Scientific law

Scientific laws are important ideas that help us understand how nature works. They are based on repeated experiments or observations of natural events. Whether we are studying stars in space, chemicals mixing, or plants growing, scientific laws give us rules to predict what will happen.

These laws are developed from real data and often use mathematics to describe patterns we see in the world. They help scientists explain why things happen and what might happen next. Even when new theories come along, the basic laws usually still hold true, though we might learn more about where they apply.

Unlike guesses or ideas that are still being tested, scientific laws have been checked many times through experiments and watching nature. They are narrower than big scientific theories but still very powerful tools for understanding our world. Scientists use these laws in many fields, from physics to biology, to make sense of everything around us.

Overview

A scientific law is a rule that describes how nature behaves the same way each time under the same conditions. It is based on experiments and observations that show patterns in the world around us. Unlike theories, which try to explain why things happen, laws tell us what happens based on what we have seen.

Many scientific laws can be written as math equations. For example, the law of conservation of energy says that the total energy in the universe stays the same. These laws help us make predictions about what will happen next, but they only work under the conditions where they were observed. Scientists test these laws to see if they still work in new situations, and sometimes they find that a law needs to be updated to fit new discoveries.

Properties

Scientific laws are ideas we learn from doing many experiments and watching things happen many times over. They tell us how some things always behave when certain conditions are right. These laws help science understand the world around us.

Scientific laws have some common features. They work well when used in the right places. They are the same everywhere in the universe. They are often easy to understand, sometimes written in just one equation. They stay the same after they are found. They also usually keep amounts the same and show patterns in space and time. Though they are mostly used in natural sciences, there are also laws in social sciences, such as Zipf's law, which talks about general patterns in how people behave.

Laws as consequences of mathematical symmetries

Main article: Symmetry (physics)

Some scientific laws come from patterns, or symmetries, in nature. For example, rules about how tiny parts of matter behave are linked to the idea that all electrons are the same. Other laws are connected to the idea that space and time are the same everywhere — meaning no moment or place is special.

Important ideas like keeping energy and movement also come from these patterns. Studying these helps scientists learn the basic rules of how nature works.

Laws of physics

Conservation laws

Conservation laws are important rules in physics. They come from the idea that space and time look the same everywhere. One key idea is called Noether's theorem. It says that if something stays the same in a certain way, there is a related quantity that never changes.

For example, conservation of mass tells us that in most everyday situations, mass stays the same during chemical reactions. However, for very small particles, mass can change into energy and vice versa.

Other important conservation laws include conservation of energy, momentum, and angular momentum. These help us understand how things move and change over time.

Laws of classical mechanics

Classical mechanics describes how things move and interact with forces. One big idea is the principle of least action. It says that the path a system takes between two points is the one where a special math rule called the "action" is smallest.

This idea connects to Newton's laws, which many people learn in school. These laws tell us how forces affect the motion of objects. There are also more advanced versions called Lagrange's equations and Hamilton's equations, which are useful for more complex situations.

Laws of gravitation and relativity

Some of the most famous physics laws come from Isaac Newton and Albert Einstein. Newton studied how objects move under the influence of gravity. Einstein developed the theory of relativity, which gives us a deeper understanding of space, time, and gravity.

Einstein's ideas led to important results like the mass–energy equivalence, shown by the famous equation E = mc². This tells us that mass and energy are two forms of the same thing.

Thermodynamics

Thermodynamics studies heat, temperature, and energy. Some key laws include:

Newton's law of cooling: How objects lose heat
Ideal gas law: How gases behave under different conditions
Dalton's law: How pressures of different gases mix together

Electromagnetism

Electromagnetism studies electric and magnetic fields. Maxwell's equations are the main rules that describe how these fields behave and how they are created by electric charges and currents.

Before Maxwell, scientists found simpler rules like Coulomb's law (how electric charges attract or repel) and Ohm's law (how electricity flows through materials).

Photonics

Photonics is the study of light. Some basic rules include:

Law of reflection: Angle of incidence equals angle of reflection
Law of refraction (Snell's law): How light bends when it passes from one material to another

Laws of quantum mechanics

Quantum mechanics studies the smallest particles, where things behave very differently than in our everyday world. Some key ideas include:

The state of a particle is described by a wavefunction
Physical quantities are linked to operators that act on this wavefunction
The wavefunction follows the Schrödinger equation, which predicts how systems change over time

Radiation laws

When we study how atoms and molecules emit or absorb light, we find rules like:

Stefan–Boltzmann law: How much energy a hot object radiates
Planck's law: The spectrum of black-body radiation
Wien's displacement law: The relationship between temperature and peak wavelength of radiation

Physics, conserved quantity	Conserved quantity q	Volume density ρ (of q)	Flux J (of q)	Equation
Hydrodynamics, fluids	m = mass (kg)	ρ = volumic mass density (kg⋅m⁻³)	ρ u, where u = velocity field of fluid (m⋅s⁻¹)	∂ ρ ∂ t = − ∇ ⋅ ( ρ u ) {\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot (\rho \mathbf {u} )}
Electromagnetism, electric charge	q = electric charge (C)	ρ = volumic electric charge density (C⋅m⁻³)	J = electric current density (A⋅m⁻²)	∂ ρ ∂ t = − ∇ ⋅ J {\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {J} }
Thermodynamics, energy	E = energy (J)	u = volumic energy density (J⋅m⁻³)	q = heat flux (W⋅m⁻²)	∂ u ∂ t = − ∇ ⋅ q {\displaystyle {\frac {\partial u}{\partial t}}=-\nabla \cdot \mathbf {q} }
Quantum mechanics, probability	Ψ\|²d³r = probability distribution	Ψ\|² = probability density function (m⁻³), Ψ = wavefunction of quantum system	j = probability current/flux	∂ \| Ψ \| 2 ∂ t = − ∇ ⋅ j {\displaystyle {\frac {\partial \|\Psi \|^{2}}{\partial t}}=-\nabla \cdot \mathbf {j} }

Laws of motion
Principle of least action: S = ∫ t 1 t 2 L d t {\displaystyle {\mathcal {S}}=\int _{t_{1}}^{t_{2}}L\,\mathrm {d} t\,\!}
The Euler–Lagrange equations are: d d t ( ∂ L ∂ q ˙ i ) = ∂ L ∂ q i {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{i}}}\right)={\frac {\partial L}{\partial q_{i}}}} Using the definition of generalized momentum, there is the symmetry: p i = ∂ L ∂ q ˙ i p ˙ i = ∂ L ∂ q i {\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}\quad {\dot {p}}_{i}={\frac {\partial L}{\partial {q}_{i}}}}	Hamilton's equations ∂ p ∂ t = − ∂ H ∂ q {\displaystyle {\dfrac {\partial \mathbf {p} }{\partial t}}=-{\dfrac {\partial H}{\partial \mathbf {q} }}} ∂ q ∂ t = ∂ H ∂ p {\displaystyle {\dfrac {\partial \mathbf {q} }{\partial t}}={\dfrac {\partial H}{\partial \mathbf {p} }}} The Hamiltonian as a function of generalized coordinates and momenta has the general form: H ( q , p , t ) = p ⋅ q ˙ − L {\displaystyle H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {p} \cdot \mathbf {\dot {q}} -L}
	Hamilton–Jacobi equation H ( q , ∂ S ∂ q , t ) = − ∂ S ∂ t {\displaystyle H\left(\mathbf {q} ,{\frac {\partial S}{\partial \mathbf {q} }},t\right)=-{\frac {\partial S}{\partial t}}}
Newton's laws Newton's laws of motion They are low-limit solutions to relativity. Alternative formulations of Newtonian mechanics are Lagrangian and Hamiltonian mechanics. The laws can be summarized by two equations (since the 1st is a special case of the 2nd, zero resultant acceleration): F = d p d t , F i j = − F j i {\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}},\quad \mathbf {F} _{ij}=-\mathbf {F} _{ji}} where p = momentum of body, F_ij = force on body i by body j, F_ji = force on body j by body i. For a dynamical system the two equations (effectively) combine into one: d p i d t = F E + ∑ i ≠ j F i j {\displaystyle {\frac {\mathrm {d} \mathbf {p} _{\mathrm {i} }}{\mathrm {d} t}}=\mathbf {F} _{\text{E}}+\sum _{i\neq j}\mathbf {F} _{ij}} in which F_E = resultant external force (due to any agent not part of system). Body i does not exert a force on itself.

Einstein field equations (EFE): R μ ν + ( Λ − R 2 ) g μ ν = 8 π G c 4 T μ ν {\displaystyle R_{\mu \nu }+\left(\Lambda -{\frac {R}{2}}\right)g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }\,\!} where Λ = cosmological constant, R_μν = Ricci curvature tensor, T_μν = stress–energy tensor, g_μν = metric tensor	Geodesic equation: d 2 x λ d t 2 + Γ μ ν λ d x μ d t d x ν d t = 0 , {\displaystyle {\frac {{\rm {d}}^{2}x^{\lambda }}{{\rm {d}}t^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {{\rm {d}}x^{\mu }}{{\rm {d}}t}}{\frac {{\rm {d}}x^{\nu }}{{\rm {d}}t}}=0\ ,} where Γ is a Christoffel symbol of the second kind, containing the metric.
GEM Equations If g the gravitational field and H the gravitomagnetic field, the solutions in these limits are: ∇ ⋅ g = − 4 π G ρ {\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho \,\!} ∇ ⋅ H = 0 {\displaystyle \nabla \cdot \mathbf {H} =\mathbf {0} \,\!} ∇ × g = − ∂ H ∂ t {\displaystyle \nabla \times \mathbf {g} =-{\frac {\partial \mathbf {H} }{\partial t}}\,\!} ∇ × H = 4 c 2 ( − 4 π G J + ∂ g ∂ t ) {\displaystyle \nabla \times \mathbf {H} ={\frac {4}{c^{2}}}\left(-4\pi G\mathbf {J} +{\frac {\partial \mathbf {g} }{\partial t}}\right)\,\!} where ρ is the mass density and J is the mass current density or mass flux.
In addition there is the gravitomagnetic Lorentz force: F = γ ( v ) m ( g + v × H ) {\displaystyle \mathbf {F} =\gamma (\mathbf {v} )m\left(\mathbf {g} +\mathbf {v} \times \mathbf {H} \right)} where m is the rest mass of the particlce and γ is the Lorentz factor.

Newton's law of universal gravitation: For two point masses: F = G m 1 m 2 \| r \| 2 r ^ {\displaystyle \mathbf {F} ={\frac {Gm_{1}m_{2}}{\left\|\mathbf {r} \right\|^{2}}}\mathbf {\hat {r}} \,\!} For a nonuniform mass distribution of local mass density ρ(r) of body of volume V, this becomes: g = G ∫ V r ρ d V \| r \| 3 {\displaystyle \mathbf {g} =G\int _{V}{\frac {\mathbf {r} \rho \,\mathrm {d} {V}}{\left\|\mathbf {r} \right\|^{3}}}\,\!}	Gauss's law for gravity: An equivalent statement to Newton's law is: ∇ ⋅ g = 4 π G ρ {\displaystyle \nabla \cdot \mathbf {g} =4\pi G\rho \,\!}
Kepler's 1st law: Planets move in an ellipse, with the star at a focus r = ℓ 1 + e cos ⁡ θ {\displaystyle r={\frac {\ell }{1+e\cos \theta }}\,\!} where e = 1 − ( b / a ) 2 {\displaystyle e={\sqrt {1-(b/a)^{2}}}} is the eccentricity of the elliptic orbit, of semi-major axis a and semi-minor axis b, and ℓ is the semi-latus rectum. This equation in itself is nothing physically fundamental; simply the polar equation of an ellipse in which the pole (origin of polar coordinate system) is positioned at a focus of the ellipse, where the orbited star is.
Kepler's 2nd law: equal areas are swept out in equal times (area bounded by two radial distances and the orbital circumference): d A d t = \| L \| 2 m {\displaystyle {\frac {\mathrm {d} A}{\mathrm {d} t}}={\frac {\left\|\mathbf {L} \right\|}{2m}}\,\!} where L is the orbital angular momentum of the particle (i.e. planet) of mass m about the focus of orbit,
Kepler's 3rd law: The square of the orbital time period T is proportional to the cube of the semi-major axis a: T 2 = 4 π 2 G ( m + M ) a 3 {\displaystyle T^{2}={\frac {4\pi ^{2}}{G\left(m+M\right)}}a^{3}\,\!} where M is the mass of the central body (i.e. star).

Laws of thermodynamics
First law of thermodynamics: The change in internal energy dU in a closed system is accounted for entirely by the heat δQ absorbed by the system and the work δW done by the system: d U = δ Q − δ W {\displaystyle \mathrm {d} U=\delta Q-\delta W\,} Second law of thermodynamics: There are many statements of this law, perhaps the simplest is "the entropy of isolated systems never decreases", Δ S ≥ 0 {\displaystyle \Delta S\geq 0} meaning reversible changes have zero entropy change, irreversible process are positive, and impossible process are negative.	Zeroth law of thermodynamics: If two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with one another. T A = T B , T B = T C ⇒ T A = T C {\displaystyle T_{A}=T_{B}\,,T_{B}=T_{C}\Rightarrow T_{A}=T_{C}\,\!} Third law of thermodynamics: As the temperature T of a system approaches absolute zero, the entropy S approaches a minimum value C: as T → 0, S → C.
For homogeneous systems the first and second law can be combined into the Fundamental thermodynamic relation: d U = T d S − P d V + ∑ i μ i d N i {\displaystyle \mathrm {d} U=T\,\mathrm {d} S-P\,\mathrm {d} V+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}\,\!}
Onsager reciprocal relations: sometimes called the fourth law of thermodynamics J u = L uu ∇ ( 1 / T ) − L ur ∇ ( m / T ) ; {\displaystyle \mathbf {J} _{\text{u}}=L_{\text{uu}}\,\nabla (1/T)-L_{\text{ur}}\,\nabla (m/T);} J r = L ru ∇ ( 1 / T ) − L rr ∇ ( m / T ) . {\displaystyle \mathbf {J} _{\text{r}}=L_{\text{ru}}\,\nabla (1/T)-L_{\text{rr}}\,\nabla (m/T).}

Maxwell's equations Gauss's law for electricity

∇ ⋅ E = ρ ε 0 {\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}}

Gauss's law for magnetism

∇ ⋅ B = 0 {\displaystyle \nabla \cdot \mathbf {B} =0}

Faraday's law

∇ × E = − ∂ B ∂ t {\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}

Ampère's circuital law (with Maxwell's correction)

∇ × B = μ 0 J + 1 c 2 ∂ E ∂ t {\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}\ }

Lorentz force law:

F = q ( E + v × B ) {\displaystyle \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)}

Quantum electrodynamics (QED): Maxwell's equations are generally true and consistent with relativity – but they do not predict some observed quantum phenomena (e.g. light propagation as EM waves, rather than photons, see Maxwell's equations for details). They are modified in QED theory.

Quantum mechanics, Quantum field theory Schrödinger equation (general form): Describes the time dependence of a quantum mechanical system. i ℏ d d t \| ψ ⟩ = H ^ \| ψ ⟩ {\displaystyle i\hbar {\frac {d}{dt}}\left\|\psi \right\rangle ={\hat {H}}\left\|\psi \right\rangle } The Hamiltonian (in quantum mechanics) H is a self-adjoint operator acting on the state space, \| ψ ⟩ {\displaystyle \|\psi \rangle } (see Dirac notation) is the instantaneous quantum state vector at time t, position r, i is the unit imaginary number, ħ = h/2π is the reduced Planck constant.	Wave–particle duality Planck–Einstein law: the energy of photons is proportional to the frequency of the light (the constant is the Planck constant, h). E = h ν = ℏ ω {\displaystyle E=h\nu =\hbar \omega } De Broglie wavelength: this laid the foundations of wave–particle duality, and was the key concept in the Schrödinger equation, p = h λ k ^ = ℏ k {\displaystyle \mathbf {p} ={\frac {h}{\lambda }}\mathbf {\hat {k}} =\hbar \mathbf {k} } Heisenberg uncertainty principle: Uncertainty in position multiplied by uncertainty in momentum is at least half of the reduced Planck constant, similarly for time and energy; Δ x Δ p ≥ ℏ 2 , Δ E Δ t ≥ ℏ 2 {\displaystyle \Delta x\,\Delta p\geq {\frac {\hbar }{2}},\,\Delta E\,\Delta t\geq {\frac {\hbar }{2}}} The uncertainty principle can be generalized to any pair of observables – see main article.
Wave mechanics Schrödinger equation (original form): i ℏ ∂ ∂ t ψ = − ℏ 2 2 m ∇ 2 ψ + V ψ {\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi }
Pauli exclusion principle: No two identical fermions can occupy the same quantum state (bosons can). Mathematically, if two particles are interchanged, fermionic wavefunctions are anti-symmetric, while bosonic wavefunctions are symmetric: ψ ( ⋯ r i ⋯ r j ⋯ ) = ( − 1 ) 2 s ψ ( ⋯ r j ⋯ r i ⋯ ) {\displaystyle \psi (\cdots \mathbf {r} _{i}\cdots \mathbf {r} _{j}\cdots )=(-1)^{2s}\psi (\cdots \mathbf {r} _{j}\cdots \mathbf {r} _{i}\cdots )} where r_i is the position of particle i, and s is the spin of the particle. There is no way to keep track of particles physically, labels are only used mathematically to prevent confusion.

Laws of chemistry

Main article: Chemical law

Chemical laws are rules that tell us how chemicals behave. These rules come from many experiments and watching how things change. One important rule is the law of conservation of mass. It tells us that the amount of matter stays the same during a chemical reaction.

Other key rules include Joseph Proust's law of definite composition. This says that chemicals are made of elements in fixed amounts. Dalton's law of multiple proportions explains that elements combine in simple whole-number ratios. These laws help scientists learn how chemicals react and make new substances.

Laws of biology

Main article: Biological rules

Laws of biology help us understand how living things grow, change, and interact with their environment. Some important ideas include the Competitive exclusion principle, which tells us that two species cannot live in the same way and share the same resources forever. Another set of rules, called Mendelian laws, explain how traits are passed from parents to offspring.

There are also ideas like the Hardy–Weinberg principle, which helps scientists study how populations stay the same under certain conditions. Natural Selection is a key idea in understanding how species change over time.

Laws of Earth sciences

Scientific laws help us understand patterns in Earth sciences like geography and geology. In geography, important ideas include Arbia's law of geography, Tobler's first law of geography, and Tobler's second law of geography.

In geology, laws such as Archie's law, Buys Ballot's law, and the Principle of original horizontality help scientists study rocks and landforms. Other key principles include the Law of superposition, which tells us that older rock layers are found below younger ones.

Other fields

Some math ideas and rules are called "laws" because they help us understand science in the real world.

Examples of ideas sometimes called laws include the Titius–Bode law of planetary positions, Zipf's law of language, and Moore's law of technology growth. Other ideas, like the law of unintended consequences, Occam's razor in philosophy, and the Pareto principle in economics, are also sometimes called "laws."

History

People have seen patterns in nature for a very long time, even before we had written history. Early ideas about nature were often mixed with beliefs about gods and spirits. It was in ancient Greece that thinkers started to study nature in a more organized way.

The idea of "laws of nature" began in ancient Rome, where people used the word "law" to describe how things worked. Later, in the 1600s, scientists like Isaac Newton and René Descartes helped change these ideas into clear scientific laws. They used careful experiments and math to learn about nature, focusing on evidence and logic instead of old beliefs.