Week 16 — Classical Mechanics and Lagrangian Dynamics

Overview

Classical mechanics is the physics under any simulation of vehicles, robots, or rigid bodies. This week covers Newtonian dynamics and free-body reasoning, oscillations (the single most reused model in engineering), conservation of energy and momentum, angular momentum and rotation, and then the Lagrangian formulation — a coordinate-free, constraint-friendly reformulation that scales to complex mechanisms far better than \(F=ma\). Accelerating/rotating frames (Coriolis) and coupled oscillators / normal modes close the week. The differential equations here are solved with the numerical methods of Weeks 3–4.

Readings

Morin Ch 2 — Using \(F=ma\). Newton’s laws, free-body diagrams, motion under forces.
Morin Ch 3 — Oscillations. Simple, damped, and driven harmonic motion; resonance.
Morin Ch 4 — Energy and momentum. Work, potential energy, conservation, collisions.
Morin Ch 5 — Angular momentum. Torque, rotational motion.
Morin Ch 6 — The Lagrangian method. Generalized coordinates, Euler–Lagrange equations, constraints.
Morin Ch 8 — Accelerating frames. Fictitious forces, rotating frames, the Coriolis force.
Morin Ch 10 — Coupled oscillators and normal modes.

Key Concepts

Newtonian dynamics

Newton’s second law \(\mathbf{F} = m\ddot{\mathbf{r}}\) turns a force model into a second-order ODE. The method is always the same: draw a free-body diagram, sum forces (and torques), write the equations of motion, integrate. This is the direct route, but it becomes unwieldy with constraints — which motivates the Lagrangian approach below.

Oscillations

The damped, driven harmonic oscillator

\[m\ddot{x} + b\dot{x} + kx = F_0\cos(\omega t)\]

is the canonical linear system. Its homogeneous solution decays (under/over/critically damped depending on \(b^2\) vs \(4mk\)); its steady-state response peaks at resonance near \(\omega \approx \sqrt{k/m}\). The same equation governs RLC circuits, suspension systems, and vibration modes — and its frequency-domain analysis is exactly Week 12’s LTI theory.

Conservation laws

Work–energy: \(W = \int \mathbf{F}\cdot d\mathbf{r} = \Delta(\tfrac12 m v^2)\). For conservative forces \(\mathbf{F} = -\nabla U\), total mechanical energy \(E = T + U\) is conserved. Momentum \(\mathbf{p} = m\mathbf{v}\) is conserved in the absence of external forces (collisions), and angular momentum \(\mathbf{L} = \mathbf{r}\times\mathbf{p}\) is conserved under zero net torque. Conservation laws give first integrals that reduce the order of the equations of motion.

Angular momentum and rotation

Torque \(\boldsymbol{\tau} = \mathbf{r}\times\mathbf{F} = \dot{\mathbf{L}}\) drives rotation; for a rigid body \(\mathbf{L} = I\boldsymbol{\omega}\) with the inertia tensor \(I\) (a symmetric matrix whose eigenvectors are the principal axes — Week 2 again). This is the basis of rigid-body simulation and attitude dynamics.

The Lagrangian method

Define the Lagrangian \(L = T - U\) (kinetic minus potential energy) in generalized coordinates \(q_i\). The equations of motion are the Euler–Lagrange equations:

\[\frac{d}{dt}\frac{\partial L}{\partial \dot q_i} - \frac{\partial L}{\partial q_i} = 0.\]

This is the stationary-action principle, handles constraints cleanly (via choice of coordinates or Lagrange multipliers), and is coordinate-free — vastly preferable for pendulums, linkages, and robot arms. It is also the structural template for optimal control and the configuration-space geometry of Week 17.

Rotating frames and normal modes

In an accelerating or rotating frame, fictitious forces appear — centrifugal and the velocity-dependent Coriolis force \(-2m\,\boldsymbol{\omega}\times\mathbf{v}\) — essential for any earth-frame or body-frame analysis. Coupled oscillators are solved by finding normal modes: diagonalize the coupled linear system (a generalized eigenvalue problem, Weeks 1–2) so the motion decouples into independent oscillations at the mode frequencies.

Connections

Forward: Week 17’s configuration spaces and geodesics generalize Lagrangian dynamics geometrically; Week 18’s wave equation is the continuum oscillator.
Across courses: vehicle physics, the bicycle model, suspension and collision dynamics (Course 2), control loops and IMU/state dynamics (Course 1).