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Classical Mechanics

Introduction

Classical mechanics is the framework used to describe the motion of macroscopic bodies when speeds are small compared with the speed of light and quantum effects are negligible. It connects kinematics, which describes motion, with dynamics, which explains motion through interactions. At university level, classical mechanics is also a training ground for mathematical modeling: you translate a physical situation into coordinates, forces, energies, constraints, differential equations, and testable predictions.

You will study three mutually consistent viewpoints. Newtonian mechanics emphasizes forces and momentum. Lagrangian mechanics organizes dynamics through generalized coordinates and the stationary-action principle. Hamiltonian mechanics describes evolution in phase space and makes symmetry, conservation laws, and links to later areas of physics especially transparent.

The Newton's cradle animation is a useful starting point for discussing momentum, kinetic energy, idealization, and the difference between an approximate model and a real dissipative system.

The MIT OpenCourseWare introduction above places classical mechanics in an undergraduate physics curriculum and emphasizes the role of problem solving.

Learning Goals

By the end of this aiMOOC, you should be able to derive and interpret equations of motion; choose appropriate coordinates and reference frames; use force, momentum, energy, and angular momentum methods; analyze oscillations, rigid-body motion, and central-force motion; formulate systems with Lagrangian and Hamiltonian methods; identify conserved quantities from symmetries; use numerical integration when analytic solutions are unavailable; and judge when a classical model is valid.

Prerequisites and Mathematical Language

You should be comfortable with single-variable and multivariable calculus, ordinary differential equations, vectors, basic linear algebra, and introductory physics. In mechanics, a vector such as $𝐫$ has both magnitude and direction. Differentiation with respect to time turns position into velocity and velocity into acceleration:

$𝐯 = \frac{d 𝐫}{d t}, 𝐚 = \frac{d 𝐯}{d t} = \frac{d^{2} 𝐫}{d t^{2}} .$

Dimensional analysis is a first check on every result. An equation must be dimensionally consistent, and limiting cases should agree with simpler situations you already understand.

Kinematics and Reference Frames

Kinematics describes trajectories without asking what causes them. A particle trajectory is a function $𝐫 (t)$ . In Cartesian coordinates, $𝐫 = x \hat{𝐢} + y \hat{𝐣} + z \hat{𝐤}$ . In polar, cylindrical, or spherical coordinates, the basis vectors can vary with position, so differentiating them requires care.

For constant acceleration in one dimension, integration gives $v (t) = v_{0} + a t$ and $x (t) = x_{0} + v_{0} t + \frac{1}{2} a t^{2}$ . These familiar formulas are special cases, not universal laws.

Ideal projectile motion near Earth's surface illustrates vector decomposition. Neglecting air resistance and treating gravitational acceleration as constant, horizontal motion is uniform while vertical motion has constant downward acceleration. The resulting trajectory is parabolic in a Cartesian frame fixed to the ground.

The Stanford lecture above develops vectors, velocity, and acceleration as mathematical foundations for particle mechanics.

Galilean Transformations

Two inertial frames moving at constant relative velocity $𝐕$ are related in Newtonian mechanics by a Galilean transformation. If $𝐫^{'} = 𝐫 - 𝐕 t$ , then $𝐯^{'} = 𝐯 - 𝐕$ and $𝐚^{'} = 𝐚$ . Because acceleration is unchanged, Newton's second law has the same form in all inertial frames when mass and force transform in the classical way.

This invariance has a limited domain of validity. At speeds that are not small compared with the speed of light, special relativity replaces Galilean kinematics.

Newtonian Dynamics

Newton's laws of motion organize the force-based description of motion. The first law identifies inertial frames. The second law is most generally written

$𝐅_{n e t} = \frac{d 𝐩}{d t},$

where $𝐩$ is linear momentum. For constant mass, this becomes $𝐅_{n e t} = m 𝐚$ . The third law states that forces between two interacting bodies occur in equal and opposite pairs in the Newtonian idealization.

A free-body diagram isolates one chosen body and shows only the external forces acting on it. This simple discipline prevents many sign errors and helps distinguish real forces from coordinate-dependent inertial forces.

Modeling Forces

Common idealized forces include gravity near Earth's surface $𝐅_{g} = m 𝐠$ , a linear spring force $𝐅_{s} = - k 𝐱$ , kinetic friction $F_{k} = μ_{k} N$ , and drag forces whose form depends on the flow regime. The universal gravitational interaction between point masses is

$𝐅 = - \frac{G m_{1} m_{2}}{r^{2}} \hat{𝐫} .$

Every force law is a model with assumptions. For example, a Hookean spring is approximately linear only over a range of extensions, and a constant coefficient of friction is an empirical approximation rather than a fundamental law.

Constraints and Degrees of Freedom

A mechanical system may be restricted by rods, surfaces, rolling conditions, or joints. A constraint reduces the number of independent coordinates. If a pendulum bob is attached to a rigid rod of fixed length, two Cartesian coordinates in a plane are replaced by one angle. The number of independent generalized coordinates is the number of degrees of freedom.

Constraint forces can be difficult to eliminate directly in Newtonian equations. This is one reason Lagrangian mechanics is powerful: suitable generalized coordinates can build many constraints into the coordinate choice itself.

Momentum, Impulse, and Collisions

For a system of particles, total linear momentum is $𝐏 = \sum_{i} 𝐩_{i}$ . The impulse delivered by a net force is

$𝐉 = \int_{t_{1}}^{t_{2}} 𝐅_{n e t} d t = Δ 𝐩 .$

If the net external force on a system is zero, total momentum is conserved. Internal forces can redistribute momentum among components without changing the system total.

In a perfectly elastic collision, both total momentum and total kinetic energy are conserved. In an inelastic collision, total momentum can still be conserved for an isolated system, while some kinetic energy is transformed into internal energy, deformation, sound, or other forms.

Center of Mass

The center of mass of particles with masses $m_{i}$ at positions $𝐫_{i}$ is

$𝐑 = \frac{1}{M} \sum_{i} m_{i} 𝐫_{i}, M = \sum_{i} m_{i} .$

Its motion obeys $M 𝐀 = 𝐅_{e x t}$ . This equation separates overall translational motion from internal motion and is central to collision theory, rocket dynamics, and many-body systems.

Work, Energy, and Potential

The work done by a force along a path is

$W = \int_{𝐫_{1}}^{𝐫_{2}} 𝐅 \cdot d 𝐫 .$

For a particle of constant mass, the work-energy theorem gives $W_{n e t} = Δ T$ , where $T = \frac{1}{2} m v^{2}$ . A conservative force can be written as $𝐅 = - \nabla V$ , where $V$ is potential energy.

If only conservative forces do work and the potential has no explicit time dependence, the mechanical energy

$E = T + V$

is conserved. Energy methods often reduce a differential-equation problem to an algebraic relation between position and speed.

Potential-Energy Diagrams and Stability

In one dimension, equilibrium points satisfy $d V / d x = 0$ . A local minimum of $V$ corresponds to stable equilibrium under small perturbations, whereas a local maximum corresponds to unstable equilibrium. Expanding a smooth potential near a stable minimum gives

$V (x) \approx V (x_{0}) + \frac{1}{2} V^{″} (x_{0}) (x - x_{0})^{2},$

which explains why small oscillations around many stable equilibria are approximately harmonic.

Rotation of Rigid Bodies

A Rigid body is an idealization in which distances between material points remain fixed. For rotation about a fixed axis, the kinetic energy is $T_{r o t} = \frac{1}{2} I ω^{2}$ , where the moment of inertia is

$I = \int r_{⊥}^{2} d m .$

The distribution of mass matters: moving mass farther from the axis increases $I$ even if the total mass is unchanged.

Torque is $𝝉 = 𝐫 \times 𝐅$ . Angular momentum is $𝐋 = 𝐫 \times 𝐩$ for a particle, and the general rotational equation is

$𝝉_{e x t} = \frac{d 𝐋}{d t} .$

When external torque vanishes, total angular momentum is conserved. This principle applies to orbiting systems, spinning bodies, and coupled systems, although the relation between $𝐋$ and angular velocity can require the full inertia tensor.

Inertia Tensor and Principal Axes

In three-dimensional rigid-body motion, angular momentum and angular velocity need not be parallel. They are related by the inertia tensor $𝐈$ :

$𝐋 = 𝐈 𝝎 .$

The principal axes are directions in which the inertia tensor is diagonal. Choosing body-fixed principal axes simplifies the rotational equations and leads to Euler's equations for torque-driven or torque-free rotation.

A spinning gyroscope under gravity can precess because the torque changes the direction of its angular momentum. The animation visualizes this geometric relationship.

Central Forces and Orbital Motion

A Central force points along the line connecting a particle to a fixed center and depends only on the radial coordinate for a spherically symmetric conservative interaction. Because $𝝉 = 𝐫 \times 𝐅 = 0$ , angular momentum is conserved. Motion therefore lies in a plane.

For a two-body problem, center-of-mass motion can be separated from relative motion by introducing the reduced mass $μ = m_{1} m_{2} / (m_{1} + m_{2})$ . The relative dynamics then resemble a one-body problem in an effective potential.

Newton's inverse-square gravitational law explains Kepler's laws of planetary motion. For bound two-body gravitational motion, the relative orbit is an ellipse with the center of force at one focus, equal areas are swept in equal times, and the orbital period satisfies $T^{2} = 4 π^{2} a^{3} / [G (m_{1} + m_{2})]$ .

Effective Potential

With conserved angular momentum magnitude $ℓ$ , radial motion in a central potential $V (r)$ can be written using

$V_{e f f} (r) = V (r) + \frac{ℓ^{2}}{2 μ r^{2}} .$

The second term is the centrifugal barrier. Plotting $V_{e f f}$ helps you identify allowed radial regions, circular orbits, turning points, and stability without first solving the full orbit.

Oscillations and Normal Modes

The ideal simple harmonic oscillator obeys

$m \ddot{x} + k x = 0,$

with angular frequency $ω_{0} = \sqrt{k / m}$ and solution $x (t) = A \cos (ω_{0} t + ϕ)$ . Harmonic motion appears broadly because smooth potentials look quadratic near stable minima.

For a simple pendulum of length $L$ , the exact equation is $\ddot{θ} + (g / L) \sin θ = 0$ . The small-angle approximation $\sin θ \approx θ$ gives simple harmonic motion with $ω_{0} = \sqrt{g / L}$ . At larger amplitudes, the period depends on amplitude and the motion is nonlinear.

Damping and periodic driving lead to the model

$m \ddot{x} + b \dot{x} + k x = F_{0} \cos ω t .$

This equation introduces transients, steady-state response, resonance, phase lag, and energy dissipation. Coupled oscillators extend the idea to normal modes, in which the entire system oscillates at characteristic frequencies with fixed relative amplitudes.

Lagrangian Mechanics

Newton's equations are often efficient when forces and Cartesian coordinates are simple. Lagrangian mechanics is usually more flexible when constraints or curvilinear coordinates are important.

Choose generalized coordinates $q_{i}$ and define the Lagrangian $L (q_{i}, {\dot{q}}_{i}, t)$ . For many conservative mechanical systems,

$L = T - V .$

The action is

$S = \int_{t_{1}}^{t_{2}} L d t .$

The physical path makes the action stationary under allowed variations with fixed endpoints. This yields the Euler-Lagrange equations

$\frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{i}}) - \frac{\partial L}{\partial q_{i}} = 0.$

For a simple pendulum, using the angle $θ$ as the single generalized coordinate automatically enforces the fixed-length constraint. The Lagrangian method then produces the nonlinear pendulum equation without solving explicitly for the rod tension.

The Stanford lecture above connects symmetry, conservation laws, stationary action, and Lagrangian methods.

Cyclic Coordinates and Conserved Momenta

The generalized momentum conjugate to $q_{i}$ is

$p_{i} = \frac{\partial L}{\partial {\dot{q}}_{i}} .$

If $L$ does not depend explicitly on a coordinate $q_{i}$ , then $q_{i}$ is cyclic and its conjugate momentum is conserved. This is a practical route from coordinate symmetry to a constant of motion.

This Stanford lecture uses particle transformations and the Lagrangian as tools for more advanced mechanical reasoning.

Hamiltonian Mechanics and Phase Space

Hamiltonian mechanics replaces generalized velocities with generalized momenta. Define

$H (q_{i}, p_{i}, t) = \sum_{i} p_{i} {\dot{q}}_{i} - L .$

When the Legendre transformation is regular, the equations of motion become Hamilton's equations:

${\dot{q}}_{i} = \frac{\partial H}{\partial p_{i}}, {\dot{p}}_{i} = - \frac{\partial H}{\partial q_{i}} .$

For many time-independent mechanical systems with standard kinetic energy, $H = T + V$ and equals the conserved total energy. This identification is common but is not true for every possible Lagrangian.

A system with $n$ degrees of freedom has a $2 n$ -dimensional phase space with coordinates $(q_{i}, p_{i})$ . A physical state corresponds to a point in phase space, and time evolution traces a trajectory. The pendulum phase portrait displays oscillatory and rotational regions separated by a separatrix.

Poisson Brackets

For functions $A (q, p, t)$ and $B (q, p, t)$ , the Poisson bracket is

${A, B} = \sum_{i} (\frac{\partial A}{\partial q_{i}} \frac{\partial B}{\partial p_{i}} - \frac{\partial A}{\partial p_{i}} \frac{\partial B}{\partial q_{i}}) .$

The time evolution of an observable can be written $\dot{A} = {A, H} + \partial A / \partial t$ . Poisson brackets compactly express canonical structure and foreshadow the algebraic language of quantum mechanics.

Symmetry and Conservation Laws

A symmetry is a transformation that leaves the relevant physical description invariant. Noether's theorem provides a deep connection between continuous symmetries of the action and conserved quantities. In familiar mechanical settings, invariance under time translations is associated with energy conservation, spatial translations with linear momentum conservation, and rotations with angular momentum conservation.

This viewpoint unifies conservation laws that may look unrelated in Newtonian calculations. It also gives you a powerful strategy: before solving differential equations, look for symmetry and constants of motion that can reduce the problem.

Non-Inertial Frames

An accelerating or rotating coordinate system is non-inertial. To use Newton-like equations in such a frame, you introduce inertial terms such as the centrifugal and Coriolis forces.

For a frame rotating with angular velocity $𝜴$ , the Coriolis inertial force on a mass moving with rotating-frame velocity $𝐯_{r o t}$ is

$𝐅_{C} = - 2 m 𝜴 \times 𝐯_{r o t} .$

The Coriolis effect is not a new fundamental interaction. It arises because motion is being described in a rotating coordinate system. It is important in geophysical fluid dynamics, rotating machinery, and laboratory turntable experiments.

Nonlinear Dynamics and Chaos

Deterministic equations do not guarantee long-term predictability. Nonlinear systems can display chaos, in which nearby initial states separate rapidly over time. A classical double pendulum is a standard example.

The labeled double pendulum has two angular degrees of freedom and coupled nonlinear equations of motion. Its state requires both angles and their associated velocities or momenta.

Datei:Double Pendulum Animation.webm

The animation shows trajectories beginning from slightly different initial conditions and developing visibly different motions. This is a demonstration of sensitivity to initial conditions, not of randomness in the underlying deterministic equations.

Numerical Mechanics

Many realistic systems cannot be solved in closed form. Numerical methods approximate trajectories by advancing the state in discrete time steps. The explicit Euler method is simple but can accumulate large errors and can behave poorly for oscillatory or Hamiltonian systems. Higher-order Runge-Kutta methods are often more accurate for a given step size, while symplectic integrators are designed to respect the geometric structure of Hamiltonian dynamics and often control long-term energy drift better.

A numerical result is not automatically a physical result. You should test convergence by reducing the time step, monitor conserved quantities when appropriate, compare against analytic limiting cases, and distinguish numerical instability from genuine physical instability.

Model Validation and Limits of Classical Mechanics

A strong mechanics solution states its assumptions. Typical idealizations include point particles, rigid bodies, massless strings, frictionless pivots, uniform gravitational fields, negligible air resistance, and perfectly elastic impacts. Their usefulness depends on the required accuracy and scale.

Classical mechanics is highly successful for many macroscopic systems at ordinary speeds, but it has limits. Special relativity becomes necessary when speeds approach the speed of light. General relativity is required when Newtonian gravity is insufficient, especially in strong gravitational fields or for high-precision relativistic effects. Quantum mechanics becomes essential when quantum behavior cannot be neglected, typically at microscopic scales or when wave properties and quantization dominate.

The most important habit is not memorizing which equation belongs to which chapter. It is learning how to choose a model, identify its symmetries and constraints, derive consequences, test them, and revise the model when observations demand it.

Worked Strategy for Mechanics Problems

A reliable workflow is to define the system boundary, sketch the geometry, choose coordinates, state approximations, identify constraints and symmetries, and then select the formulation that reduces complexity. Use Newton's laws when forces are direct, conservation laws when symmetries eliminate unnecessary detail, Lagrange's equations when constraints and generalized coordinates dominate, and Hamiltonian methods when phase-space structure or canonical variables are central.

After deriving equations, check dimensions, signs, limiting cases, equilibrium behavior, and conservation laws. If you solve numerically, verify convergence and compare the numerical trajectory with at least one analytic or qualitative prediction.

Interactive Tasks

Quiz: Test Your Knowledge

Which equation gives Newton's second law in its most general momentum form? (Net force equals the time derivative of momentum) (!Net force equals momentum divided by time) (!Net force equals kinetic energy times acceleration) (!Net force always equals mass times velocity)

What is conserved for an isolated system with zero net external force? (Total linear momentum) (!Total speed) (!Total kinetic energy in every collision) (!Total acceleration)

What condition characterizes a conservative force in terms of potential energy? (The force is the negative gradient of potential energy) (!The force is always constant) (!The force is proportional to velocity) (!The force is independent of position)

What quantity is conserved when the net external torque on a system is zero? (Total angular momentum) (!Angular velocity) (!Moment of inertia) (!Rotational kinetic energy in every process)

What approximation turns the simple pendulum equation into a harmonic oscillator equation? (The small angle approximation) (!The high speed approximation) (!The rigid body approximation) (!The constant momentum approximation)

For many conservative mechanical systems, what is the standard Lagrangian? (Kinetic energy minus potential energy) (!Kinetic energy plus potential energy) (!Momentum minus force) (!Potential energy divided by time)

What does a cyclic generalized coordinate imply? (Its conjugate momentum is conserved) (!Its acceleration must vanish) (!Its value must be zero) (!Its kinetic energy must be constant)

What variables form canonical coordinates in Hamiltonian phase space? (Generalized coordinates and conjugate momenta) (!Forces and accelerations) (!Masses and energies) (!Positions and temperatures)

Which symmetry is associated with conservation of angular momentum in standard mechanical applications of Noether's theorem? (Rotational symmetry) (!Time translation symmetry) (!Scale symmetry) (!Reflection symmetry)

What does chaos in a deterministic mechanical system imply? (Strong sensitivity to initial conditions) (!Absence of governing equations) (!Violation of energy conservation in every case) (!Truly random forces at every instant)

Memory Game

Impulse	Change in linear momentum
Torque	Rate of change of angular momentum
Lagrangian	Function commonly formed from kinetic energy minus potential energy
Hamiltonian	Generator of canonical time evolution
Separatrix	Phase space boundary separating qualitatively different motions
Reduced mass	Effective mass used in the relative two body problem

Drag and Drop

Match the correct terms.	Topic
Free body diagram	Shows external forces on one isolated body
Energy method	Uses kinetic and potential energy to reduce dynamical calculations
Generalized coordinate	Describes an independent degree of freedom
Phase portrait	Displays trajectories in a state space of coordinates and momenta
Symplectic integrator	Numerically preserves Hamiltonian geometric structure

...

Crossword Puzzle

Momentum	Which conserved vector equals mass times velocity for a nonrelativistic particle of constant mass?
Lagrangian	Which function is used in the Euler Lagrange equations?
Hamiltonian	Which function generates canonical time evolution in phase space?
Inertia	Which property measures resistance to changes in rotational motion about an axis?
Resonance	What phenomenon produces a large steady response near a system's natural frequency?
Separatrix	What phase space curve divides qualitatively different kinds of pendulum motion?

LearningApps

Cloze Text

Classical mechanics describes the

of macroscopic bodies under suitable nonrelativistic conditions. Newton's second law connects net force with the time derivative of

. A force derived from a potential is called

. When external torque vanishes, total

is conserved. Near a stable equilibrium, many systems behave approximately like a

. The Lagrangian formulation uses generalized coordinates and the stationary

principle. A coordinate absent explicitly from the Lagrangian is called

. Hamiltonian mechanics represents a state in

. Continuous symmetries of the action are related to conserved quantities by

. Nonlinear deterministic systems can show sensitive dependence on initial conditions known as

.

Open-Ended Tasks

Easy

Projectile motion investigation: Record a short video of a safe tossed object, track several positions frame by frame, compare the trajectory with the ideal constant-gravity model, and state two likely sources of error.
Free body diagram portfolio: Create four original free-body diagrams for everyday or laboratory situations and explain why each force belongs on the chosen body.
Energy landscape sketch: Draw a potential-energy curve with at least two equilibrium points, classify their stability, and explain the expected motion for three different total energies.
Pendulum measurement: Measure the periods of a simple pendulum at several small amplitudes, estimate gravitational acceleration from your data, and discuss uncertainty.

Standard

Collision analysis project: Film a low-speed collision between carts or pucks, estimate velocities before and after impact, test momentum conservation, and evaluate whether kinetic energy is conserved within uncertainty.
Rigid body inertia experiment: Compare the rotational acceleration of two objects with similar mass but different mass distributions, model their moments of inertia, and relate the measurements to torque.
Numerical oscillator study: Implement a numerical solution of a damped driven oscillator, vary driving frequency, plot steady-state amplitude, and interpret resonance and phase behavior.
Rotating frame demonstration: Design a safe turntable or rotating-platform demonstration, document the observed path from inertial and rotating viewpoints, and explain the Coriolis term.

Advanced

Lagrangian derivation project: Choose a constrained system with at least two generalized coordinates, derive its Lagrangian and Euler-Lagrange equations, and compare the derivation with a Newtonian force analysis.
Central force simulation: Numerically integrate a gravitational two-body or reduced one-body orbit, vary energy and angular momentum, and classify the resulting trajectories using the effective potential.
Hamiltonian phase space study: Construct a phase portrait for a nonlinear oscillator or pendulum, identify fixed points and separatrices, and explain how energy contours organize the motion.
Chaos experiment: Build or simulate a double pendulum, compare trajectories with nearly identical initial conditions, quantify their separation over time, and produce a short scientific video discussing predictability.

Learning Assessment

Model selection assessment: Given a constrained rotating system, justify whether a Newtonian, Lagrangian, or Hamiltonian formulation is most efficient and identify what information each formulation emphasizes.
Conservation law assessment: Analyze a collision followed by rotational motion, determine which quantities are conserved in each stage, and explain why a conserved quantity in one stage may fail to be conserved in another.
Orbital reasoning assessment: Use angular momentum and effective-potential arguments to predict whether a given central-force trajectory can reach the origin, remain circular, or encounter a turning point.
Approximation assessment: Derive the small-angle pendulum approximation, estimate when it becomes inaccurate, and explain how the exact nonlinear equation changes the qualitative prediction for the period.
Symmetry transfer assessment: Identify a continuous symmetry in a mechanical model and use Noether-style reasoning to predict the associated conserved quantity before solving the equations of motion.
Numerical validation assessment: Compare two numerical integrations of the same oscillator at different time steps, diagnose whether disagreement comes from physical sensitivity or numerical error, and propose a validation test.
Limits of theory assessment: For several physical scenarios involving high speed, microscopic scale, strong gravity, and ordinary laboratory motion, decide where classical mechanics is adequate and justify when another theory is required.

Evidence of Learning

Knowledge: You can explain the relationships among force, momentum, work, energy, torque, angular momentum, generalized coordinates, action, and phase space.
Analytical skills: You can derive equations of motion, use conservation laws, linearize near equilibrium, and interpret effective potentials and phase portraits.
Modeling skills: You can state assumptions, choose coordinates and reference frames, estimate uncertainties, and identify the domain in which an idealized model is credible.
Computational skills: You can implement and validate numerical time integration, inspect conserved quantities, test step-size convergence, and distinguish numerical artifacts from physical behavior.
Products: Your evidence can include derivations, laboratory reports, annotated diagrams, simulations, data visualizations, scientific videos, and reproducible computational notebooks.
Transfer: You can apply the same mechanics principles to unfamiliar systems in engineering, astronomy, biomechanics, robotics, geophysics, and later courses in theoretical physics.

OERs on the Topic

For further university-level study, the open MIT course MIT OpenCourseWare provides lecture videos, worked examples, and problem sets for undergraduate classical mechanics: MIT 8.01SC Classical Mechanics.

The Stanford lectures embedded in this aiMOOC provide a complementary route into vectors, action principles, and analytical mechanics.