Conservation laws — energy, momentum, angular momentum

Intuition [Beginner]

Newton's laws tell you how force produces acceleration. Conservation laws tell you what does not change — quantities that remain constant throughout the motion regardless of the details. Three such quantities pervade all of physics: energy, momentum, and angular momentum. Each one is a bookkeeping tool that lets you solve problems without tracking every detail of the force.

Momentum is mass times velocity. When two objects collide, the total momentum before the collision equals the total momentum after. A small, fast object can have the same momentum as a large, slow one. A 70 kg person running at 5 m/s has momentum 350 kg m/s. A 1400 kg car moving at 0.25 m/s has the same momentum. In a collision between them, the momentum transfers from one to the other, but the total never changes.

Energy (specifically, mechanical energy) comes in two forms: kinetic (the energy of motion) and potential (stored energy due to position). A ball at the top of a hill has potential energy. As it rolls down, potential energy converts to kinetic. At the bottom, the ball is moving fast and has no potential energy left — all of it became kinetic. The total stays constant at every point along the slope.

Angular momentum is the rotational analogue of momentum. A spinning figure skater pulls her arms in and spins faster. Her mass does not change, her speed increases, but the angular momentum — which depends on both the distribution of mass and the spin rate — stays fixed. The total angular momentum of an isolated system never changes.

Why these three? Because each one corresponds to a symmetry of space and time. Momentum conservation comes from the fact that the laws of physics are the same everywhere in space (translation symmetry). Energy conservation comes from the fact that the laws are the same at all times (time-translation symmetry). Angular momentum conservation comes from the fact that the laws are the same no matter which direction you face (rotation symmetry). This deep connection between symmetries and conservation laws is Noether's theorem, developed in 09.03.01 pending.

Visual [Beginner]

Figure: A ball rolls without friction down a curved hill, starting from rest at height h. At the top (left), the ball is stationary — a tall blue bar shows potential energy, and the kinetic energy bar (green) is at zero. Halfway down the slope, both bars are equal height. At the bottom (right), the potential energy bar has shrunk to zero and the kinetic energy bar is at its maximum — equal to the original potential energy. A dashed horizontal line across all three positions marks the constant total energy.

Worked example [Beginner]

A 60 g tennis ball is thrown horizontally at 25 m/s and is struck by a racket. The ball returns at 30 m/s in the opposite direction. The racket contact lasts 0.004 seconds.

Momentum check. Before the hit, the ball's momentum is 0.060 kg times 25 m/s = 1.5 kg m/s (let us call this the positive direction). After the hit, the ball's momentum is 0.060 times (-30) = -1.8 kg m/s. The change in the ball's momentum is -1.8 minus 1.5 = -3.3 kg m/s.

By conservation of total momentum, the racket (and the player holding it) gains +3.3 kg m/s of momentum. If the player plus racket have a combined mass of 75 kg, the recoil velocity is 3.3 divided by 75 = 0.044 m/s — small but real.

Force. The average force on the ball is the momentum change divided by the contact time: 3.3 divided by 0.004 = 825 N. The ball experiences a force comparable to the weight of an 84 kg person, but concentrated into a region the size of the racket face and lasting only 4 milliseconds.

Energy check. Kinetic energy before: half times 0.060 times 625 = 18.75 J. After: half times 0.060 times 900 = 27 J. The kinetic energy increased from 18.75 J to 27 J. This is not a violation of energy conservation — the extra 8.25 J came from the player's muscles doing work on the racket during the swing. Mechanical energy of the ball alone is not conserved because an external force (the racket) did work on it.

What this tells us: momentum conservation holds for the combined ball-plus-racket system even though individual momenta change. Energy conservation holds when you account for all energy inputs (the player's work), not just the ball's kinetic energy.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let a system of $N$ particles have masses $m_i$, positions ${\mathbf{r}}_i$, and velocities ${\mathbf{v}}_i$ for $i=1,\dots ,N$.

Linear momentum. The total linear momentum is

$$\mathbf{P}:=\sum _{i=1}^N{m}_i{\mathbf{v}}_i.$$

Theorem (Conservation of linear momentum). If the net external force on the system is zero, then $d\mathbf{P}/dt=0$.

Proof: $\dot{\mathbf{P}}={\sum }_i{m}_i{\dot{\mathbf{v}}}_i={\sum }_i{\mathbf{F}}_i={\sum }_i{\mathbf{F}}_i^{\text{ext}}+{\sum }_i{\sum }_{j\ne i}{\mathbf{F}}_{ij}$. The internal forces cancel in pairs by Newton's third law (${\mathbf{F}}_{ij}+{\mathbf{F}}_{ji}=0$). With no external forces, $\dot{\mathbf{P}}=0$.

Kinetic energy. The total kinetic energy is

$$K:=\sum _{i=1}^N\frac{1}{2}{m}_i|{\mathbf{v}}_i{|}^2.$$

Work-energy theorem. For a single particle acted on by a force $\mathbf{F}$, the change in kinetic energy equals the work done by the force:

$$\Delta K={\int }_{{\mathbf{r}}_1}^{{\mathbf{r}}_2}\mathbf{F}\cdot d\mathbf{r}.$$

Proof: $\frac{dK}{dt}=\frac{d}{dt}(\frac{1}{2}m|\mathbf{v}{|}^2)=m\mathbf{v}\cdot \dot{\mathbf{v}}=\mathbf{v}\cdot (m\dot{\mathbf{v}})=\mathbf{v}\cdot \mathbf{F}=\mathbf{F}\cdot \frac{d\mathbf{r}}{dt}$. Integrating: $K_2-K_1={\int }_{t_1}^{t_2}\mathbf{F}\cdot \mathbf{v}dt={\int }_{{\mathbf{r}}_1}^{{\mathbf{r}}_2}\mathbf{F}\cdot d\mathbf{r}$.

Potential energy and conservative forces. A force $\mathbf{F}$ is conservative if the work done depends only on the endpoints, not the path. Equivalently, $\nabla \times \mathbf{F}=0$ (on a simply connected domain). Then there exists a potential energy function $V(\mathbf{r})$ with

$$\mathbf{F}=-\nabla V.$$

Conservation of mechanical energy. If all forces are conservative, the total mechanical energy $E=K+V$ is constant:

$$K+V=\text{constant}.$$

Proof: $\frac{dE}{dt}=\frac{dK}{dt}+\frac{dV}{dt}=\mathbf{F}\cdot \mathbf{v}+\nabla V\cdot \mathbf{v}=\mathbf{F}\cdot \mathbf{v}+(-\mathbf{F})\cdot \mathbf{v}=0$.

Angular momentum. The total angular momentum about the origin is

$$\mathbf{L}:=\sum _{i=1}^N{\mathbf{r}}_i\times m_i{\mathbf{v}}_i.$$

Theorem (Conservation of angular momentum). If the net external torque on the system is zero, then $d\mathbf{L}/dt=0$. For a single particle under a central force $\mathbf{F}=f(r)\hat{\mathbf{r}}$, $\mathbf{L}$ is conserved.

Proof: $\dot{\mathbf{L}}={\sum }_i({\dot{\mathbf{r}}}_i\times m_i{\mathbf{v}}_i+{\mathbf{r}}_i\times m_i{\dot{\mathbf{v}}}_i)={\sum }_i{\mathbf{r}}_i\times {\mathbf{F}}_i$. For internal forces satisfying the strong third law: ${\sum }_i{\sum }_{j\ne i}{\mathbf{r}}_i\times {\mathbf{F}}_{ij}={\sum }_{i<j}({\mathbf{r}}_i-{\mathbf{r}}_j)\times {\mathbf{F}}_{ij}=0$ since ${\mathbf{F}}_{ij}$ is parallel to ${\mathbf{r}}_i-{\mathbf{r}}_j$. With no external torque, $\dot{\mathbf{L}}=0$.

Counterexamples to common slips

• "Energy is always conserved in collisions." Momentum is always conserved for isolated systems; kinetic energy is conserved only in elastic collisions. In inelastic collisions, kinetic energy is converted to thermal energy and deformation. The total energy (kinetic + thermal + all other forms) is always conserved, but the mechanical energy need not be.

• Conflating conservation with constancy. A quantity is "conserved" if it is constant for an isolated system. A ball's kinetic energy changes as it falls — but the total energy $K+V$ (ball plus Earth) is conserved. The individual kinetic energy is not conserved; the total energy of the isolated system is.

• Angular momentum conservation requires zero net torque, not zero net force. A particle in a uniform gravitational field experiences a net force (its weight) but zero torque about the centre of mass. Angular momentum about the centre of mass is conserved even though the particle accelerates.

Key theorem with proof [Intermediate+]

Theorem (Centre-of-mass decomposition). For a system of $N$ particles with total mass $M={\sum }_i{m}_i$, define the centre-of-mass position $\mathbf{R}=\frac{1}{M}{\sum }_i{m}_i{\mathbf{r}}_i$ and velocity $\mathbf{V}=\dot{\mathbf{R}}$. Then:

(i) $M\dot{\mathbf{V}}={\mathbf{F}}^{\text{ext}}$ (the centre of mass moves as a single particle of mass $M$ under the total external force).

(ii) $\mathbf{P}=M\mathbf{V}$ (total momentum equals total mass times centre-of-mass velocity).

(iii) $\mathbf{L}=\mathbf{R}\times M\mathbf{V}+{\sum }_i{\mathbf{r}}_i^{\prime }\times m_i{\mathbf{v}}_i^{\prime }$ (total angular momentum decomposes into orbital part about the origin and spin part about the centre of mass), where primed quantities are relative to the centre of mass.

Proof. (i) $M\ddot{\mathbf{R}}={\sum }_i{m}_i{\ddot{\mathbf{r}}}_i={\sum }_i{\mathbf{F}}_i={\sum }_i{\mathbf{F}}_i^{\text{ext}}+{\sum }_i{\sum }_{j\ne i}{\mathbf{F}}_{ij}$. Internal forces cancel in pairs, leaving $M\ddot{\mathbf{R}}={\mathbf{F}}^{\text{ext}}$.

(ii) $\mathbf{P}={\sum }_i{m}_i{\mathbf{v}}_i={\sum }_i{m}_i{\dot{\mathbf{r}}}_i=M\dot{\mathbf{R}}=M\mathbf{V}$.

(iii) Write ${\mathbf{r}}_i=\mathbf{R}+{\mathbf{r}}_i^{\prime }$, ${\mathbf{v}}_i=\mathbf{V}+{\mathbf{v}}_i^{\prime }$, where ${\sum }_i{m}_i{\mathbf{r}}_i^{\prime }=0$ and ${\sum }_i{m}_i{\mathbf{v}}_i^{\prime }=0$ by definition of $\mathbf{R}$. Then:

$$\mathbf{L}=\sum _i(\mathbf{R}+{\mathbf{r}}_i^{\prime })\times m_i(\mathbf{V}+{\mathbf{v}}_i^{\prime })=\mathbf{R}\times M\mathbf{V}+\sum _i{\mathbf{r}}_i^{\prime }\times m_i{\mathbf{v}}_i^{\prime },$$

where the cross terms vanish: $\mathbf{R}\times {\sum }_i{m}_i{\mathbf{v}}_i^{\prime }=0$ and ${\sum }_i{m}_i{\mathbf{r}}_i^{\prime }\times \mathbf{V}=0$. ∎

Bridge. The centre-of-mass decomposition is the foundational reason that the translational motion of any system — regardless of internal complexity — reduces to single-particle dynamics: $\mathbf{R}(t)$ obeys $M\ddot{\mathbf{R}}={\mathbf{F}}^{\mathrm{ext}}$, which is exactly Newton's second law for a point mass $M$. The decomposition (iii) identifies the orbital angular momentum with the motion of the centre of mass and the spin angular momentum with the internal dynamics, and this is exactly the split that persists in rigid-body mechanics and in the Euler equations for torque-free rotation. The bridge is toward the Hamiltonian reformulation in 09.04.02 pending, where the centre-of-mass coordinates $(\mathbf{R},\mathbf{P})$ become canonical conjugate pairs, and toward Noether's theorem 09.03.01 pending, which generalises the momentum-conservation result (ii) to every continuous symmetry of the Lagrangian.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib has the mathematical machinery for conservation laws in ODE systems: first integrals 02.12.12, the derivative of kinetic energy $\frac{1}{2}m|\mathbf{v}{|}^2$, and gradient calculus for potential functions. It does not formalise the physical concepts of momentum conservation, energy conservation, or angular momentum conservation as named theorems tied to Newton's laws. The gap is a physics-layer formalisation building on existing analysis. lean_status: none.

Conservation laws as first integrals of Newtonian mechanics [Master]

The intermediate treatment establishes three conservation laws as consequences of Newton's laws. The deeper perspective is that each conservation law is a first integral 02.12.12 of the equations of motion — a function $f(\mathbf{r},\mathbf{v})$ that is constant along every trajectory.

First integrals and integrability. A system with $n$ degrees of freedom has a $2n$-dimensional phase space. Each independent first integral restricts trajectories to a $(2n-1)$-dimensional submanifold. Finding $k$ independent first integrals reduces the effective dimension to $2n-k$. If $k=2n-1$, the trajectory is completely determined (up to time parametrisation) by the intersection of the level sets — the system is integrable.

For the Kepler problem ($n=3$, so $2n=6$): the conserved quantities are energy $E$ (1 scalar), angular momentum $\mathbf{L}$ (3 components, but $|\mathbf{L}|$ and two angles), and the Laplace-Runge-Lenz vector $\mathbf{A}$ (3 components, with constraints $\mathbf{L}\cdot \mathbf{A}=0$ and $|\mathbf{A}|$ fixed by $E$ and $|\mathbf{L}|$). The total number of independent conserved quantities is 5, making the system superintegrable (more first integrals than degrees of freedom), which is why Kepler orbits are closed ellipses.

The work-energy theorem as a first-integral statement. For a conservative system, $E=\frac{1}{2}m|\mathbf{v}{|}^2+V(\mathbf{r})$ is a first integral: $\dot{E}=0$ along trajectories. This means trajectories lie on the level set $E(\mathbf{r},\mathbf{v})=E_0$, a $(2n-1)$-dimensional hypersurface in phase space. For $n=1$ (one-dimensional motion), this single equation completely determines the trajectory — you can solve for $v(r)$ or $r(t)$ from $E$ alone, without solving Newton's equation directly.

Angular momentum and the reduction to the orbital plane. The conservation of $\mathbf{L}$ restricts motion to a fixed plane (the plane perpendicular to $\mathbf{L}$). This is a geometric consequence: $\mathbf{L}=\mathbf{r}\times m\mathbf{v}$ is constant, so $\mathbf{r}$ and $\mathbf{v}$ always lie in the same plane. For the two-body problem, this reduces the effective degrees of freedom from $2n=6$ to $2n=4$ (motion in a plane), and further to $2n=2$ after using energy conservation and the constancy of $|\mathbf{L}|$ — reducing to a one-dimensional radial problem in the effective potential $V_{\text{eff}}(r)$.

Poisson brackets and the algebraic detection of first integrals. In the Hamiltonian formulation 09.04.02 pending, a function $f$ on phase space is a first integral if and only if its Poisson bracket with the Hamiltonian vanishes: $\{f,H\}=0$. The Poisson bracket is defined by

$$\{f,g\}:=\sum _{i=1}^n\left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right).$$

The three conserved quantities — total energy $H$, total momentum $\mathbf{P}$, and total angular momentum $\mathbf{L}$ — satisfy $\{H,H\}=0$ (identically), $\{P_i,H\}=0$ (when the Hamiltonian is translation-invariant), and $\{L_i,H\}=0$ (when the Hamiltonian is rotation-invariant). Moreover, the conserved quantities themselves close under the Poisson bracket: $\{P_i,P_j\}=0$, $\{L_i,L_j\}={ϵ}_{ijk}{L}_k$, and $\{P_i,L_j\}={ϵ}_{ijk}{P}_k$. This is the Lie algebra of the Galilean group, and the correspondence between Poisson brackets of conserved quantities and the Lie brackets of the symmetry generators is the content of Noether's theorem at the Hamiltonian level 09.03.01 pending.

The Liouville-Arnold theorem and complete integrability. A Hamiltonian system with $n$ degrees of freedom is completely integrable in the sense of Liouville if it admits $n$ independent first integrals $f_1=H,f_2,\dots ,f_n$ that are pairwise in involution: $\{f_i,f_j\}=0$ for all $i,j$. The Liouville-Arnold theorem 09.06.01 pending then states that the level sets $f_1=c_1,\dots ,f_n=c_n$ are invariant tori, and the motion on each torus is quasi-periodic with $n$ frequencies. The Kepler problem ($n=3$) has three commuting integrals ($E$, $|\mathbf{L}|$, $L_z$) and is therefore completely integrable; its invariant tori are the ellipses parametrised by the Keplerian orbital elements. The superintegrability (two additional integrals from the Laplace-Runge-Lenz vector) collapses the tori to closed curves, which is why all bounded Kepler orbits close.

Proposition (Independence of Newtonian first integrals). For an $N$-particle system with a conservative, translation-invariant, rotation-invariant potential $V({\mathbf{r}}_1,\dots ,{\mathbf{r}}_N)$, the quantities $E$, $\mathbf{P}$, $\mathbf{L}$, and $\mathbf{R}-\mathbf{P}t/M$ (centre-of-mass initial position) furnish 10 independent first integrals.

Proof. Energy is a scalar (1 integral). Momentum is a 3-vector (3 integrals). Angular momentum is a 3-vector (3 integrals). The Galilean boost gives the centre-of-mass integral $\mathbf{K}={\sum }_i{m}_i({\mathbf{r}}_i-{\mathbf{v}}_{i}t)$, which is a 3-vector (3 integrals). The total is 10. Independence: the quantities involve different combinations of positions, velocities, and masses, and no one is a function of the others on the full phase space. The constant $E$ restricts to a $6N-1$ dimensional surface; $\mathbf{P}$ restricts further to $6N-4$; $\mathbf{L}$ to $6N-7$; $\mathbf{K}$ to $6N-10$. These 10 integrals generate the Galilean algebra under Poisson brackets, and their independence is the statement that the Galilean algebra is 10-dimensional. $\square$

The virial theorem: from time-averaging to astrophysical mass estimation [Master]

The virial theorem connects the time-averaged kinetic and potential energies of a bound system. It applies to any long-lived bound configuration — a planet orbiting a star, a star cluster, a galaxy — and provides a relationship between observable quantities that does not require solving the equations of motion.

Proposition (Scalar virial theorem). For an $N$-particle system with a homogeneous potential $V$ of degree $k$ (that is, $V(\lambda \mathbf{r})={\lambda }^{k}V(\mathbf{r})$ for all $\lambda >0$), the time-averaged kinetic and potential energies satisfy

$$2⟨K⟩=k⟨V⟩,$$

where $⟨\cdot ⟩$ denotes the time average over a sufficiently long interval. For a gravitational system ($V\propto r^{-1}$, so $k=-1$), this gives $2⟨K⟩=-⟨V⟩$.

Proof. Define the virial $G:={\sum }_i{\mathbf{p}}_i\cdot {\mathbf{r}}_i$. Differentiate:

$$\frac{dG}{dt}=\sum _i{\dot{\mathbf{p}}}_i\cdot {\mathbf{r}}_i+\sum _i{\mathbf{p}}_i\cdot {\dot{\mathbf{r}}}_i=\sum _i{\mathbf{F}}_i\cdot {\mathbf{r}}_i+2K.$$

The first term ${\sum }_i{\mathbf{F}}_i\cdot {\mathbf{r}}_i$ is called the virial of the forces. For a homogeneous potential of degree $k$, Euler's theorem for homogeneous functions gives ${\sum }_i{\mathbf{F}}_i\cdot {\mathbf{r}}_i=-{\sum }_i{\nabla }_{i}V\cdot {\mathbf{r}}_i=-kV$. Hence $dG/dt=-kV+2K$. Time-averaging over a period $\tau$: $⟨dG/dt⟩=(G(\tau )-G(0))/\tau$. For a bound system, $G$ is bounded (positions and velocities are bounded), so $⟨dG/dt⟩\to 0$ as $\tau \to \infty$. Therefore $2⟨K⟩-k⟨V⟩=0$. $\square$

Gravitational applications. For gravitationally bound systems ($k=-1$): $⟨E⟩=⟨K⟩+⟨V⟩=⟨K⟩+(-2⟨K⟩)=-⟨K⟩=\frac{1}{2}⟨V⟩$. The total energy is negative (the system is bound) and its magnitude equals the average kinetic energy. Equivalently, the system satisfies the equipartition-like relation that the kinetic energy is half the magnitude of the (negative) potential energy.

This result is foundational in astrophysics ^{[Binney-Tremaine 2008]}. Given a galaxy cluster, an observer measures the line-of-sight velocities of the member galaxies (giving $⟨K⟩$) and applies the virial theorem to infer the total gravitational mass $⟨V⟩=-2⟨K⟩$. Zwicky's 1933 application of this method to the Coma cluster produced a mass estimate far exceeding the luminous mass — the first quantitative evidence for dark matter ^{[Zwicky 1933 *Helv. Phys. Acta* 6]}.

The tensor virial theorem. The scalar virial theorem has a tensor generalisation. Define the second-moment tensor $I_{jk}={\sum }_i{m}_i{r}_{ij}{r}_{ik}$ (the moment of inertia tensor). Then $\frac{1}{2}{\ddot{I}}_{jk}=2K_{jk}+W_{jk}$, where $K_{jk}=\frac{1}{2}{\sum }_i{m}_i{v}_{ij}{v}_{ik}$ is the kinetic-energy tensor and $W_{jk}=\frac{1}{2}{\sum }_i(F_{ij}{r}_{ik}+F_{ik}{r}_{ij})$ is the potential-energy tensor. Time-averaging: $2⟨K_{jk}⟩+⟨W_{jk}⟩=0$. The trace recovers the scalar virial theorem. The off-diagonal components constrain the shape of the system: an axisymmetric system has $K_{xx}=K_{yy}\ne K_{zz}$, which determines the flattening from observed velocity dispersions.

Clausius and the origin of the virial concept. The virial theorem was introduced by Rudolf Clausius in 1870 ^{[Clausius 1870 *Phil. Mag.* 40]}, who defined the "virial" as the sum ${\sum }_i{\mathbf{F}}_i\cdot {\mathbf{r}}_i$ and used it to study the thermal properties of gases. Clausius's original formulation was restricted to systems in a steady state; the generalisation to time-averaged bound systems is due to Poincare and later textbook treatments. The virial theorem is a purely classical result, but it generalises to quantum mechanics (where time averages become expectation values and the virial becomes $⟨\hat{G}⟩=⟨{\sum }_i{\hat{\mathbf{p}}}_i\cdot {\hat{\mathbf{r}}}_i⟩$) and to statistical mechanics (where time averages are replaced by ensemble averages).

Proposition (Quantum virial theorem). For a Hamiltonian $\hat{H}=\hat{T}+\hat{V}$ with $\hat{V}$ homogeneous of degree $k$, any stationary state $|\psi ⟩$ satisfies $2⟨\hat{T}⟩=k⟨\hat{V}⟩$.

Proof. In a stationary state $\hat{H}|\psi ⟩=E|\psi ⟩$, the expectation value of any operator is constant: $d⟨\hat{G}⟩/dt=(i/\hslash )⟨[\hat{H},\hat{G}]⟩=0$, where $\hat{G}={\sum }_i{\hat{\mathbf{p}}}_i\cdot {\hat{\mathbf{r}}}_i$. Compute $[\hat{H},\hat{G}]=[\hat{T},\hat{G}]+[\hat{V},\hat{G}]=2i\hslash \hat{T}-i\hslash k\hat{V}$ (using the canonical commutator $[{\hat{p}}_j,{\hat{r}}_j]=-i\hslash$ and the homogeneity of $\hat{V}$). Setting the commutator to zero: $2⟨\hat{T}⟩=k⟨\hat{V}⟩$. $\square$

This quantum version appears in the hydrogen atom 14.04.01 pending, where it gives $⟨\hat{T}⟩=-\frac{1}{2}⟨\hat{V}⟩$ for the Coulomb potential, and the relationship $E_n=-⟨\hat{T}⟩=\frac{1}{2}⟨\hat{V}⟩$ reproduces the Bohr energy levels.

Galilean invariance and the ten conserved quantities of Newtonian space-time [Master]

Newtonian mechanics takes place in Galilean space-time — the affine space $\mathbb{R}\times {\mathbb{R}}^3$ (time $\times$ space) equipped with the Euclidean distance on spatial slices and the absolute time difference $t_2-t_1$. The Galilean group $G(3)$ is the group of transformations that preserve this structure: spatial rotations, spatial translations, time translations, and Galilean boosts. The group has 10 continuous parameters (3 rotations, 3 spatial translations, 1 time translation, 3 boosts), and each parameter generates a conserved quantity.

The ten generators and their conservation laws. The correspondence between symmetries and conserved quantities is:

Symmetry	Generator	Conserved quantity
Time translation $t\mapsto t+\tau$	$H$ (Hamiltonian)	Energy $E$
Spatial translation $\mathbf{r}\mapsto \mathbf{r}+\mathbf{a}$	$\mathbf{P}$ (total momentum)	Linear momentum $\mathbf{P}$
Rotation $\mathbf{r}\mapsto R\mathbf{r}$	$\mathbf{L}$ (total angular momentum)	Angular momentum $\mathbf{L}$
Galilean boost $\mathbf{r}\mapsto \mathbf{r}+\mathbf{u}t$	$\mathbf{K}={\sum }_i{m}_i({\mathbf{r}}_i-{\mathbf{v}}_{i}t)$	Centre-of-mass uniform motion

The first three rows give the conservation laws of this unit. The fourth — the Galilean boost symmetry — gives the conservation of $\mathbf{K}=M\mathbf{R}-\mathbf{P}t$, which encodes the fact that the centre of mass moves at constant velocity: $\mathbf{R}(t)=\mathbf{R}(0)+(\mathbf{P}/M)t$. This is the content of the centre-of-mass theorem proved in the Intermediate tier.

The Galilean Lie algebra. The ten conserved quantities are not algebraically independent. Under Poisson brackets, they close to form the Galilean Lie algebra $\mathfrak{gal}(3)$, a 10-dimensional real Lie algebra with the bracket relations:

$$\{L_i,L_j\}={ϵ}_{ijk}{L}_k,\{L_i,P_j\}={ϵ}_{ijk}{P}_k,\{L_i,K_j\}={ϵ}_{ijk}{K}_k,$$ $$\{P_i,P_j\}=0,\{K_i,K_j\}=0,\{P_i,K_j\}={\delta }_{ij}M,\{H,P_i\}=0,\{H,K_i\}=P_i,\{H,L_i\}=0.$$

The relation $\{P_i,K_j\}={\delta }_{ij}M$ is distinctive: the mass $M$ enters as a central charge of the Galilean algebra. This is the algebraic reason that mass is a conserved, additive quantity in Newtonian mechanics — it is the central extension parameter of the Galilean group. In the relativistic (Poincare) limit, this central extension disappears and mass becomes the Casimir $M^2=H^2-|\mathbf{P}{|}^2{c}^2/c^4$.

Noether's theorem as the unifying principle. The intermediate-tier proofs of the three conservation laws — momentum from Newton's third law, energy from conservative forces, angular momentum from central forces — are each special arguments. Noether's theorem 09.03.01 pending unifies all three (and the fourth, the boost symmetry) into a single principle: for every continuous one-parameter symmetry of the action functional

$$S={\int }_{t_1}^{t_2}L({\mathbf{r}}_i,{\dot{\mathbf{r}}}_i,t)dt,$$

there exists a conserved quantity. The correspondence is:

• $\partial L/\partial t=0$ (time-translation invariance) $\Rightarrow$ energy conservation.
• $\partial L/\partial {\mathbf{r}}_i=0$ for all $i$ (spatial-translation invariance) $\Rightarrow$ momentum conservation.
• Rotational invariance of $L$ $\Rightarrow$ angular momentum conservation.

The Newtonian proofs are recovered as special cases: Newton's third law (${\mathbf{F}}_{ij}=-{\mathbf{F}}_{ji}$) is the dynamical manifestation of translation invariance, and the conservative-force condition ($\nabla \times \mathbf{F}=0$) is the dynamical manifestation of time-translation invariance.

Proposition (Conservation from the Lagrangian). Let $L=T-V$ be the Lagrangian for an $N$-particle system. (i) If $L$ does not depend explicitly on $t$, then $H={\sum }_i{\dot{\mathbf{r}}}_i\cdot \partial L/\partial {\dot{\mathbf{r}}}_i-L$ is conserved. (ii) If $L$ is invariant under spatial translations ${\mathbf{r}}_i\mapsto {\mathbf{r}}_i+ϵ\hat{\mathbf{e}}$, then $\mathbf{P}\cdot \hat{\mathbf{e}}$ is conserved. (iii) If $L$ is invariant under rotations about axis $\hat{\mathbf{n}}$, then $\mathbf{L}\cdot \hat{\mathbf{n}}$ is conserved.

Proof. (i) $\frac{dH}{dt}=-\frac{\partial L}{\partial t}=0$ by hypothesis. (ii) The infinitesimal translation ${\mathbf{r}}_i\mapsto {\mathbf{r}}_i+ϵ\hat{\mathbf{e}}$ changes $L$ by $\delta L=ϵ{\sum }_i\frac{\partial L}{\partial {\mathbf{r}}_i}\cdot \hat{\mathbf{e}}$. Invariance means ${\sum }_i\frac{\partial L}{\partial {\mathbf{r}}_i}\cdot \hat{\mathbf{e}}=0$. By the Euler-Lagrange equations, $\frac{\partial L}{\partial {\mathbf{r}}_i}=\frac{d}{dt}\frac{\partial L}{\partial {\dot{\mathbf{r}}}_i}$, so $\frac{d}{dt}{\sum }_i\frac{\partial L}{\partial {\dot{\mathbf{r}}}_i}\cdot \hat{\mathbf{e}}=0$, hence $\mathbf{P}\cdot \hat{\mathbf{e}}={\sum }_i{m}_i{\mathbf{v}}_i\cdot \hat{\mathbf{e}}$ is conserved. (iii) Analogous, with $\delta L=ϵ{\sum }_i\frac{\partial L}{\partial {\mathbf{r}}_i}\cdot (\hat{\mathbf{n}}\times {\mathbf{r}}_i)=0$ giving $\frac{d}{dt}{\sum }_i{\mathbf{r}}_i\times m_i{\mathbf{v}}_i\cdot \hat{\mathbf{n}}=0$. $\square$

This proposition is the restricted (Newtonian-Lagrangian) version of Noether's theorem; the full version 09.03.01 pending applies to arbitrary field theories and yields an infinite-dimensional algebra of conservation laws for systems with gauge symmetries.

Central-force motion: reduction, orbit classification, and superintegrability [Master]

The two-body central-force problem is the paradigmatic application of all three conservation laws simultaneously. The reduction procedure demonstrates how first integrals simplify dynamics: starting from a 12-dimensional phase space (two particles in three dimensions), successive use of conservation laws reduces the problem to a single quadrature.

Step 1: Centre-of-mass separation. As shown in Exercise 8 and the Key theorem, the two-body Lagrangian separates into $\frac{1}{2}M|\dot{\mathbf{R}}{|}^2+\frac{1}{2}\mu |\dot{\mathbf{r}}{|}^2-V(r)$ where $\mu =m_1{m}_2/(m_1+m_2)$ is the reduced mass, $\mathbf{R}$ is the centre-of-mass coordinate, and $\mathbf{r}={\mathbf{r}}_1-{\mathbf{r}}_2$ is the relative coordinate. The centre-of-mass moves at constant velocity (momentum conservation). The relative motion is a single particle of mass $\mu$ in the potential $V(r)$. This uses conservation of total momentum $\mathbf{P}$.

Step 2: Reduction to the orbital plane. The angular momentum $\mathbf{L}=\mu \mathbf{r}\times \dot{\mathbf{r}}$ is conserved (central force). The constancy of $\mathbf{L}$ means: $\mathbf{r}$ and $\dot{\mathbf{r}}$ always lie in the plane perpendicular to $\mathbf{L}$. Choose coordinates with $\mathbf{L}=L\hat{z}$; then motion is confined to the $xy$-plane. In polar coordinates $(r,\theta )$: $\mu r^2\dot{\theta }=L=\text{constant}$ (angular momentum conservation in component form). This uses conservation of total angular momentum $\mathbf{L}$, reducing 3D relative motion to 2D planar motion.

Step 3: Reduction to one-dimensional radial motion. The energy of the relative motion is

$$E=\frac{1}{2}\mu ({\dot{r}}^2+r^2{\dot{\theta }}^2)+V(r)=\frac{1}{2}\mu {\dot{r}}^2+\underset{V_{\text{eff}}(r)}{\underbrace{\frac{L^2}{2\mu r^2}+V(r)}}.$$

The effective potential $V_{\text{eff}}(r)=L^2/(2\mu r^2)+V(r)$ combines the true potential with the centrifugal barrier $L^2/(2\mu r^2)$. Energy conservation fixes the radial motion: $\dot{r}=\pm \sqrt{2(E-V_{\text{eff}}(r))/\mu }$. This is a one-dimensional problem in $r$, with turning points at $E=V_{\text{eff}}(r)$.

Orbit classification. For the gravitational potential $V(r)=-GM\mu /r$:

• $E<0$: the radial motion oscillates between two turning points $r_{\min }$ and $r_{\max }$ (periapsis and apoapsis). The orbit is an ellipse with semi-major axis $a=-GM\mu /(2E)$ and eccentricity $e=\sqrt{1+2E{L}^2/(GM\mu {)}^2\mu }$. Kepler's first law.
• $E=0$: one turning point at finite $r$, the other at $r\to \infty$. The orbit is a parabola.
• $E>0$: no outer turning point; the particle escapes. The orbit is a hyperbola.

The orbit shape depends on $E$ and $L$ through the eccentricity formula. Circular orbits occur at $e=0$, i.e., at the minimum of $V_{\text{eff}}$: $r_0=L^2/(GM{\mu }^2)$.

The Laplace-Runge-Lenz vector. The Kepler problem has an additional conserved quantity beyond $E$ and $\mathbf{L}$:

$$\mathbf{A}:=\mathbf{p}\times \mathbf{L}-GM{\mu }^2\hat{\mathbf{r}}.$$

A direct computation using $\dot{\mathbf{p}}=-GM\mu \hat{\mathbf{r}}/r^2$ and $\dot{\mathbf{L}}=0$ shows $\dot{\mathbf{A}}=0$. The vector $\mathbf{A}$ points from the focus to the periapsis, with magnitude $|\mathbf{A}|=GM{\mu }^{2}e$. Its conservation constrains the orbit to be a fixed ellipse — the orientation of the ellipse does not precess.

The existence of $\mathbf{A}$ is the reason the Kepler problem is superintegrable: with 3 degrees of freedom, a generic integrable system has 3 independent first integrals in involution ($E$, $L^2$, $L_z$). The Kepler problem has 5 independent integrals ($E$, $\mathbf{L}$ with 2 independent components after fixing $|\mathbf{L}|$, and $\mathbf{A}$ with 1 independent component after fixing $|\mathbf{A}|$ and $\mathbf{L}\cdot \mathbf{A}=0$). The extra integrals collapse the invariant tori to periodic orbits — all bounded Kepler orbits close.

Bertrand's theorem. The superintegrability of the Kepler problem is exceptional. Bertrand's theorem (1873) states that the only central potentials for which all bounded orbits are closed are $V(r)=-k/r$ (Kepler) and $V(r)=\frac{1}{2}k{r}^2$ (isotropic harmonic oscillator). Both have an additional conserved vector (the LRL vector for Kepler; the Fradkin tensor for the oscillator) that makes them superintegrable. All other central potentials produce precessing (open) orbits.

Proposition (Kepler's third law from conservation laws). For a circular orbit of radius $r_0$ in the gravitational potential $V=-GM\mu /r$, the orbital period is $T=2\pi \sqrt{r_0^3/(GM)}$, independent of the orbiting mass.

Proof. For a circular orbit, $V_{\text{eff}}^{\prime }(r_0)=0$: $-L^2/(\mu r_0^3)+GM\mu /r_0^2=0$, giving $L=\mu \sqrt{GM{r}_0}$. Angular momentum conservation: $L=\mu r_0^2\dot{\theta }$, so $\dot{\theta }=L/(\mu r_0^2)=\sqrt{GM/r_0^3}$. The period is $T=2\pi /\dot{\theta }=2\pi \sqrt{r_0^3/(GM)}$. $\square$

Kepler's third law is a consequence of the inverse-square force law (which sets the relationship between $L$ and $r_0$) and angular momentum conservation (which sets $\dot{\theta }$). It appeared originally as an empirical law ^{[Kepler 1619 *Harmonices Mundi*]}; Newton's derivation from the conservation laws and the gravitational force law was one of the central achievements of the Principia ^{[Newton 1687]}.

Synthesis. The conservation laws of Newtonian mechanics are the foundational reason that the Kepler problem is solvable in closed form: the three conservation laws — energy, momentum, angular momentum — are the first-integral shadows of the 10-dimensional Galilean symmetry group, and this is exactly the algebraic structure that Noether's theorem 09.03.01 pending generalises to arbitrary Lagrangian systems. The central insight is that each independent first integral removes one degree of freedom from the dynamics; putting these together, the 6-dimensional phase space of the relative Kepler problem is reduced to a 1-dimensional radial problem by the five independent integrals $E$, $|\mathbf{L}|$, $L_z$, $A_x$, $A_y$. The bridge is between the Newtonian proofs of this unit (each using a special property of the forces) and the unified Noether framework, which identifies conservation of energy with time-translation invariance, conservation of momentum with spatial-translation invariance, and conservation of angular momentum with rotational invariance. The pattern recurs throughout physics: in the Hamilton-Jacobi theory 09.05.02 pending, the conserved quantities become the new canonical momenta that trivialise the Hamiltonian; in action-angle variables 09.06.01 pending, the integrals of motion label the invariant tori of the Liouville-Arnold theorem; and in the KAM theorem 09.08.01 pending, the persistence of these tori under small perturbations determines the boundary between regular and chaotic motion.

Connections [Master]

• Kinematics 09.01.01 defines velocity and acceleration, which enter the definitions of kinetic energy and momentum.

• Newton's laws 09.01.02 pending are the dynamical foundation from which the conservation laws are derived. Each conservation theorem in this unit is a consequence of a specific feature of Newton's laws (third law for momentum, conservative forces for energy, central forces for angular momentum).

• Noether's theorem 09.03.01 pending generalises the three conservation laws of this unit: every continuous symmetry of the action functional generates a conserved quantity. The Galilean-algebraic treatment in this unit provides the concrete Newtonian instance that Noether's theorem elevates to a general principle.

• The action principle 09.02.01 pending reformulates mechanics so that the conservation laws emerge naturally from the variational structure rather than from Newton's laws.

• Hamilton's equations 09.04.02 pending re-express the conservation laws in the language of phase space: conserved quantities are functions $f$ with $\{f,H\}=0$. The Poisson-bracket algebra of the conserved quantities reproduces the Galilean Lie algebra.

• Action-angle variables 09.06.01 pending develop the Liouville-Arnold theory of completely integrable systems, where the conserved quantities label invariant tori and the motion on each torus is quasi-periodic. The Kepler problem treated here is the canonical example.

• The KAM theorem 09.08.01 pending studies the persistence of the invariant tori (described by the conserved quantities of this unit) under small perturbations, establishing the boundary between integrable and chaotic dynamics.

• Thermodynamics — first law 11.01.01 pending extends energy conservation to include heat and internal energy, taking mechanical energy conservation as its starting point. The virial theorem developed here has a direct statistical-mechanical analogue in the equipartition theorem.

• Chemical thermodynamics 14.06.01 applies energy conservation to molecular-scale processes, with free energy as the bookkeeping quantity. The quantum virial theorem connects to the hydrogen atom energy levels in 14.04.01 pending.

• Oxidative phosphorylation 17.04.02 pending converts proton-gradient energy to ATP bond energy, governed by energy conservation as the accounting principle.

Historical & philosophical context [Master]

The concept of conservation in physics predates Newton. Descartes, in Principia Philosophiae (1644) ^{[Descartes 1644]}, asserted that the total "quantity of motion" ($\sum m|v|$) in the universe is conserved by God. This was incorrect — it is the vector momentum $\sum m\mathbf{v}$ that is conserved, not the sum of speeds. Huygens (1669) and Wallis (1669) independently established the correct momentum conservation law for collisions, published in the Philosophical Transactions of the Royal Society ^{[Huygens 1669]}.

Leibniz introduced the concept of vis viva ($m{v}^2$, twice the kinetic energy) in 1686 ^{[Leibniz 1686]}, arguing that this — not Descartes' $m|v|$ — was the conserved quantity. The ensuing "vis viva controversy" lasted decades, resolved only when it was understood that both momentum and energy are conserved, but they are different quantities with different conservation conditions. The distinction between elastic collisions (both conserved) and inelastic collisions (only momentum conserved) was clarified by d'Alembert in Traite de dynamique (1743) ^{[d'Alembert 1743]}.

The term "energy" in its modern sense was introduced by Thomas Young in 1807. The general principle of conservation of energy — including thermal, chemical, and mechanical forms — was established by Joule (1843, mechanical equivalent of heat), Mayer (1842), and Helmholtz (1847, Uber die Erhaltung der Kraft) ^{[Helmholtz 1847]}. The mechanical energy conservation of this unit is the special case restricted to kinetic and potential energy.

Angular momentum conservation emerged from the study of planetary motion. Kepler's second law (equal areas in equal times, 1609) is a statement of angular momentum conservation for orbits under a central force. Newton derived it from his law of gravitation in the Principia (1687, Book I, Proposition I) ^{[Newton 1687]}. The general theorem — that central forces conserve angular momentum — appears in the work of Euler and d'Alembert. The Laplace-Runge-Lenz vector was studied by Laplace, Lagrange, and Jacobi in the context of perturbation theory; its modern interpretation as the generator of a hidden SO(4) symmetry of the Kepler problem is due to Fock (1935) and Bargmann (1936) ^{[Fock 1935 *Z. Phys.* 98]}.

The virial theorem was introduced by Clausius in 1870 ^{[Clausius 1870]}. Its application to galaxy-cluster mass estimation by Zwicky (1933) provided the first quantitative evidence for dark matter ^{[Zwicky 1933]}.

The unification of all three conservation laws under a single symmetry principle is due to Emmy Noether (1918, "Invariante Variationsprobleme") ^{[Noether 1918]}, whose theorem connects continuous symmetries of the action to conserved quantities. The Noether perspective is the one that survives in modern physics: conservation laws are not separate empirical facts but mathematical consequences of the symmetries of the physical theory.

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