Legendre Transformation Explained: From Lagrangian to Hamiltonian | Lecture 2

The Legendre transformation: the mathematical operation that converts a function of one set of variables into a function of its conjugate variables (its partial derivatives). It's the key technical step that takes you from the Lagrangian L(q, q̇) to the Hamiltonian H(q, p) — and it shows up across physics, from classical mechanics to thermodynamics (Helmholtz, Gibbs, and enthalpy are all Legendre transforms of each other). In this lecture we develop the Legendre transformation abstractly with a geometric interpretation (a tangent-line construction), prove it's involutive (applying it twice returns you to where you started), and then use it to derive Hamilton's canonical equations from the Euler-Lagrange equations. We close with the simplest non-trivial example: the spring-mass system (the harmonic oscillator), seen first as a Lagrangian and then transformed to its Hamiltonian formulation in (q, p) phase space. This is Lecture 2 of an online course on Hamiltonian and Nonlinear Dynamics. ▶️ Chapters 0:00 Recap from Lecture 1: why Hamiltonian mechanics? 1:30 Today's plan: the Legendre transformation, Hamilton's equations, examples 3:45 Where the Legendre transformation shows up (thermodynamics, stat mech) 5:16 Formal definition of the Legendre transformation 10:10 New function g(v) = u·v − f(u) and involutive property 13:46 Active & passive variables 19:04 Applying the LT to go from Lagrangian to Hamiltonian 22:28 Geometric interpretation of the Legendre transformation 24:53 The 1D geometric picture: slope and y-intercept of the tangent line 34:29 Derivation of Hamilton's canonical equations from the LT 40:25 Hamilton's canonical equations: q̇ᵢ = ∂H/∂pᵢ, ṗᵢ = −∂H/∂qᵢ 42:48 Discussion: convexity condition, non-canonical equations 45:21 Worked example: spring-mass system (simple harmonic oscillator) 49:28 Computing the Hamiltonian H = T + V (kinetic plus potential) 52:53 Hamilton's equations → first-order ODEs (advantage over Lagrange's equations) 53:44 The x-notation: writing Hamilton's equations in matrix form 57:13 Matrix exponential e^(At) for linear Hamiltonian systems 1:01:15 Interpreting the matrix: shrink, rotate, expand (area-preserving) 1:04:50 Phase portrait interpretation: H as a height function 1:06:19 Gradient ∇H and motion along level sets of H 1:11:20 Higher dimensions: 4D phase space from 2-DOF systems 1:12:02 Canonical symplectic matrix J and J² = −I 1:13:19 Mysterious connection between J and imaginary numbers 📘 What you'll learn The Legendre transformation as a tangent-line construction Why it's involutive: applying it twice returns you to the original function How to compute the conjugate momentum p from the Lagrangian How to construct H(q, p) from L(q, q̇) How Hamilton's canonical equations emerge from this construction How to apply everything to the spring-mass harmonic oscillator How the same Legendre transformation appears in thermodynamics 🎓 Course Hamiltonian and Nonlinear Dynamics (full online course)    • Hamiltonian Mechanics: Full Graduate Cours...   ▶️ Previous lecture (Lecture 1) — Introduction to Hamiltonian Systems    • Hamiltonian Mechanics Explained: Why Study...   ▶️ Next lecture (Lecture 3) — Hamiltonian System Properties, Classical Uncertainty Principle, 2D Fluid Stream Functions    • Hamiltonian System Properties | Phase Spac...   🔰 New to this topic? Start here A shorter, gentler introduction to Hamiltonian systems in 2D    • Hamiltonian Systems: Stream Functions, Cen...   📄 Course lecture notes PDF: https://drive.google.com/drive/folder... 📖 References Hamill — A Student's Guide to Lagrangians and Hamiltonians (accessible) Hand & Finch — Analytical Dynamics (standard graduate text) Mark Levi — Classical Mechanics with Calculus of Variations & Optimal Control: An Intuitive Introduction 📚 Math references for additional background Verhulst — Nonlinear Differential Equations and Dynamical Systems Wiggins — Introduction to Applied Nonlinear Dynamical Systems and Chaos Marsden & Ratiu — Introduction to Mechanics and Symmetry ⸻ 👨‍🏫 Instructor Dr. Shane Ross Professor of Aerospace Engineering, Virginia Tech (Caltech PhD, former NASA/JPL and Boeing) Research: https://ross.aoe.vt.edu Follow: https://x.com/RossDynamicsLab Subscribe: https://www.youtube.com/user/RossDyna... ⸻ 🔗 Related courses Analytical (Lagrangian) Dynamics —    • Lagrangian Mechanics & 3D Rigid Body Dynam...   Nonlinear Dynamics & Chaos —    • Nonlinear Dynamics & Chaos — Full Course F...   Center Manifolds, Normal Forms & Bifurcations —    • Local Bifurcation Theory: Center Manifolds...   3-Body Problem Orbital Dynamics —    • Three-Body Problem: Trajectory Design & Lo...   Recorded: 2021-06-22 #LegendreTransformation #HamiltonianMechanics #LagrangianMechanics #HamiltonsEquations #ClassicalMechanics #HarmonicOscillator #PhaseSpace #ConjugateMomentum #AnalyticalDynamics #VirginiaTech