Theoretical Classical Mechanics: From Beginner to Expert

Explore kinematics in theoretical classical mechanics, focusing on velocities, positions, accelerations, and trajectories on straight lines or circles, without discussing forces, with prerequisites in differentiation, integration, and vectors.

Explore kinematics by describing motion with the position vector r(t) and its derivatives, and learn to characterize trajectories using velocity and acceleration.

Explore uniform motion in one dimension: an object on the x axis moves with constant velocity, so displacement grows linearly with time and acceleration is zero.

Define velocity as the derivative of position with respect to time and use average velocity from delta x over delta t; use a tangent line for instantaneous velocity. Define acceleration as the derivative of velocity, or the second derivative of position.

Shows how derivatives give velocity from position and how integrals recover position from velocity via area under the velocity curve, with integration constants.

Explore how differentiation and integration link velocity, position, and acceleration in a kinematics example, using a polynomial velocity model to compute position and analyze the graphs.

Explore motion with constant acceleration, showing velocity grows linearly as v0 + a t and displacement grows quadratically as x0 + v0 t + (1/2) a t^2.

Explore the superposition principle to analyze motion in multiple dimensions, decoupling x and y components under gravity to derive velocity, position, and the throw parabola.

Practice what you’ve learned with a multiple choice quiz and exercises, and optionally review the solution video to reinforce how to understand classical mechanics through doing.

Apply kinematics to a roller coaster by deriving acceleration from the given velocity function and determine if it exceeds one g, using analytic calculation or plotting.

Explore circular motion in polar coordinates with a constant radius, using the polar angle phi, angular velocity omega, and angular acceleration alpha to relate angular and linear motion.

Explore uniform circular motion with constant angular velocity and zero angular acceleration. Derive x equals r cos phi and y equals r sin phi, and emphasize periodic motion.

Explore constant angular acceleration in circular motion, deriving omega = alpha t + omega0 and theta = 1/2 alpha t^2 + omega0 t + theta0.

Explore a difficult pendulum in circular motion, reveal kinematics limits, derive angular acceleration from gravity in polar coordinates, discuss differential equations and starting conditions, including overshoot and the harmonic oscillator.

Explore the harmonic oscillator through a simplified angle transformation, derive the small-angle differential equation, and reveal how sinusoidal solutions arise under Newtonian mechanics.

Explore rotational kinematics on a roundabout with a 5 m radius, starting at 10 m/s and constant angular deceleration of -0.1 rad/s²; calculate stop time and total rotations.

Analyze a roundabout with constant angular acceleration and starting angular velocity omega0; derive omega(t) = omega0 + alpha t and compute the total angle phi to show about three rotations.

Describe how to characterize an object's trajectory using position, velocity, and acceleration, and relate these through derivatives and integrals, with straight-line and circular motion examples.

Learn differentiation and integration in multiple dimensions, including line integrals and non-cartesian coordinate systems. Prepare for dynamics by mastering these math basics, with polar coordinates and circular-motion concepts.

Explore derivatives in functions of multiple variables, from one-dimensional derivatives to partial derivatives, directional derivatives, and the gradient, using examples and the novela operator.

Learn how the nabla operator enables gradient, divergence, curl, and the Laplace operator, with vector field examples and physical interpretations like radial fields and currents.

Derive and verify gradient, curl, and divergence for three-dimensional functions, and prove curl of gradient and divergence of curl vanish. Analyze the vector field H = R/|R|^3 and related identities.

Extend one-dimensional integrals to multidimensional cases by using variable-dependent boundaries, turning area under a curve into volume and mass via density functions.

Explore line integrals in three-dimensional space, projecting paths into one dimension to evaluate scalar and vector integrals along curves, including closed paths.

Explore Cartesian, polar, cylindrical, and spherical coordinate systems, highlighting when to use each for simple integration and spherical symmetry, and apply spherical integration to a sphere's volume and surface area.

Derive the spherical volume element and surface element, and apply them to compute the sphere’s volume and area, illustrating the benefits of spherical coordinates in integration.

Learn how Taylor expansion uses derivatives and factorials to form a convergent series that approximates functions, like e^x around zero, with truncation and applications to potential energy.

Consolidate your understanding of multi dimensional differentiation and integration, explore alternative coordinate systems and tailor expansions, and preview the dynamics of forces and their properties.

explore the dynamical approach to classical mechanics through forces, work, energy, power, and conservation laws. study momentum conservation in straight line motion, with circular motion to follow.

Explore how mass, inertia, and forces drive motion, derive Newton's axioms of dynamics, and see how gravity produces mass-independent free fall without friction.

Explore Newton's laws of motion through the rocket example, linking inertia, forces, and momentum; learn how force drives acceleration and momentum evolves via derivatives and integration.

Explore weight and gravity in theoretical classical mechanics, using the inverse-square law, near-earth g ≈ 9.81 m/s^2, and a Taylor expansion around earth’s radius.

Explore how pulleys reorient gravity and reduce the force needed to lift a weight, comparing fixed and movable configurations and introducing the trade-off between applied force and rope length.

Explore gravity and the normal force on an inclined plane with no friction, yielding a resultant force g sin alpha and the corresponding acceleration via vector addition and kinematics.

Pendulum, a mass on a fixed rope, acts as a tangential restoring force and becomes a harmonic oscillator with frequency sqrt(g/L). Spring-mass system shows the same SHM frequency sqrt(k/m).

Analyze how friction on inclined planes arises from surface irregularities, governing static and kinetic friction and the threshold set by the normal force and material state.

Explore dry friction on a ten-degree inclined plane, comparing steel on steel and steel on ice, and determine stationary versus sliding, critical angle, and kinetic friction effects.

Explore how conservative forces like gravity and the harmonic oscillator derive from potentials, compute potentials for various forces, and use gradients to recover forces, noting friction lacks a potential.

Explore how work relates to conservative forces and potentials, linking gravity, potential energy, and kinetic energy through path-independent work and boundary differences.

Explore how work and energy relate to pulleys by comparing fixed and movable systems. A movable pulley doubles end-point displacement, halving the pull needed to lift the mass.

Explain how work and potential energy relate to energy and how energy is conserved through transformations between kinetic and potential forms, using pendulums and harmonic oscillators as examples.

Learn how power, the rate of energy change or work per unit time, equals force times velocity for constant conditions, with a lift example illustrating watts and horsepower.

This exercise analyzes the energy and gravity of a 100,000 kg spaceship launching from earth, comparing constant gravity, first-order Taylor, and exact 1/r^2 models, with escape velocity and booster power.

Examine gravitational forces at r1 and r2 in three approximations, and compute the work and potential energy changes for a spaceship. Derive the second escape velocity and booster power.

Use momentum conservation to analyze inelastic collisions where deforming objects form a single body with a common velocity, illustrating energy conversion to heat and deformation during crashes.

Explore elastic collisions where two bodies exchange velocities while conserving energy and momentum, derive v1 and v2 from coupled equations, and illustrate with Newton's cradle.

Analyze a low-speed car collision to determine velocity at impact in an inelastic collision, then use a 10-meter stop under kinetic friction to assess if either car exceeds 50 km/h.

Determine the impact velocity from kinetic friction, then apply momentum conservation for a 45-degree inelastic collision to compute post-collision speeds and assess which driver exceeded the speed limit.

Review the origin of accelerations, the forces behind dynamics, and work, energy, and momentum, applying conservation laws to solve advanced problems as you move to the circular motion dynamics section.

Explore the dynamics of circular motion by comparing centripetal and centrifugal forces from different frames of reference, and connect these ideas to work, energy, and the Coriolis force.

Clarifies centripetal force as the radial force keeping a particle on a circular path, deriving its magnitude as v^2/r using polar coordinates with e_r and e_phi, and noting radius constancy.

Explain that centripetal and centrifugal forces have equal magnitude but opposite directions, with centripetal toward the center sustaining circular motion and centrifugal outward due to inertia from the moving perspective.

Clarify that centripetal and centrifugal forces are the same viewed differently, and show that centripetal force does no work on a circular path, while tangential forces do work during acceleration.

Apply circular motion concepts to a roller coaster loop, analyze starting heights with and without attached rails, required top velocity to avoid falling, radial acceleration, and omega versus phi.

The solution uses energy conservation to relate starting height and top velocity to loop radius through potential and kinetic energy, with no friction, considering centrifugal and centripetal forces.

Explore rotational energy as the kinetic energy of rotating bodies, derived from tangential acceleration and angular velocity omega, with work from tangential forces and moment of inertia for extended objects.

Learn how the moment of inertia quantifies extended objects by summing r_perp^2 dm, deriving I = ∫ density r_perp^2 dV, and linking to the rotational energy 1/2 I omega^2.

Compute the moment of inertia for a stick using length density and r^2 integration about the rotation axis, comparing edge and center rotations and their kinetic energy with omega.

Derive the moment of inertia for a solid sphere using cylindrical coordinates, with constant density, and integrate over the perpendicular distance to obtain I = 2/5 m r^2.

Apply energy conservation with rotational energy to compare a sphere and a cylinder, using cylinder's i = 1/2 m r^2 and speeds with 0.1 m radius, 10 m, 45-degree incline.

Determine the cylinder and sphere moments of inertia, then use energy conservation on an incline to show the sphere rolls faster due to lower rotational inertia.

Explore torque as the cross product of position and force and how lever arm length and applied force influence the work required to rotate an object in circular motion.

Explain how levers reach equilibrium by balancing torques around a fixed point, using lever arms and masses to illustrate the relation R1/R2 = F2/F1.

Learn how angular momentum, defined as r cross p, changes under torque via L dot = torque, and how it is conserved in rotational motion.

Compute net torque and angular acceleration for a two-mass lever, then relate angular momentum to I omega and explain how arm spread changes spin.

Compute the net torque from two lever arms under gravity, then derive angular acceleration from the total moment of inertia; explain angular momentum L = I ω and skater conservation.

Compare translation and rotation in circular motion, linking arc length, polar angle, velocity, and angular velocity through vector relations and torque with angular momentum.

Explain how a spinning top shows fast rotation and slow precession under gravity. Relate spin and precession frequencies via torque and angular momentum.

Explore how inertial and accelerated frames of reference affect velocity measurements, and derive centrifugal and Coriolis forces from rotating coordinates using displacement and cross-product relations.

Compare inertial and accelerated frames to derive how forces transform in moving and rotating coordinate systems, including fictitious forces like centrifugal and Coriolis.

Explore how the Coriolis force emerges in rotating frames, affecting trajectories from long-range shooting and rocket launches to hurricanes and the Foucault pendulum.

Kepler's first law shows planets move on elliptical orbits around the sun, deriving the ellipse from energy conservation and angular momentum in polar coordinates.

Explore Kepler's second law via angular momentum conservation, showing how the area of the orbiting triangle remains constant, so velocity speeds up near the sun and slows farther away.

Derive Kepler's third law by equating centripetal and gravitational forces for bodies on elliptical orbits, showing that T squared is proportional to R cubed.

Explain how centripetal and centrifugal forces orient oppositely in circular motion. Link moment of inertia and angular momentum conservation to spinning tops and planets orbiting the sun to illustrate dynamics.

Learn theoretical classical mechanics using Lambert's and Hamilton's principles, introduce the Lagrangian, and derive motion equations while handling constraints such as objects on a table under gravity.

The lecture introduces lagrangian mechanics as a framework for handling constraints. It uses pendulum and plane examples to show generalized coordinates and holonomic constraints shape the trajectory and distinguish forces.

D'Alembert's principle uses generalized coordinates to account for constraints, showing constraint forces do no work and only external forces drive motion to yield the effective equations of motion.

Apply D'Alembert's principle to a pendulum using generalized coordinates; show constraint forces do no work, derive the phi equation of motion via chain rule, and discuss small-angle harmonic motion.

Apply d'Alembert's principle to derive the equation of motion for an object on an inclined plane using a generalized coordinate along the slope, yielding the acceleration g sin alpha.

This lecture applies D'Alembert's principle to an inclined plane, deriving that the acceleration equals g sin alpha while showing constraints do not generate work.

Express generalized forces as the external forces in terms of generalized coordinates, with constraint forces doing no virtual work, and derive the Lagrange equation from this framework.

Derive the Lagrange equation from total and generalized forces, using the kinetic energy and chain rule to relate q, q-dot, and q-double-dot, leading to Euler Lagrange equation.

Learn how the Euler-Lagrange equation (second kind) arises from the Lagrangian L = T − U under conservative forces, and why it cannot address non-conservative forces like friction.

Apply the Lagrange equation to the harmonic oscillator using the Lagrangian T minus U, yielding z̈ = -(k/m) z; the solution is sine/cosine with initial-condition constants.

Apply the Euler-Lagrange method to a pendulum and a skater on a half pipe, deriving the equations of motion. Then treat the Kepler problem with r and phi.

Derive the pendulum's lagrangian with a fixed radius, define phi as the generalized coordinate, and obtain the equation of motion phi double dot = -(g/L) sin(phi).

Explore the Kepler problem in lagrangian mechanics using generalized coordinates r and phi; derive the lagrangian and the two coupled equations of motion, and discuss their numerical solution.

Explore Hamilton's principle: the actual trajectory minimizes the action, the integral of the Lagrangian (T minus U) along the path.

Derive the Euler-Lagrange equation from Hamilton's principle by varying the trajectory and using integration by parts, showing stationary action yields the same result as the Lagrange approach.

We solve a vertical throw using the Lagrangian and Euler-Lagrange equation, then compare three trajectories by action to show the smallest action balances kinetic and potential energy.

Apply calculus of variation to a problem by minimizing a length integral with the Euler-Lagrange equation from the action, proving the shortest path between two points is a straight line.

Explore the Euler–Lagrange equation of the first kind, contrasting it with the second kind, using actual coordinates and Lagrange multipliers, with an Atwood machine example and a constraint F=0.

Explore the Atwood machine through the Euler–Lagrange equation of the first kind, using a constraint and a Lagrange multiplier to derive the coupled two-mass acceleration and effective coordinate.

Noether theorem states that continuous symmetries lead to conserved quantities, shown via infinitesimal variations of the action in the lagrangian framework, yielding the Noether charge pi times F.

Explore how continuous symmetries yield conserved quantities via Noether’s theorem, shown by rotation about the z axis, yielding the conserved z component of angular momentum.

Derive time invariance as a continuous symmetry yielding the Noether charge, the Hamiltonian, a conserved quantity; express it as p_i qdot - L and as total energy.

Embrace the theoretical physics approach to classical mechanics by starting from Lambert's principle and Hamilton's principle of minimal action, introducing the Lagrangian to derive equations of motion and simplify constraints.

The lecture presents the Hamiltonian approach as a framework for the Lagrangian approach, defining the Hamiltonian as the sum of kinetic and potential energy and deriving the equations of motion.

Derive the Hamiltonian from the Lagrangian framework and Noether’s theorem, linking time invariance to H, the total energy T plus U, with H = p·qdot − L.

Explore how Hamiltonian and Lagrangian mechanics relate through the Legendre transformation, with momentum P and velocity Q dot as inverse functions, illustrating their equivalence.

Derive Hamilton's equations of motion from the Lagrangian and Hamiltonian, showing ∂H/∂P = q dot and ∂H/∂Q = -p dot, to solve coupled equations with the harmonic oscillator example.

Explore how Hamilton's equations solve the harmonic oscillator, visualize energy-conserving trajectories in phase space, and understand how friction breaks time invariance and reshapes orbits.

Apply Hamilton's equations to pendulum and Kepler two-body problems, transforming the Lagrangian to the Hamiltonian, compute generalized momenta, and verify equivalence with the Lagrangian approach.

Derive the pendulum's Hamiltonian by defining the generalized momentum pi_phi and rewriting the Lagrangian in terms of phi and pi_phi, then apply Hamilton's equations.

Derive the Hamiltonian for a two-body system in radius and angle with momenta pi_r and pi_phi; show pi_phi equals z-component of angular momentum and is conserved by rotational invariance.

Explore how Hamiltonian mechanics uses the chain rule and Hamilton's equations to express time evolution. Introduce the Poisson bracket and its relation to conserved quantities and quantum mechanics.

Explore the equivalence of Lagrangian, Hamiltonian, and Hamilton-Jacobi formulations in classical mechanics, and how a canonical transformation to q dash and p dash yields a zero Hamiltonian, revealing conserved quantities.

Conclude section by highlighting that the grounding and Hamiltonian approaches are equivalent and often one is more convenient, both deriving equations of motion from fundamental laws rather than Newtonian phenomenology.

begin level four with advanced mathematics, exploring complex numbers and the imaginary unit to handle square roots of negatives, then introduce matrices and the eigen system for later differential equations.

Explore what complex numbers are, why we need them, and how the imaginary unit i and the complex plane express real and imaginary parts in solving quadratic zeros.

Learn how complex numbers, written as real plus imaginary parts, are added and subtracted in the complex plane, and how the conjugate reveals real and imaginary parts and magnitude.

Explore multiplication and division of complex numbers, using polar representation, complex conjugates, and reciprocal methods to compute products and quotients with real and imaginary parts.

Compute the sum, difference, and product of complex numbers; derive conjugates, absolute values, and inverses; and visualize results as vectors in the complex plane using Euler form.

Define what a matrix is by extending vectors to a rectangle of numbers, identify rows and columns, and introduce diagonal, identity, and hessian matrices, with tensors and rotations as applications.

Add and subtract matrices by summing or subtracting corresponding elements; ensure the same rank and apply element-wise operations, noting commutativity, while matrix multiplication is more difficult.

Learn how scalar multiplication and matrix multiplication work, including dot-product interpretation, matrix rank requirements, and why AB differs from BA through a practical 3x3 example.

Learn to compute determinants of 2x2, 3x3, and 4x4 matrices using cofactors and sign patterns with practical examples.

Learn to compute eigenvalues and eigenvectors to form the eigen system of a matrix, using determinants and the characteristic polynomial. See a numerical example with three coupled oscillators.

Practice matrix operations including addition and products with an identity matrix, perform matrix–vector multiplication, and solve the eigen system of a 2×2 matrix to obtain eigenvalues ±√2 and corresponding eigenvectors.

Present methods for solving differential equations of motion analytically, from free fall with constant acceleration to the harmonic oscillator, then introduce numerical approaches for harder cases.

Explore differential equations through the harmonic oscillator, deriving x'' = -(k/m) x from Newton's law. Learn to express solutions with sine, cosine, exponentials, and damped cases.

Classify differential equations into ordinary and partial forms, illustrating with one-dimensional odes like the harmonic and damped oscillator, and multidimensional pdes such as the heat equation.

Classify ordinary differential equations by order, degree, autonomy, and linearity using the harmonic oscillator example, highlighting constant coefficients, homogeneous form, and when numerical methods become necessary.

Explore the trivial direct integration of differential equations with a single derivative, find the antiderivative and a constant using one initial condition, and extend to higher derivatives.

Explore free fall by direct integration of the second-order differential equation for the z coordinate under gravity, deriving velocity and position from initial conditions v0 and z0.

Solve homogeneous linear odes with constant coefficients by the exponential ansatz, convert to a polynomial and find roots; build the general solution via superposition, then determine coefficients from initial conditions.

Apply the exponential ansatz to the harmonic oscillator, turning the linear differential equation into an algebraic one, derive complex roots, and obtain a cosine solution via initial conditions.

Solve linear differential equations using direct integration and exponential, cosine, and sine forms, then apply boundary conditions to determine constants and final solutions.

Solve the damped harmonic oscillator using exponential ansatz, define mu and omega tilde, and distinguish strong damping (real roots) from weak damping (damped oscillations) for a complete solution.

Explore solving inhomogeneous linear ordinary differential equations with constant coefficients by first solving the homogeneous part, then adding a specific solution, illustrated with a driven harmonic oscillator.

Solve a second-order differential equation by first solving the homogeneous part with complex exponentials, then obtain a specific solution by a trial function and apply boundary conditions to determine constants.

Solve the inhomogeneous equation y'' + 100 y = 100 x^2 + 10 x - 8 by combining the homogeneous solution with a quadratic particular solution. Use boundary conditions to determine constants.

Continue exploring differential equations by revisiting the ode and pde classification, contrasting analytic with numerical solutions, and introducing Euler methods using Mathematica and Python.

Complete this section on solving differential equations analytically to reveal the position versus time relationship. Some cases are too difficult mathematically, so we will rely on a computer next.

Learn to solve differential equations numerically with Python 3, starting with simple code and using SciPy Runge-Kutta to model earth–moon orbit, spaceship trajectories, and eigenvalues from coupled oscillators.

Learn to download and install Python via the Anaconda individual edition with a graphical installer, select Python 3.9, and launch Jupyter Notebook through the Anaconda Navigator.

Implement the Euler method in a Jupyter notebook to solve first-order differential equations, illustrate with radioactive decay, and compare numerical results to the analytical solution.

The lecture demonstrates solving radioactive decay with an exponential function using a general function for first-order differential equations, implemented via the Euler method, and shows how to generalize the approach.

Convert second-order differential equations to two first-order equations and solve free fall using the Euler method, updating position and velocity under gravity and showing both numerical and analytical results.

Explore the pendulum as a harmonic oscillator under the small-angle approximation with zero damping, solving its second-order differential equation via Euler method and comparing to the analytical cosine solution.

Explore the actual pendulum solution beyond the small-angle approximation, solving with sine of theta and comparing numerical results to the harmonic oscillator.

Discover how damping causes exponential amplitude decay in a pendulum and oscillator, with energy leaving the pendulum but total energy conserved, and how driving forces enable synchronization over time.

Explore solving differential equations with SciPy's solve_ivp using the rk45 method, comparing to Euler's method, and converting a second-order system to first-order form for accurate simulations.

Simulate a ball in a rotationally symmetric bowl as two uncoupled harmonic oscillators with damping, then solve via solve_ivp for x, y, vx, vy and plot time-series and top-down trajectories.

Model a ball in a bowl with Wolfram Mathematica 12.2, solving the second-order equations of motion from the potential, visualizing with 3d and density plots, and animating the spiral trajectory.

Explore the three-body problem with sun, earth, and moon by deriving coupled differential equations from Newton's law of gravitation and applying numerical methods to simulate motion and plan lunar orbits.

Program the sun-earth-moon three-body differential equations in vector form, define masses and distances, and prepare initial conditions for the solver.

Solve the sun-earth-moon differential equations with an ivp solver, set time bounds and initial conditions, and plot equidistant trajectories while tightening tolerances to reveal the sun-centered motion.

Examine the three-body problem with sun, Earth, and Moon by solving differential equations and analyzing their trajectories, distances, and the Earth's year-long orbit and the Moon's orbit around Earth.

Explore the three-body problem in three dimensions by examining the moon's inclination and adding a small z component to the starting velocity, then update the plots to 3d representations.

Add a spaceship as the fourth body to the three-body problem, solving differential equations for gravity from Sun, Earth, and Moon to model a moon mission.

Increase starting velocity to 1.25x to create an elliptical orbit around Earth, analyze distance variations and velocity changes, and note possible Moon encounters or Earth escape.

Explore how boosting a satellite beyond Earth's circular orbit creates elliptical paths, escapes Earth's gravity, and transitions to sun-dominated trajectories, with Moon and Earth-sun reference frames analyzed.

Vary the starting velocity from circular to elliptical, creating a Moon encounter. The maneuver shows earth escape, Moon influence, and potential to reach Mars.

Plan a brake maneuver to reach moon orbit by timing engine thrust with the Heaviside function, updating the four-body equations of motion and simulating the trajectory.