PU-101 · Newtonian Mechanics
Starting from Newton's three laws and the notion of an inertial frame, the unit builds the full deductive machinery of classical particle and rigid-body dynamics, showing how the conservation laws of energy, momentum, and angular momentum follow from force law and symmetry. It culminates in the central-force solution of the Kepler problem and in d'Alembert's principle, which reformulates the whole edifice as the Euler-Lagrange equations and hands the student off to analytical mechanics.
Lectures
| L01 | Space, Time, and the Program of Mechanics — |
| L02 | Kinematics in One and Three Dimensions — |
| L03 | Newton's Laws as Axioms and Inertial Frames |
| L04 | Galilean Relativity and the Limits of Newton |
| L05 | Forces, Free-Body Diagrams, and Constraints — |
| L06 | Work and the Work-Energy Theorem |
| L07 | Conservative Forces and Potential Energy |
| L08 | Energy Conservation and Potential Landscapes |
| L09 | Momentum and Newton's Third Law |
| L10 | Systems of Particles and the Center of Mass |
| L11 | Variable-Mass Systems and the Rocket Equation |
| L12 | Angular Momentum and Torque |
| L13 | Central Forces and the Two-Body Problem |
| L14 | The Effective Potential and the Orbit Equation |
| L15 | Gravitation and Kepler's Laws |
| L16 | Small Oscillations and Simple Harmonic Motion |
| L17 | Damping, Driving, and Resonance |
| L18 | Coupled Oscillators and Normal Modes |
| L19 | Rigid Bodies and the Inertia Tensor |
| L20 | Principal Axes and Rigid-Body Rotation |
| L21 | Euler's Equations and Torque-Free Precession |
| L22 | Non-Inertial Frames and Fictitious Forces |
| L23 | The Coriolis Force on the Rotating Earth |
| L24 | From Newton to Lagrange: d'Alembert's Principle |
| L25 | The Euler-Lagrange Equations and What Comes Next |
| L26 | Synthesis: Conservation Laws and the Shape of Mechanics |
Derivations homed in this unit
Galilean Invariance of Newton's Laws
Newton's second law is form-invariant under Galilean boosts, translations, and rotations, which is what singles out the inertial frames.
The Work-Energy Theorem
Integrating F = ma along a trajectory shows the work done by the net force equals the change in kinetic energy.
Conservative Forces and Potential Energy
A force field is path-independent iff it is curl-free iff it is the negative gradient of a scalar potential energy.
Conservation of Mechanical Energy
For systems acted on only by conservative forces the sum of kinetic and potential energy is a constant of the motion.
Momentum Conservation from Newton's Third Law
Pairwise internal forces cancel, so the total linear momentum of an isolated system of particles is conserved.
The Center-of-Mass Theorem
The center of mass of a system accelerates as though the total external force acted on the total mass concentrated there.
Angular Momentum, Torque, and Central Forces
The time derivative of angular momentum equals the net torque, so angular momentum is conserved under central forces.
The Two-Body Problem and Reduced Mass
Separating center-of-mass and relative coordinates reduces the two-body central problem to one particle of reduced mass in a fixed potential.
Effective Potential and the Orbit Equation
Angular-momentum conservation collapses a central-force problem to one-dimensional radial motion in an effective potential, yielding the Binet orbit equation.
Kepler's Three Laws from Gravity
Solving the inverse-square orbit equation gives elliptical orbits, the equal-areas law, and the period-semimajor-axis relation.
Small Oscillations and Simple Harmonic Motion
Linearizing any smooth potential about a stable equilibrium yields the harmonic-oscillator equation and its characteristic frequency.
Damped, Driven Oscillators and Resonance
Solving the driven damped oscillator gives the steady-state amplitude, phase lag, and the resonance peak with its quality factor.
Coupled Oscillators and Normal Modes
Diagonalizing the coupled linearized equations of motion gives normal modes as eigenvectors and normal frequencies as eigenvalues.
The Tsiolkovsky Rocket Equation
Applying momentum conservation to a variable-mass body gives the logarithmic relation between velocity gain, exhaust speed, and mass ratio.
The Moment-of-Inertia Tensor
The angular momentum of a rigid body is a linear map of its angular velocity, defining the symmetric inertia tensor and its principal axes.
Euler's Equations and Torque-Free Precession
Writing angular-momentum balance in the body frame yields Euler's equations and predicts free precession and the stability of rotation axes.
Rotating Frames: Centrifugal and Coriolis Forces
Transforming Newton's law into a rotating frame produces the centrifugal, Coriolis, and Euler fictitious forces.
Euler-Lagrange Equations from d'Alembert
d'Alembert's principle in generalized coordinates converts Newton's equations into the Euler-Lagrange equations for the Lagrangian.