Newton's laws of motion

Intuition [Beginner]

Kinematics gives you the language to describe motion. Newton's three laws of motion give you the rules that govern it — they connect what the object is doing (acceleration) to what is being done to it (force). Without these laws, you can measure a trajectory; with them, you can predict one.

Newton's first law says: an object at rest stays at rest, and an object moving in a straight line at constant speed keeps doing that, unless something pushes or pulls it. This sounds obvious, but it is a rejection of thousands of years of Aristotelian thinking that said motion requires a cause. The first law says the opposite: motion is the default. Change in motion is what requires a cause.

Newton's second law is the workhorse: force equals mass times acceleration. Written as an equation, $F=ma$. If you know the total force acting on an object and you know its mass, you can compute its acceleration. From the acceleration, you can work out the velocity and position (using the kinematic tools from the previous unit). The second law is a recipe: given force, find motion. It is the bridge from dynamics to kinematics.

Newton's third law says: if object A pushes on object B, then object B pushes back on object A with the same size force in the opposite direction. You push on the ground with your foot; the ground pushes back on your foot. A rocket expels gas downward; the gas pushes the rocket upward. Every force is one half of a pair. There are no unpaired forces in nature.

Why mass matters: a 1 newton force on a 1 kilogram object produces 1 metre per second squared of acceleration. The same 1 newton force on a 100 kilogram object produces only 0.01 metres per second squared of acceleration. Mass is the measure of how much an object resists being accelerated — its inertia. Heavy things are hard to speed up or slow down not because the force is weaker, but because the same force produces less acceleration.

Visual [Beginner]

Figure: Three scenarios side by side. On the left, a hockey puck glides across frictionless ice — no horizontal forces act, so the puck moves in a straight line at constant speed (first law). In the centre, a person pushes a shopping cart with a constant forward force — the cart accelerates forward, and a second arrow below the cart shows the reaction force the cart exerts backward on the person's hands (third law). On the right, a small car and a large truck are both pushed with the same-size force arrow; the car has a large acceleration arrow while the truck has a small one — same force, different masses, different accelerations (second law).

Worked example [Beginner]

A 1200 kg car sits on a flat road. The driver presses the accelerator, producing a forward force of 3600 N from the engine on the tyres. A constant friction-plus-drag force of 600 N opposes the motion. What is the acceleration, and how fast is the car going after 5 seconds starting from rest?

Step 1. Find the net force. The forward force is 3600 N, the opposing force is 600 N. Net force = 3600 − 600 = 3000 N, forward.

Step 2. Apply Newton's second law. $F=ma$, so $a=F/m=3000/1200=2.5$ m/s² forward.

Step 3. Find the velocity after 5 seconds. Starting from rest ($v_0=0$), with constant acceleration: $v=v_0+at=0+2.5\times 5=12.5$ m/s, which is about 45 km/h.

Step 4. Check with the third law. The engine drives the tyres forward against the road surface. The road surface pushes forward on the tyres (that is the 3600 N force on the car). Simultaneously, the tyres push backward on the road surface with 3600 N — the reaction force that the road feels. The road does not move because it is attached to the Earth, whose enormous mass makes its acceleration undetectable.

What this tells us: the second law converts a known force into a known acceleration. Combined with kinematics, this gives the complete motion. The third law reminds us that the force on the car is one side of an interaction pair — the road participates too.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $\mathbf{r}(t)\in {\mathbb{R}}^3$ denote the position of a point particle of mass $m>0$ in an inertial reference frame, parametrised by Newtonian absolute time $t\in \mathbb{R}$. Denote velocity $\mathbf{v}(t)=\dot{\mathbf{r}}(t)$ and acceleration $\mathbf{a}(t)=\ddot{\mathbf{r}}(t)$.

Newton's first law (law of inertia). In the absence of net external force, a free particle moves with constant velocity: $\dot{\mathbf{v}}=0$.

The first law defines the class of reference frames — inertial frames — in which the second and third laws hold. A frame that accelerates relative to an inertial frame is non-inertial; in such a frame, free particles appear to accelerate even without forces, and the second law requires correction terms (fictitious forces) developed in 09.01.01 Exercise 8.

Newton's second law. The net force $\mathbf{F}$ on a particle equals the product of its mass and its acceleration:

$$\mathbf{F}=m\mathbf{a}=m\ddot{\mathbf{r}}.$$

More generally, for a system of $N$ particles with positions ${\mathbf{r}}_1,\dots ,{\mathbf{r}}_N$, masses $m_1,\dots ,m_N$, and forces ${\mathbf{F}}_1,\dots ,{\mathbf{F}}_N$:

$$m_i{\ddot{\mathbf{r}}}_i={\mathbf{F}}_i({\mathbf{r}}_1,\dots ,{\mathbf{r}}_N,{\dot{\mathbf{r}}}_1,\dots ,{\dot{\mathbf{r}}}_N,t),i=1,\dots ,N.$$

The force ${\mathbf{F}}_i$ is typically decomposed into external forces (gravity, electromagnetic, applied) and internal forces (inter-particle interactions: springs, contact forces, gravitational attraction between the particles themselves).

Newton's third law (action-reaction). If particle $j$ exerts a force ${\mathbf{F}}_{ij}$ on particle $i$, then particle $i$ exerts a force ${\mathbf{F}}_{ji}$ on particle $j$ satisfying

$${\mathbf{F}}_{ji}=-{\mathbf{F}}_{ij}.$$

The strong form additionally requires ${\mathbf{F}}_{ij}$ to lie along the line connecting ${\mathbf{r}}_i$ and ${\mathbf{r}}_j$ (central forces). The strong form is needed for conservation of angular momentum; the weak form alone guarantees only conservation of linear momentum.

Definition of mass. Mass enters the second law as the proportionality constant relating force to acceleration. Operationally, mass is measured by comparing accelerations: if the same force produces acceleration $a_1$ on object 1 and $a_2$ on object 2, then $m_1/m_2=a_2/a_1$. This defines mass as a ratio; a unit mass (the kilogram, since 2019 defined via Planck's constant $h=6.62607015\times 10^{-34}$ J·s) fixes the scale.

Momentum form. Defining the linear momentum of a particle as $\mathbf{p}=m\mathbf{v}$, Newton's second law takes the form

$$\mathbf{F}=\frac{d\mathbf{p}}{dt}.$$

For constant mass (the usual case in Newtonian mechanics), this reduces to $\mathbf{F}=m\mathbf{a}$. The momentum form is more general: it applies to systems with variable mass (rockets expelling fuel, raindrops accumulating mass) and extends naturally to the Lagrangian and Hamiltonian formulations.

Counterexamples to common slips

• The third law does not imply equilibrium. The book-on-table example: the book's weight $\mathbf{W}=-mg\hat{\mathbf{z}}$ and the normal force $\mathbf{N}=+mg\hat{\mathbf{z}}$ balance on the book, but these are not a third-law pair. The third-law partner of the book's weight is the gravitational force the book exerts on the Earth (upward, on the Earth). The third-law partner of the normal force on the book is the force the book exerts on the table (downward, on the table). Third-law pairs act on different objects; they cannot balance on the same object.

• Force is not stored in objects. A common misconception is that a moving object "has force." It does not. It has momentum $m\mathbf{v}$. Force is an interaction — a push or pull between two objects at a moment in time. The second law relates the net force acting on the object to the rate of change of its momentum.

• The second law is not a definition of force. It is a physical law: it asserts that the quantity $\mathbf{F}$ (defined independently by force laws: gravity, electromagnetism, spring forces) equals $m\mathbf{a}$. If it were a definition, it would be vacuous. Its content is that the force laws (which you can determine from geometry and material properties) predict the correct accelerations.

Key theorem with proof [Intermediate+]

Theorem (Newton's second law determines trajectories uniquely). Let $m>0$ and let $\mathbf{F}(\mathbf{r},\mathbf{v},t)$ be a continuously differentiable vector-valued function on ${\mathbb{R}}^3\times {\mathbb{R}}^3\times \mathbb{R}$. Then for any initial position ${\mathbf{r}}_0$ and initial velocity ${\mathbf{v}}_0$, the equation $m\ddot{\mathbf{r}}=\mathbf{F}(\mathbf{r},\dot{\mathbf{r}},t)$ has a unique solution $\mathbf{r}(t)$ defined on a maximal interval $I\ni t_0$ satisfying $\mathbf{r}(t_0)={\mathbf{r}}_0$ and $\dot{\mathbf{r}}(t_0)={\mathbf{v}}_0$.

Proof. Rewrite the second-order equation $m\ddot{\mathbf{r}}=\mathbf{F}(\mathbf{r},\dot{\mathbf{r}},t)$ as a first-order system on phase space $(\mathbf{r},\mathbf{v})\in {\mathbb{R}}^6$:

$$\dot{\mathbf{r}}=\mathbf{v},\dot{\mathbf{v}}=\frac{1}{m}\mathbf{F}(\mathbf{r},\mathbf{v},t).$$

Define the vector field $\mathbf{X}:{\mathbb{R}}^6\times \mathbb{R}\to {\mathbb{R}}^6$ by $\mathbf{X}(\mathbf{r},\mathbf{v},t)=(\mathbf{v},\mathbf{F}(\mathbf{r},\mathbf{v},t)/m)$. Since $\mathbf{F}$ is continuously differentiable, $\mathbf{X}$ is continuously differentiable in $(\mathbf{r},\mathbf{v})$ and continuous in $t$. By the Picard-Lindelof theorem 02.12.01, the initial value problem $\dot{\mathbf{y}}=\mathbf{X}(\mathbf{y},t)$, $\mathbf{y}(t_0)=({\mathbf{r}}_0,{\mathbf{v}}_0)$ has a unique solution defined on a maximal interval $I\ni t_0$. This solution $\mathbf{y}(t)=(\mathbf{r}(t),\mathbf{v}(t))$ satisfies $m\ddot{\mathbf{r}}=\mathbf{F}(\mathbf{r},\dot{\mathbf{r}},t)$ by construction. ∎

Corollary (Determinism of Newtonian mechanics). Under the hypotheses of the theorem, specifying the initial position and velocity of every particle in an $N$-particle system, together with the force law ${\mathbf{F}}_i$, uniquely determines the entire past and future trajectory of the system (up to the maximal interval of existence).

This is Laplace's determinism thesis in its precise mathematical form. The corollary assumes the force law is known exactly and the initial data are specified exactly; both conditions fail in practice, and sensitivity to initial conditions (chaos) makes the resulting predictions unreliable on long timescales even when the mathematical uniqueness holds.

Bridge. The uniqueness theorem builds toward 09.02.01 pending, where Newton's second-order equation $m\ddot{\mathbf{r}}=\mathbf{F}$ emerges as the Euler-Lagrange condition on an action functional $S[\gamma ]=\int Ldt$ over curves in configuration space. The foundational reason uniqueness matters physically is that it guarantees the initial-value problem has exactly one solution — the central insight that makes Newtonian mechanics a predictive theory rather than a merely descriptive one. This is exactly the bridge between the physical law $F=ma$ and the mathematical theory of dynamical systems on manifolds. Putting these together with the Galilean symmetries developed in the Master tier, Newton's laws identify classical mechanics with the study of integral curves of smooth vector fields on cotangent bundles of Riemannian manifolds, appearing again in 09.04.02 pending as Hamilton's first-order system on $T^{\ast }Q$.

Worked example: the Atwood machine

Two masses $m_1=5$ kg and $m_2=3$ kg hang from a massless, inextensible string over a frictionless pulley. Find the acceleration of the system and the string tension.

Take the heavier mass $m_1$ as descending and $m_2$ as ascending. Newton's second law on each mass (positive direction: up for $m_2$, down for $m_1$):

$$m_{1}g-T=m_{1}a,T-m_{2}g=m_{2}a.$$

Adding eliminates $T$: $(m_1-m_2)g=(m_1+m_2)a$, so

$$a=\frac{m_1-m_2}{m_1+m_2}g=\frac{2}{8}\times 9.8=2.45{\text{ m/s}}^2.$$

The tension: $T=m_1(g-a)=5\times (9.8-2.45)=36.75$ N. Check: $T=m_2(g+a)=3\times (9.8+2.45)=36.75$ N — consistent.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib has the ODE existence and uniqueness machinery needed to formalise the theorem that Newton's second law determines unique trajectories (via Analysis.Calculus.FDeriv, Analysis.ODE). It does not have a formalisation of "force" as a physical concept, inertial frames, or Newton's laws stated as axioms of a physical theory. Such a formalisation would sit on top of the existing ODE theory and would be a physics-layer contribution rather than a mathematical one. lean_status: none.

Conservative forces, the work-energy theorem, and the virial theorem [Master]

Newton's second law relates force to acceleration. Integrating it along a trajectory yields relations between force, work, and energy that constrain the motion without requiring a full solution of the ODE.

Work and the work-energy theorem. For a particle of mass $m$ moving along a curve $\gamma$ from position ${\mathbf{r}}_1$ to ${\mathbf{r}}_2$ under force $\mathbf{F}$, the work done by $\mathbf{F}$ is

$$W_{12}={\int }_{{\mathbf{r}}_1}^{{\mathbf{r}}_2}\mathbf{F}\cdot d\mathbf{r}={\int }_{t_1}^{t_2}\mathbf{F}(\mathbf{r}(t),\dot{\mathbf{r}}(t),t)\cdot \dot{\mathbf{r}}(t)dt.$$

Substituting Newton's second law $\mathbf{F}=m\ddot{\mathbf{r}}$ and using $\frac{d}{dt}|\dot{\mathbf{r}}{|}^2=2\dot{\mathbf{r}}\cdot \ddot{\mathbf{r}}$ gives the work-energy theorem: $W_{12}=\frac{1}{2}m|{\dot{\mathbf{r}}}_2{|}^2-\frac{1}{2}m|{\dot{\mathbf{r}}}_1{|}^2=T_2-T_1$, where $T=\frac{1}{2}m{v}^2$ is the kinetic energy. The work-energy theorem is a direct integral consequence of Newton's second law; it is not an independent principle.

Conservative forces. A force $\mathbf{F}$ is conservative if the work $W_{12}$ depends only on the endpoints ${\mathbf{r}}_1,{\mathbf{r}}_2$ and not on the path connecting them. For simply connected domains in ${\mathbb{R}}^3$, the following are equivalent:

1. $\mathbf{F}$ is conservative (path-independent work).
2. ${\oint }_{\gamma }\mathbf{F}\cdot d\mathbf{r}=0$ for every closed curve $\gamma$.
3. $\nabla \times \mathbf{F}=0$ (irrotational).
4. $\mathbf{F}=-\nabla V$ for some scalar potential $V:{\mathbb{R}}^3\to \mathbb{R}$.

The equivalence of (1)–(3) is Stokes' theorem. The implication (3) $\Rightarrow$ (4) is the Poincare lemma for simply connected domains: a closed 1-form is exact. In the language of differential forms, a conservative force is an exact 1-form $-dV$, and the work integral is the evaluation of this 1-form along the path.

When $\mathbf{F}=-\nabla V$, the work-energy theorem becomes $T_2-T_1=V({\mathbf{r}}_1)-V({\mathbf{r}}_2)$, or equivalently $T+V=E=\text{const}$. This is conservation of mechanical energy — the first integral of Newton's equations that does not follow from the Galilean symmetries alone (energy conservation is associated with time-translation invariance via Noether's theorem 09.03.01 pending).

Equilibrium and stability. An equilibrium point of a conservative system is a point ${\mathbf{r}}_0$ where $\nabla V({\mathbf{r}}_0)=0$ (zero force). The equilibrium is stable (in the sense of Lagrange) if $V$ has a strict local minimum at ${\mathbf{r}}_0$: for sufficiently small perturbations, the total energy $E=T+V$ constrains the particle to remain near ${\mathbf{r}}_0$. The sufficient condition is that the Hessian $H_{ij}={\partial }^{2}V/\partial r_i\partial r_j$ evaluated at ${\mathbf{r}}_0$ is positive definite — then small oscillations about equilibrium have real, positive frequencies ${\omega }_{\alpha }$ satisfying $\det (H_{ij}-{\omega }^{2}m{\delta }_{ij})=0$ (the secular equation). When the Hessian has a negative eigenvalue, the equilibrium is unstable: perturbations along the corresponding eigenvector grow exponentially. The case of zero eigenvalues (flat directions) requires higher-order analysis.

The virial theorem. For an $N$-particle system with positions ${\mathbf{r}}_i$ and total kinetic energy $T=\frac{1}{2}{\sum }_i{m}_i|{\dot{\mathbf{r}}}_i{|}^2$, the virial is $G:={\sum }_i{\mathbf{r}}_i\cdot {\mathbf{p}}_i={\sum }_i{m}_i{\mathbf{r}}_i\cdot {\dot{\mathbf{r}}}_i$. Its time derivative is

$$\dot{G}=\sum _i{m}_i|{\dot{\mathbf{r}}}_i{|}^2+\sum _i{\mathbf{r}}_i\cdot {\mathbf{F}}_i=2T+\sum _i{\mathbf{r}}_i\cdot {\mathbf{F}}_i.$$

For a bound system in a steady state (periodic or quasi-periodic motion), the long-time average $⟨\dot{G}⟩=0$ (because $G$ is bounded), giving the virial theorem: $⟨T⟩=-\frac{1}{2}⟨{\sum }_i{\mathbf{r}}_i\cdot {\mathbf{F}}_i⟩$.

For a system bound by an inverse-power-law central force $F\propto r^{-n}$, the virial theorem gives $⟨T⟩=-\frac{n}{2}⟨V⟩$. The physically central case is the gravitational or Coulomb potential ($n=1$, $V\propto -1/r$): $⟨T⟩=-\frac{1}{2}⟨V⟩$, or equivalently $⟨E⟩=⟨T⟩+⟨V⟩=\frac{1}{2}⟨V⟩=-⟨T⟩$. A gravitationally bound system has negative total energy, positive kinetic energy, and the kinetic energy is half the magnitude of the (negative) potential energy. This result underpins the astrophysical mass-estimation technique: measuring velocity dispersions (hence $⟨T⟩$) yields the gravitational binding energy and hence the total mass, including dark matter.

Proposition (Work-energy theorem as a first integral). For a single particle of mass $m$ in a conservative force field $\mathbf{F}=-\nabla V$, the total mechanical energy $E=\frac{1}{2}m|\dot{\mathbf{r}}{|}^2+V(\mathbf{r})$ is a first integral of Newton's second law: $\frac{dE}{dt}=0$ along every solution trajectory.

Proof. $\frac{dE}{dt}=m\dot{\mathbf{r}}\cdot \ddot{\mathbf{r}}+\nabla V\cdot \dot{\mathbf{r}}=\dot{\mathbf{r}}\cdot (m\ddot{\mathbf{r}}+\nabla V)=\dot{\mathbf{r}}\cdot (\mathbf{F}-\mathbf{F})=0$, where the last equality uses $m\ddot{\mathbf{r}}=\mathbf{F}=-\nabla V$. $\square$

The central-force two-body problem and Kepler orbits [Master]

The two-body problem — two point masses interacting via a central force — is the most important exactly solvable system in Newtonian mechanics. Its solution produces Kepler's laws of planetary motion and provides the conceptual template for perturbation theory, scattering theory, and the transition from Newtonian gravity to general relativity.

Reduction to one body. Consider masses $m_1,m_2$ at positions ${\mathbf{r}}_1,{\mathbf{r}}_2$ interacting via a central force ${\mathbf{F}}_{12}=-f(r)\hat{\mathbf{r}}$, where $\mathbf{r}={\mathbf{r}}_1-{\mathbf{r}}_2$ and $r=|\mathbf{r}|$. Define the centre of mass $\mathbf{R}=(m_1{\mathbf{r}}_1+m_2{\mathbf{r}}_2)/(m_1+m_2)$ and the relative position $\mathbf{r}={\mathbf{r}}_1-{\mathbf{r}}_2$. Newton's second law for each particle, combined with the third law ${\mathbf{F}}_{21}=-{\mathbf{F}}_{12}$, gives

$$(m_1+m_2)\ddot{\mathbf{R}}=0,\mu \ddot{\mathbf{r}}=-f(r)\hat{\mathbf{r}},$$

where $\mu =m_1{m}_2/(m_1+m_2)$ is the reduced mass. The centre of mass moves at constant velocity (conservation of total momentum). The relative motion reduces to a one-body problem: a single particle of mass $\mu$ in the central potential $V(r)=-\int f(r)dr$.

Planarity and angular momentum conservation. From the intermediate-tier result (Exercise 5), the angular momentum $\mathbf{L}=\mathbf{r}\times \mu \dot{\mathbf{r}}$ is conserved for any central force. Since $\mathbf{L}$ is constant and perpendicular to both $\mathbf{r}$ and $\dot{\mathbf{r}}$, the motion lies in a fixed plane. Using polar coordinates $(r,\theta )$ in this plane, the angular momentum is $L=\mu r^2\dot{\theta }$ (constant), and Newton's radial equation becomes

$$\mu (\ddot{r}-r{\dot{\theta }}^2)=-f(r).$$

Substituting $\dot{\theta }=L/(\mu r^2)$ yields the radial equation in effective potential form:

$$\mu \ddot{r}=-f(r)+\frac{L^2}{\mu r^3}=-\frac{d{V}_{\text{eff}}}{dr},$$

where $V_{\text{eff}}(r)=V(r)+L^2/(2\mu r^2)$. The second term is the centrifugal barrier: even when $V(r)$ is purely attractive, the effective potential has a repulsive $1/r^2$ core that prevents the orbiting particle from reaching $r=0$ (unless $L=0$, i.e., radial infall). The effective potential provides an immediate classification of orbits: $E<0$ gives bound orbits (ellipses or circles for $V=-k/r$), $E=0$ gives marginally unbound orbits (parabolae), $E>0$ gives unbound orbits (hyperbolae).

The orbit equation (Binet's formula). Eliminating time in favour of the angle $\theta$ via $u(\theta )=1/r$ and using $d/dt=(L/\mu r^2)d/d\theta$ transforms the radial equation into the Binet equation:

$$\frac{d^{2}u}{d{\theta }^2}+u=\frac{\mu }{L^2{u}^2}f\left(\frac{1}{u}\right).$$

For the inverse-square force $f(r)=k/r^2$ (gravitational: $k=G{m}_1{m}_2$; Coulomb: $k=|q_1{q}_2|/(4\pi {ϵ}_0)$), the right-hand side is $\mu k/L^2$ = constant. The solution is a conic section:

$$r(\theta )=\frac{p}{1+e\cos (\theta -{\theta }_0)},$$

with semi-latus rectum $p=L^2/(\mu k)$ and eccentricity $e=\sqrt{1+2E{L}^2/(\mu k^2)}$. The three cases are: $e<1$ (ellipse, bound), $e=1$ (parabola, $E=0$), $e>1$ (hyperbola, unbound). A circular orbit has $e=0$, requiring $L^2=\mu ka$ where $a$ is the orbital radius.

Kepler's three laws as theorems of Newtonian mechanics. Kepler stated his laws empirically (1609, 1619). Newton's laws plus the inverse-square force law derive them:

1. Elliptical orbits ($e<1$, $E<0$): the orbit equation gives a closed ellipse with the centre of force at one focus. This follows from the Binet solution above.

2. Equal areas in equal times: the areal velocity is $\frac{dA}{dt}=\frac{1}{2}{r}^2\dot{\theta }=L/(2\mu )$ = constant. This is angular momentum conservation — it holds for any central force, not only inverse-square.

3. The period-orbit relation ($T^2\propto a^3$): for an ellipse with semi-major axis $a$, the area is $\pi ab=\pi a^2\sqrt{1-e^2}$. The period is $T=\text{Area}/(dA/dt)=2\pi \mu a^2\sqrt{1-e^2}/L$. Using $p=a(1-e^2)=L^2/(\mu k)$ to eliminate $L$ gives $T^2=4{\pi }^2\mu a^3/k$. For the gravitational case $k=G{m}_1{m}_2$ and $\mu \approx m_2$ when $m_1\gg m_2$ (Sun-planet), this reduces to Kepler's third law: $T^2=4{\pi }^2{a}^3/(G{m}_1)$.

Proposition (Bertrand's theorem — statement). Among all central-force potentials $V(r)$ producing bound orbits, only $V\propto -1/r$ (Kepler) and $V\propto r^2$ (harmonic oscillator) yield closed orbits for all initial conditions. The proof, due to Bertrand 1873 ^{[Bertrand 1873]}, analyses the stability of nearly-circular orbits and shows that the precession rate of the perihelion vanishes for the entire range of radii only for these two potentials. For all other potentials, generic bound orbits are open rosettes that never exactly repeat — a fact with direct physical consequences: the observed precession of Mercury's perihelion, after accounting for perturbations from other planets, was the key anomaly that general relativity resolved.

The Laplace-Runge-Lenz vector. The Kepler problem possesses a conserved quantity beyond energy and angular momentum: the Laplace-Runge-Lenz vector

$$\mathbf{A}=\mathbf{p}\times \mathbf{L}-\mu k\hat{\mathbf{r}},$$

where $\mathbf{p}=\mu \dot{\mathbf{r}}$ and $k=G{m}_1{m}_2$ (gravitational) or $k=|q_1{q}_2|/(4\pi {ϵ}_0)$ (Coulomb). A direct computation using $\dot{\mathbf{p}}=-k\hat{\mathbf{r}}/r^2$ and $\dot{\mathbf{L}}=0$ confirms $\dot{\mathbf{A}}=0$. The vector $\mathbf{A}$ points from the force centre to the perihelion of the orbit, and its magnitude is $A=\mu ke$ where $e$ is the eccentricity. For circular orbits ($e=0$), $\mathbf{A}$ vanishes.

The existence of this additional conserved vector is not a consequence of a spacetime symmetry (Galilean or rotational). Instead, it reflects the dynamical symmetry of the inverse-square problem: the Hamiltonian is invariant under the group $\mathrm{SO}(4)$ (for $E<0$) or $\mathrm{SO}(3,1)$ (for $E>0$), larger than the rotational $\mathrm{SO}(3)$. The six generators of $\mathrm{SO}(4)$ are the three components of $\mathbf{L}$ and the three components of a suitably normalised $\mathbf{A}$; the algebra they satisfy — $[L_i,L_j]=i\hslash {ϵ}_{ijk}{L}_k$ and $[A_i,L_j]=i\hslash {ϵ}_{ijk}{A}_k$ in the quantum version — is $\mathfrak{so}(4)\cong \mathfrak{su}(2)\oplus \mathfrak{su}(2)$. This larger symmetry explains the "accidental degeneracy" of the hydrogen atom energy levels (the $n^2$-fold degeneracy that depends only on the principal quantum number $n$, not on the angular momentum $l$), and appears again in 09.04.01 pending the Legendre transform treatment of the Kepler problem. The Runge-Lenz vector is the classical counterpart of the quantum-mechanical degeneracy observed in the hydrogen spectrum 12.06.01 pending.

Rigid body dynamics and Euler's equations [Master]

A rigid body is a system of $N$ particles (in the continuum limit, a mass distribution $\rho (\mathbf{r})$) subject to the holonomic constraint that all inter-particle distances are constant. Newton's laws apply to each particle; the constraints introduce internal reaction forces that maintain the rigidity condition. The resulting dynamics are governed by a remarkably rich structure — the configuration space is the rotation group $\mathrm{SO}(3)$, and the equations of motion are Euler's equations on its Lie algebra $\mathfrak{so}(3)$.

Angular velocity and angular momentum. Fix a body-fixed frame $\{e_1^{\prime },e_2^{\prime },e_3^{\prime }\}$ rigidly attached to the body and a space-fixed inertial frame $\{e_1,e_2,e_3\}$. The orientation of the body is encoded by a rotation matrix $R(t)\in \mathrm{SO}(3)$ mapping body coordinates to space coordinates: ${\mathbf{r}}_{\text{space}}=R(t){\mathbf{r}}_{\text{body}}$. The angular velocity $\mathbf{\omega }$ is defined by $\dot{R}=[\mathbf{\omega }{]}_{\times }R$, where $[\mathbf{\omega }{]}_{\times }$ is the $3\times 3$ skew-symmetric matrix with $[\mathbf{\omega }{]}_{\times }\mathbf{v}=\mathbf{\omega }\times \mathbf{v}$. Equivalently, $\mathbf{\omega }$ is the unique vector satisfying ${\dot{\mathbf{r}}}_i=\mathbf{\omega }\times {\mathbf{r}}_i$ for every point ${\mathbf{r}}_i$ fixed in the body.

The total angular momentum about the origin is

$$\mathbf{L}=\sum _i{m}_i{\mathbf{r}}_i\times {\dot{\mathbf{r}}}_i=\sum _i{m}_i{\mathbf{r}}_i\times (\mathbf{\omega }\times {\mathbf{r}}_i).$$

Using the vector identity $\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\mathbf{b}(\mathbf{a}\cdot \mathbf{c})-\mathbf{c}(\mathbf{a}\cdot \mathbf{b})$, this becomes $\mathbf{L}=\mathbf{I}\mathbf{\omega }$, where $\mathbf{I}$ is the inertia tensor with components

$$I_{jk}=\sum _i{m}_i(r_i^2{\delta }_{jk}-r_{i,j}{r}_{i,k}),\text{or continuously}{I}_{jk}=\int \rho (\mathbf{r})(r^2{\delta }_{jk}-r_j{r}_k)d^{3}r.$$

The inertia tensor is a real symmetric positive-semidefinite $3\times 3$ matrix. By the spectral theorem, there exists an orthonormal basis — the principal axes — in which $\mathbf{I}=\mathrm{diag}(I_1,I_2,I_3)$ with $I_1\ge I_2\ge I_3>0$ (for a body with nonzero volume). The diagonal entries are the principal moments of inertia. Choosing the body-fixed frame aligned with the principal axes simplifies all subsequent calculations.

Euler's equations. Newton's second law for angular motion is $\dot{\mathbf{L}}=\mathbf{\tau }$, where $\mathbf{\tau }={\sum }_i{\mathbf{r}}_i\times {\mathbf{F}}_i$ is the total external torque. In the body frame, the time derivative acquires a transport term: ${\dot{\mathbf{L}}}_{\text{space}}={\dot{\mathbf{L}}}_{\text{body}}+\mathbf{\omega }\times \mathbf{L}$. Writing $\mathbf{L}=\mathbf{I}\mathbf{\omega }$ in the principal-axis frame yields Euler's equations:

$$I_1{\dot{\omega }}_1=(I_2-I_3){\omega }_2{\omega }_3+{\tau }_1,I_2{\dot{\omega }}_2=(I_3-I_1){\omega }_3{\omega }_1+{\tau }_2,I_3{\dot{\omega }}_3=(I_1-I_2){\omega }_1{\omega }_2+{\tau }_3.$$

These are the equations of motion for the angular velocity of a rigid body in its own body frame. For a torque-free rigid body ($\mathbf{\tau }=0$, i.e., no external forces or forces acting at the centre of mass), the right-hand sides contain only the nonlinear coupling terms between angular velocity components.

Stability of torque-free rotation. The three equilibrium solutions of torque-free Euler's equations are rotation about each principal axis at constant angular velocity. Linearising about rotation around axis $k$ with ${\omega }_k=\Omega$, ${\omega }_j=0$ for $j\ne k$, gives a system whose eigenvalues determine stability:

• Rotation about the axis of largest moment $I_1$: stable (oscillatory perturbations, eigenvalues pure imaginary).
• Rotation about the axis of smallest moment $I_3$: stable (oscillatory perturbations).
• Rotation about the axis of intermediate moment $I_2$: unstable (one positive real eigenvalue, exponential divergence).

This is the intermediate-axis theorem (also called the tennis racket theorem or Dzhanibekov effect). A rigid body spinning freely about its intermediate axis cannot maintain that rotation: any perturbation, no matter how small, grows exponentially and the body tumbles. The effect was dramatically observed in microgravity by cosmonaut Vladimir Dzhanibekov in 1985 and is a direct consequence of the structure of Euler's equations — the nonlinear $(I_j-I_k){\omega }_j{\omega }_k$ coupling terms that make the intermediate axis a saddle point of the energy surface in angular-velocity space.

Proposition (Euler's equations conserve energy and angular momentum magnitude). For torque-free motion, the kinetic energy $T=\frac{1}{2}(I_1{\omega }_1^2+I_2{\omega }_2^2+I_3{\omega }_3^2)$ and the squared angular momentum $L^2=I_1^2{\omega }_1^2+I_2^2{\omega }_2^2+I_3^2{\omega }_3^2$ are both constants of motion.

Proof. $\dot{T}=I_1{\omega }_1{\dot{\omega }}_1+I_2{\omega }_2{\dot{\omega }}_2+I_3{\omega }_3{\dot{\omega }}_3$. Substituting Euler's equations (with ${\tau }_i=0$): $2\dot{T}=(I_2-I_3){\omega }_1{\omega }_2{\omega }_3+(I_3-I_1){\omega }_1{\omega }_2{\omega }_3+(I_1-I_2){\omega }_1{\omega }_2{\omega }_3=0$. Similarly, $\frac{d}{dt}(L^2)=2(I_1^2{\omega }_1{\dot{\omega }}_1+\cdots )=2(I_1(I_2-I_3)+I_2(I_3-I_1)+I_3(I_1-I_2)){\omega }_1{\omega }_2{\omega }_3=0$. $\square$

The intersection of the constant-$T$ ellipsoid and the constant-$L^2$ sphere in $({\omega }_1,{\omega }_2,{\omega }_3)$-space determines the polhode curves — the trajectories of the angular velocity vector in the body frame. These curves are closed (periodic motion) for stable axes and open (asymptotic to the unstable axis) for the intermediate axis. The corresponding trajectory of the angular momentum vector in the space frame — the herpolhode — gives the wobbling motion observed from outside. For an axisymmetric body ($I_1=I_2\ne I_3$), the polhode is a circle and the body precesses uniformly about its symmetry axis: this is the mechanism behind the steady precession of a spinning top, the gyroscope, and (to first approximation) the Earth's axial precession with period $\sim 26,000$ years.

Newtonian mechanics as a geometric theory [Master]

The intermediate treatment phrases Newton's second law as $m\ddot{\mathbf{r}}=\mathbf{F}(\mathbf{r},\dot{\mathbf{r}},t)$ in ${\mathbb{R}}^3$. The geometric reformulation replaces ${\mathbb{R}}^3$ by a Riemannian manifold $(Q,g)$ where $Q$ is the configuration space and $g$ encodes the kinetic energy via $T=\frac{1}{2}g(\dot{\gamma },\dot{\gamma })$.

Newton's equations on a Riemannian manifold. Let $\gamma :[t_0,t_1]\to Q$ be a curve in the configuration manifold and $\nabla$ the Levi-Civita connection of $g$. The Newton equation is

$$m{\nabla }_{\dot{\gamma }}\dot{\gamma }=\mathbf{F}(\gamma ,\dot{\gamma },t),$$

where ${\nabla }_{\dot{\gamma }}\dot{\gamma }$ is the covariant acceleration. In local coordinates $(q^i)$ with Christoffel symbols ${\Gamma }_{jk}^i$:

$$m({\ddot{q}}^i+{\Gamma }_{jk}^i{\dot{q}}^j{\dot{q}}^k)=F^i.$$

When $\mathbf{F}=0$, the equation reduces to ${\nabla }_{\dot{\gamma }}\dot{\gamma }=0$: free particles follow geodesics of $(Q,g)$. Newton's first law is the geodesic equation on a flat (Euclidean) manifold. On a curved manifold, the Christoffel terms encode the "fictitious forces" — they are the geometric manifestation of using non-Cartesian coordinates or of genuine manifold curvature.

Galilean invariance. Newton's equations are invariant under the Galilean group: time translations $t\to t+t_0$, spatial rotations $\mathbf{r}\to R\mathbf{r}$ ($R\in \mathrm{SO}(3)$), spatial translations $\mathbf{r}\to \mathbf{r}+\mathbf{a}$, and Galilean boosts $\mathbf{r}\to \mathbf{r}+{\mathbf{v}}_{0}t$. The boost invariance requires forces to depend on relative positions and velocities, not absolute ones. This is the physical content of Galileo's relativity principle: the laws of mechanics take the same form in all inertial frames.

The Galilean group is a 10-dimensional Lie group (1 time translation + 3 rotations + 3 spatial translations + 3 boosts). By Noether's theorem (developed in 09.03.01 pending), each continuous symmetry corresponds to a conserved quantity: time translation gives energy, rotations give angular momentum, spatial translations give linear momentum. Galilean boosts give the theorem that the centre of mass of an isolated system moves at constant velocity.

The third law as a statement about forces, not geometry. Unlike the first two laws, the third law is not naturally a geometric statement. It is a constraint on the form of inter-particle forces: internal forces come in equal-opposite pairs along the line of centres. In the geometric formulation on $Q$, the third law is encoded by requiring the total force 1-form to be the differential of a potential: $\mathbf{F}=-dV$. When this holds, the system is conservative, and the equations of motion are the geodesic equation of the Jacobi metric on the energy surface — the entry point to the Lagrangian formulation in 09.02.01 pending.

Limits of Newtonian mechanics. Newton's laws presuppose: (i) instantaneous action at a distance (gravitational force propagates without delay); (ii) absolute time (simultaneity is frame-independent); (iii) point particles with definite positions and velocities. All three assumptions are violated in special relativity (no instantaneous action, no absolute simultaneity) and quantum mechanics (no definite joint position and velocity). The Newtonian framework is recovered as the low-velocity, macroscopic limit of both SR and QM, and its domain of validity is the regime where $v\ll c$ and action $\gg \hslash$.

Synthesis. The geometric reformulation is the foundational reason that Newton's laws generalise beyond ${\mathbb{R}}^3$ to arbitrary configuration manifolds — the central insight is that the second law $m{\nabla }_{\dot{\gamma }}\dot{\gamma }=\mathbf{F}$ identifies dynamics with geodesic motion corrected by external force 1-forms on a Riemannian manifold. This is exactly the bridge to the Lagrangian formulation 09.02.01 pending, where the geodesic-plus-force structure emerges as the Euler-Lagrange equation for the action functional on $TQ$. Putting these together with the Galilean symmetry group (10-dimensional Lie group acting on space-time), the conserved quantities of 09.01.03 follow from Noether's theorem 09.03.01 pending and the pattern recurs in Hamiltonian mechanics 09.04.02 pending, where the second law becomes the flow of the Hamiltonian vector field $X_H$ on the symplectic manifold $T^{\ast }Q$. The rigid body, the two-body Kepler problem, and the virial theorem are all instances of this single geometric framework: specify the configuration manifold, the metric (kinetic energy), and the force 1-form (potential), and Newton's law determines the integral curves.

Connections [Master]

• Kinematics 09.01.01 provides the language (position, velocity, acceleration) that Newton's second law relates to force. Without the kinematic definitions, $F=ma$ has no content.

• Conservation laws 09.01.03 (pending) follow from Newton's second and third laws applied to isolated systems: the third law guarantees internal forces cancel in the total momentum sum, giving conservation of total momentum.

• The action principle 09.02.01 pending (pending) reformulates Newton's second law as the Euler-Lagrange equation of a variational principle, generalising $F=ma$ to configuration manifolds with constraints.

• Noether's theorem 09.03.01 pending (pending) explains the conserved quantities (energy, momentum, angular momentum) as consequences of the Galilean symmetries of Newton's equations, rather than as separate empirical facts.

• Hamilton's equations 09.04.02 pending restate Newton's second law as a first-order system on phase space, replacing force by the Hamiltonian and enabling the transition to quantum mechanics.

• Maxwell's equations 10.04.01 pending and the Lorentz force extend Newton's second law to charged particles in electromagnetic fields: $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$.

• The measurement problem 20.03.01 pending contrasts Newtonian determinism (exact prediction from initial data) with quantum indeterminacy (probabilistic outcomes from identical preparations).

Historical & philosophical context [Master]

Newton published the three laws in Philosophiae Naturalis Principia Mathematica (1687), under the heading Axiomata sive Leges Motus (Axioms, or Laws of Motion). The first law (inertia) was not original to Newton; it appears in Galileo's Discorsi (1638) and was stated by Descartes in Principia Philosophiae (1644, Part II, §37). Newton's contribution was to place it as an axiom alongside the second and third laws, forming a complete deductive framework for mechanics.

The second law ($F=ma$, or more precisely $F=dp/dt$) was the genuine innovation. In the Principia Newton states it in terms of impulses: "Mutationem motus proportionalem esse vi motrici impressae" — the change of motion is proportional to the motive force impressed. The modern differential form emerged through the work of Euler (Mechanica, 1736), who first wrote Newton's second law as a differential equation, and d'Alembert (Traite de dynamique, 1743), who reformulated it in terms of accelerations and constraints.

The third law has the deepest physical content. Newton tested it experimentally using colliding pendulums, demonstrating that momentum transfers were equal and opposite. The strong form (forces directed along the line of centres) was needed to explain the observed conservation of angular momentum in planetary orbits. The weak form alone (equal and opposite, not necessarily central) is sufficient for linear momentum conservation but not for angular momentum conservation.

The philosophical significance of the three laws is that they reduce the problem of motion to the problem of force. Once you know the force law (gravitational, electromagnetic, spring, friction), the second law determines the motion uniquely from initial conditions. This reduction — dynamics reduced to kinematics plus force specification — set the template for all subsequent physical theories. Lagrangian and Hamiltonian mechanics reformulate the template; Maxwell's equations specify the electromagnetic force law; general relativity modifies the gravitational one; quantum mechanics reinterprets the determinism. But the basic architecture — specify the interaction, derive the motion — is Newton's.

Bibliography [Master]

• Newton, I., Philosophiae Naturalis Principia Mathematica (1687), Axiomata sive Leges Motus. [Originator.]
• Galilei, G., Discorsi e dimostrazioni matematiche intorno a due nuove scienze (1638).
• Descartes, R., Principia Philosophiae (1644), Part II.
• Euler, L., Mechanica sive motus scientia analytice exposita (1736).
• d'Alembert, J. le R., Traite de dynamique (1743).
• Taylor, J. R., Classical Mechanics (University Science Books, 2005), Ch. 1–2.
• Susskind, L. & Hrabovsky, G., The Theoretical Minimum: Classical Mechanics (Basic Books, 2014), Lecture 2–3.
• Arnold, V. I., Mathematical Methods of Classical Mechanics, 2nd ed. (Springer GTM 60, 1989).
• Landau, L. D. & Lifshitz, E. M., Mechanics, 3rd ed. (Course of Theoretical Physics Vol. 1, Pergamon, 1976).
• Goldstein, H., Poole, C. P. & Safko, J., Classical Mechanics, 3rd ed. (Pearson, 2002), Ch. 1.
• Tong, D., Classical Dynamics (DAMTP Cambridge lecture notes), §1.
• Feynman, R. P., Leighton, R. B. & Sands, M., The Feynman Lectures on Physics, Vol. I, Ch. 9–10 (Addison-Wesley, 1963).