Classical Mechanics: Newtonian Mechanics & Conservation Laws

Classical Mechanics » Newtonian Mechanics & Conservation Laws

Newton’s three laws, kinematics and dynamics, work and energy, conservation laws, rotational motion, gravitation and central forces, and a first look at oscillations.

The Foundation: Newton’s Revolution

Newton’s insight was that motion follows precise mathematical laws. This section builds the force-based picture in the order it is most naturally learned: the three laws that define force and inertia, the kinematics and dynamics they govern, the conservation laws that act as shortcuts, the extension from point particles to spinning bodies, and finally gravitation and the central-force problem that crowned Newton’s program — with a closing glimpse of oscillations.

Newton’s Laws of Motion

Paper: Philosophiæ Naturalis Principia Mathematica - Isaac Newton

Video: Newton's Laws of Motion Explained

The foundation of classical mechanics is built upon Newton’s three laws of motion:

Article: Newton's Laws of Motion - Wikipedia

First Law (Law of Inertia)

An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

Mathematical Expression: $\text{If } \sum \vec{F} = 0, \text{ then } \vec{v} = \text{constant}$

The first law is more than a special case of the second — it asserts the existence of inertial reference frames in which it holds, the privileged frames against which all of Newtonian mechanics is written.

Second Law (Law of Acceleration)

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

Mathematical Expression: $\vec{F} = m\vec{a}$ Where:

$\vec{F}$ = net force (N)
$m$ = mass (kg)
$\vec{a}$ = acceleration (m/s²)

In its more fundamental form the second law equates force with the rate of change of momentum, $\vec{F} = d\vec{p}/dt$, which reduces to $m\vec{a}$ for constant mass and remains correct for systems that gain or lose mass (rockets).

Third Law (Action-Reaction)

For every action, there is an equal and opposite reaction.

Mathematical Expression: $\vec{F}_{12} = -\vec{F}_{21}$

The third law is exactly what makes the total momentum of an isolated system conserved: internal force pairs cancel, so only external forces can change the system’s momentum.

Why Newton’s Laws Matter

These three laws explain a wide range of phenomena: why you lurch forward when a car brakes (First Law), how rockets work in vacuum (Third Law), and why heavier objects don’t fall faster (Second Law with gravity). Their limitation is that you must know all the forces. When forces are complicated, or when you care only about motion, the energy methods and conservation laws below — and ultimately the analytical Lagrangian and Hamiltonian reformulations — become more powerful.

Kinematics: Describing Motion

From Observation to Mathematics

Kinematics describes where things are and how fast they move, without asking why (that is dynamics). It is the language of motion: position, velocity, and acceleration.

One-Dimensional Motion

Position: $x(t)$

Velocity: $v = \dfrac{dx}{dt}$

Acceleration: $a = \dfrac{dv}{dt} = \dfrac{d^2x}{dt^2}$

Equations of Motion (Constant Acceleration)

$v = v_0 + at$
$x = x_0 + v_0 t + \tfrac{1}{2}at^2$
$v^2 = v_0^2 + 2a(x - x_0)$
$x = x_0 + \tfrac{1}{2}(v + v_0)t$

Projectile Motion

Interactive: Projectile Motion Simulator - PhET

For projectile motion under constant gravitational acceleration:

Horizontal Motion: $\begin{aligned} x &= v_{0x}\,t \\ v_x &= v_{0x} \quad (\text{constant}) \end{aligned}$

Vertical Motion: $\begin{aligned} y &= v_{0y}\,t - \tfrac{1}{2}gt^2 \\ v_y &= v_{0y} - gt \end{aligned}$

Range Formula: $R = \frac{v_0^2 \sin(2\theta)}{g}$

Maximum Height: $H = \frac{v_0^2 \sin^2\theta}{2g}$

Dynamics: Forces in Action

Connecting Force to Motion

Dynamics explains motion, with Newton’s second law ($F = ma$) as the primary tool. Often, though, thinking in terms of energy is more elegant — and it leads directly to the conservation laws of the next section.

Work and Energy

Work: The energy transferred to or from an object via the application of force along a displacement.

\[W = \vec{F}\cdot\vec{d} = Fd\cos\theta\]

For variable force: $W = \int \vec{F}\cdot d\vec{r}$

The work–energy theorem ties this directly back to kinematics: the net work done on a particle equals its change in kinetic energy,

\[W_{\text{net}} = \Delta KE = \tfrac{1}{2}mv_f^2 - \tfrac{1}{2}mv_i^2.\]

Power

Power is the rate at which work is done:

\[P = \frac{dW}{dt} = \vec{F}\cdot\vec{v}\]

Conservative Forces and Potential Energy

A force is conservative if the work it does is independent of the path — equivalently, if it can be written as the gradient of a potential energy, $\vec{F} = -\nabla U$. Gravity near the surface ($U = mgh$) and an ideal spring ($U = \tfrac{1}{2}kx^2$) are the archetypes. For conservative forces the work–energy theorem becomes the statement that mechanical energy is conserved, the bridge to the next section.

Conservation Laws: Nature’s Hidden Symmetries

Lecture: Conservation of Energy - Feynman Lectures

Rather than tracking forces at every instant, conservation laws give quantities that stay constant throughout the motion — a powerful shortcut. These laws are not arbitrary; each reflects a deep symmetry of nature:

Click to expand: Interactive Conservation Laws Demonstration

Conservation laws are fundamental principles that remain constant in isolated systems:

Energy Conservation: Total energy (kinetic + potential) remains constant
Momentum Conservation: Total momentum is conserved in collisions
Angular Momentum Conservation: Rotational momentum is preserved

Interactive: Energy Skate Park Simulation

Conservation of Energy

The total energy of an isolated system remains constant over time.

Types of Energy:

Kinetic Energy: $KE = \tfrac{1}{2}mv^2$
Potential Energy: $PE = mgh$ (gravitational)
Elastic Potential Energy: $PE = \tfrac{1}{2}kx^2$

Conservation Equation: $\begin{aligned} E_{\text{initial}} &= E_{\text{final}} \\ KE_i + PE_i &= KE_f + PE_f \end{aligned}$

Conservation of Momentum

Tutorial: Momentum Conservation in Collisions

The total momentum of an isolated system remains constant.

Linear Momentum: $\vec{p} = m\vec{v}$

Conservation Equation: $\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}}$

This follows directly from Newton’s third law: internal forces cancel in pairs, so in the absence of external forces the system’s total momentum cannot change. Momentum conservation is the master tool for analyzing collisions, recoil, and rocket propulsion.

Conservation of Angular Momentum

The total angular momentum of an isolated system remains constant.

Angular Momentum: $\vec{L} = I\vec{\omega} = \vec{r} \times \vec{p}$

Where:

$I$ = moment of inertia
$\vec{\omega}$ = angular velocity
$\vec{r}$ = position vector
$\vec{p}$ = linear momentum

(The rotational quantities $I$ and $\vec{\omega}$ are developed in the next section; we list angular momentum here to keep the three conservation laws together.)

These conservation laws are not coincidences; each reflects a symmetry of nature — a connection made precise by Noether’s theorem in the Lagrangian and Hamiltonian formulation. Energy conservation reflects symmetry under translation in time, linear momentum reflects symmetry under translation in space, and angular momentum reflects symmetry under rotation.

The Power of Conservation

Conservation laws often provide shortcuts to solving problems. Consider a figure skater spinning: when she pulls her arms in, she spins faster. Why? Angular momentum ($L = I\omega$) must stay constant. Decreasing $I$ (moment of inertia) means $\omega$ (angular velocity) must increase. No forces needed — just conservation!

But even conservation laws have limitations when dealing with constraints and complex systems. This motivates the analytical formulations of Lagrange and Hamilton.

Rotational Motion: Beyond Point Particles

Why Rotation Matters

Real objects have size and shape: they spin, roll, and tumble. The key discovery is that rotational motion follows the same patterns as linear motion, with each linear quantity having a rotational analog:

Mermaid Diagram: Linear vs Rotational Motion Analogy

```mermaid graph LR subgraph Linear Motion A[Position x] --> B[Velocity v = dx/dt] B --> C[Acceleration a = dv/dt] D[Mass m] --> E[Force F = ma] F[Momentum p = mv] end subgraph Rotational Motion G[Angular Position θ] --> H[Angular Velocity ω = dθ/dt] H --> I[Angular Acceleration α = dω/dt] J[Moment of Inertia I] --> K[Torque τ = Iα] L[Angular Momentum L = Iω] end A -.->|Analogy| G B -.->|Analogy| H C -.->|Analogy| I D -.->|Analogy| J E -.->|Analogy| K F -.->|Analogy| L ```

Rotational Kinematics

Analogous to linear motion:

Angular position: θ
Angular velocity: ω = dθ/dt
Angular acceleration: α = dω/dt

Moment of Inertia

Reference: Moment of Inertia Tables

The rotational equivalent of mass:

Point Mass: $I = mr^2$

Common Shapes:

Solid sphere: $I = \tfrac{2}{5}MR^2$
Solid cylinder: $I = \tfrac{1}{2}MR^2$
Thin rod (about center): $I = \tfrac{1}{12}ML^2$

Torque

The rotational equivalent of force:

\[\tau = r \times F = rF \sin(\theta)\]

Rotational Newton’s Second Law: $\tau = I\alpha$

Just as $\vec{F} = d\vec{p}/dt$ for translation, the rotational law in its general form is $\vec{\tau} = d\vec{L}/dt$: a torque is the rate of change of angular momentum, and zero net torque means angular momentum is conserved — the spinning-skater result above.

For a full treatment of three-dimensional rotation — the inertia tensor, Euler’s equations, gyroscopic precession, and rigid bodies — see Rigid Body Dynamics.

Gravitation and Central Forces: The First Unified Theory

From Falling Apples to Orbiting Planets

Newton’s greatest triumph wasn’t just explaining how things move — it was recognizing that the force pulling an apple to Earth is the same force keeping the Moon in orbit. This was humanity’s first “unified theory,” connecting terrestrial and celestial mechanics. Because gravitation is a central force, the rotational machinery just developed — angular momentum and torque — is exactly what makes it tractable.

Newton’s Law of Universal Gravitation

Paper: Principia - Book III: The System of the World

Tutorial: Understanding Universal Gravitation

Every particle attracts every other particle with a force:

\[F = \frac{Gm_1m_2}{r^2}\]

Where:

G = 6.674 × 10⁻¹¹ N·m²/kg² (gravitational constant)
m₁, m₂ = masses of the objects
r = distance between centers

Orbital Motion

For circular orbits:

Orbital Velocity: $v = \sqrt{\frac{GM}{r}}$

Orbital Period: $T = 2\pi\sqrt{\frac{r^3}{GM}}$

This is Kepler’s Third Law for circular orbits.

The Effective Potential: Reducing Orbits to One Dimension

Real orbits are rarely circular. The deeper insight is that any motion in a central force $-\dfrac{GMm}{r^2}\hat{r}$ conserves angular momentum $L = mr^2\dot{\phi}$ — the conservation law from the previous section, applied to a torque-free central force — and that single fact lets us collapse a two-dimensional orbit into an equivalent one-dimensional problem in the radius alone. Substituting $\dot{\phi} = L/(mr^2)$ into the energy gives:

\[E = \tfrac{1}{2}m\dot{r}^2 + V_{\text{eff}}(r), \qquad V_{\text{eff}}(r) = -\frac{GMm}{r} + \frac{L^2}{2mr^2}\]

The first term is the genuine gravitational attraction; the second is the centrifugal barrier — a repulsive contribution that arises purely from angular momentum and keeps the orbit from collapsing to the center. A particle now behaves like a bead rolling in the one-dimensional landscape $V_{\text{eff}}(r)$:

Total energy $E$	Shape of orbit
$E < 0$, at minimum of $V_{\text{eff}}$	Circular
$E < 0$, above minimum	Ellipse (bound)
$E = 0$	Parabola (escape, marginally unbound)
$E > 0$	Hyperbola (unbound flyby)

This single picture unifies planets, comets, and interstellar flybys, and it reproduces all of Kepler’s laws. It is also the template for the radial Schrödinger equation of the hydrogen atom, where the same $-1/r$ attraction and $L^2/2mr^2$ barrier reappear — a preview of how classical structure carries into quantum mechanics.

A First Look at Oscillations

Whenever a system sits near a stable equilibrium, a small displacement produces a restoring force, and the system oscillates. The simplest case is Hooke’s law, $F = -kx$, which gives simple harmonic motion:

\[x(t) = A\cos(\omega t + \varphi), \qquad \omega = \sqrt{\frac{k}{m}}, \qquad T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}.\]

Here $A$ is the amplitude and $\varphi$ the phase. Because any smooth potential looks quadratic near a minimum, the harmonic oscillator is the universal lowest-order description of small motions — which is why it reappears throughout physics, from molecular vibrations to quantum field theory.

This is only the entry point. The full development — energy in oscillation, damped and driven (resonant) oscillators, coupled oscillators and normal modes, the chain-to-continuum limit, the wave equation, standing versus traveling waves, dispersion and group velocity, Fourier decomposition, and nonlinear waves and solitons — now lives on its own page:

➡️ Oscillations & Waves — the complete treatment, from one mass on a spring all the way to solitons.

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