on this list but it is elegant and has been applied to things such as sound waves (instruments etc.), waves in fluids, light waves, quantum mechanics and general relativity.

Einstein's field equations.
Einstein's field equations.

7. The Einstein Field Equations

Only fitting that that the greatest physicist has a second equation in this list and one arguably more important than his first. It gives the fundamental reason for gravity, mass curving spacetime (a four dimensional combination of 3D space and time).

The earth bending nearby spacetime, hence objects such as the moon would be attracted towards it.
The earth bending nearby spacetime, hence objects such as the moon would be attracted towards it. | Source

The equation actually hides 10 partial differential equations by using tensor notation (everything with indices is a tensor). The left hand side contains the Einstein tensor (G) which tells you the curvature of spacetime and this is related to the stress-energy tensor (T) which tells you the distribution of energy in the universe on the right hand side. A cosmological constant term (Λ) can be included in the equation to attribute for our expanding universe, although physicists are unsure of what is actually causing this expansion. This theory completely changed our understanding of the universe and has since been experimentally validated, a beautiful example being the bending of light around stars or planets.

Heisenberg's uncertainty principle.
Heisenberg's uncertainty principle.

8. Heisenberg's Uncertainty Principle

Introduced by Werner Heisenberg in 1927, the uncertainty principle is a limit on quantum mechanics. It states that the more certain you are about a particle's momentum (P) the less certain you are about the particle's position (x) ie. momentum and position can never both be known exactly. A common misconception is that this effect is due to a problem with the measuring procedure. This is incorrect, it is a limit on accuracy fundamental to quantum mechanics. The right hand side involves Plank's constant (h) which is equal to a tiny value (a decimal with 33 zeros), which is why this effect isn't observed in our everyday, "classical", experience.

Quantisation of radiation.
Quantisation of radiation.

9. Quantisation of Radiation

A law initially introduced by Max Plank to solve a problem with black body radiation (specifically to do with efficient lightbulbs) that led to quantum theory. This law states that electromagnetic energy can only be emitted/absorbed in specific (quantised) amounts. This is now known to be due to electromagnetic radiation not being a continuous wave but actually many photons, "packets of light". The energy of a photon (E) is proportional to the frequency (f). At the time it was only a mathematical trick used by Plank to solve a frustrating problem and he both considered it unphysical and struggled with the implications. However, Einstein would link this concept to photons and this equation is now remembered as the birth of quantum theory.

Boltzmann's entropy equation.
Boltzmann's entropy equation.

10. Boltzmann Entropy

A key equation for statistical mechanics formulated by Ludwig Boltzmann. It relates the entropy of a macrostate (S) to the number of microstates corresponding to that macrostate (W). A microstate describes a system by specifying the properties of each particle, this involves microscopic properties such as particle momentum and particle position. A macrostate specifies collective properties of a group of particles, such as temperature, volume and pressure. The key thing here is that multiple different microstates can correspond to the same macrostate. Therefore, a simpler statement would be that the entropy is related to the arrangement of particles within the system (or the 'probability of the macrostate'). This equation can then be used to derive thermodynamic equations such as the ideal gas law.

Ludwig Boltzmann's grave in Vienna, with his equation carved above his bust.
Ludwig Boltzmann's grave in Vienna, with his equation carved above his bust. | Source

Bonus: Feynman Diagrams

Feynman diagrams are very simple pictorial representations of particle interactions. They can be appreciated superficially as a pretty picture of particle physics but do not underestimate them. Theoretical physicists use these diagrams as a key tool in complex calculations. There are rules to drawing a Feynman diagram, a particular one to note is that any particle traveling backwards in time is an antiparticle (corresponding to a standard particle but with the opposite of its electrical charge). Feynman did win a noble prize for quantum electrodynamics and did lots of great work but perhaps his most well known legacy are his diagrams that every physics students learns to draw and study. Feynman even painted these diagrams all over his van.

An example of a Feynman diagram, an electron and a positron annihilate into a photon which then  produces a quark and an antiquark (which then radiates a gluon).
An example of a Feynman diagram, an electron and a positron annihilate into a photon which then produces a quark and an antiquark (which then radiates a gluon). | Source

What is your favourite physics equation?

See results

Questions & Answers

  • Where have we applied the Maxwell's equations?

    Maxwell's equations form the basis of our understanding of electricity and magnetism and are therefore invoked by a huge range of modern technologies. For example: electric motors, power generation, radio communication, microwaves, lasers and all modern electronics.

  • What are the applications of relativity today?

    Relativistic effects only become significant at very large energies and hence they don't have an impact on everyday life. However, taking relativistic effects into account is essential for studies on the frontiers of scientific understanding, such as cosmology and particle physics.

  • What's an example of an energy-mass equation?

    As mentioned in the article, nuclear weapons starkly demonstrate what the energy-mass equivalence equation is telling us, a small amount of mass contains the potential to produce a huge amount of energy. The "Little Boy" bomb dropped on Hiroshima contained 64 kilograms of Uranium-235 fuel. Due to an inefficient design less than a kilogram actually underwent nuclear fission, this still released around 63 terajoules of energy (equivalent to detonating 15,000 tonnes of TNT).

  • Is there any equation for electromagnetic levitation?

    An extremely idealized equation for electromagnetic levitation would be to balance the Lorentz force experienced by an object within electromagnetic fields against its gravitational force, this would give 'q(E + vB) = mg'. In the real world, things are more complex but there are real examples of this technology, for example, maglev trains utilize magnets to levitate trains above the track.

  • Would you consider the Standard Model of particle physics one of the greatest equations ever?

    The standard model of particle physics is certainly on a par in significance with any of the equations mentioned in this article, forming the basis of all study in the exciting field of particle physics. However, when the theory is condensed into a single equation the result is long and complex, in contrast with the equations listed here (that summarise significant theories into surprisingly elegant equations).

© 2016 Sam Brind


    0 of 8192 characters used
    Post Comment
    • profile image


      4 months ago

      So useful

    • profile image

      musa stone 

      14 months ago

      very interested equations I will choose it use it love


    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

    Show Details
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)
    ClickscoThis is a data management platform studying reader behavior (Privacy Policy)