Principle of Least Action & Lagrange's Equations | Calculus of Variations | ESM/AOE 6314 L6

Derive the variational foundation of mechanics: the Principle of Least Action (also called stationary action, critical action, or Hamilton's principle). This global principle is equivalent to the local Lagrange's equations. We start with a physics-style derivation recovering Newton's equations, then develop the calculus of variations rigorously to obtain the Euler-Lagrange equations. The action integral S = ∫L dt is minimized (more precisely, made stationary) by the true path of a system. Working out the local consequences of this global principle yields Lagrange's equations. We then apply the calculus of variations to two classic problems: the Brachistochrone (the curve of fastest descent, whose answer is a cycloid) and cubic spline curves used in data fitting — showing how the same mathematical machinery solves problems far beyond mechanics. This is Lecture 6 of Advanced Dynamics (ESM/AOE 6314): Hamiltonian & Nonlinear Dynamics, Virginia Tech (graduate-level). ▶️ Chapters: 0:00 Canonical transformations come from generating functions via variational principles 2:21 The principle of least action 8:21 Physics approach: How least action leads to Newton's equations 30:32 Calculus of variations: The general approach to critical action and Euler-Lagrange 49:00 Euler-Lagrange equations: Example applications 49:59 The Brachistochrone problem (curve of fastest descent) 54:29 Cubic spline curves (data fitting) 📘 What you'll learn: State the Principle of Least Action and the action integral S = ∫L dt Understand why a global variational principle is equivalent to local Lagrange's equations Derive the Euler-Lagrange equations using the calculus of variations Solve the Brachistochrone problem and recognize the cycloid solution Apply variational methods beyond mechanics (cubic splines, data fitting) 🎓 Course: Advanced Dynamics (ESM/AOE 6314): Hamiltonian & Nonlinear Dynamics (Virginia Tech, graduate-level) 🔗 Full course playlist:    • Hamiltonian Mechanics: Full Graduate Cours...   📄 Lecture notes (PDF): https://drive.google.com/drive/folder... 🧮 Interactive action integral calculator (courtesy Cleon Teunissen): http://cleonis.nl/physics/phys256/ene... 📖 References: Sanz-Serna & Calvo, Numerical Hamiltonian Problems Hand & Finch, Analytical Mechanics Hamill, A Student's Guide to Lagrangians and Hamiltonians Levi, Classical Mechanics with Calculus of Variations and Optimal Control Brachistochrone video clip from @Vsauce — "The Brachistochrone"    • The Brachistochrone   👨‍🏫 Instructor: Dr. Shane Ross, Virginia Tech (Caltech PhD) Instructor intro:    • Professor Shane Ross Introduction   Research: https://ross.aoe.vt.edu Follow on X: https://x.com/RossDynamicsLab Subscribe: https://www.youtube.com/c/RossDynamic... ▶️ Previous: Canonical Transformations — Hamiltonian Allowable Changes of Variables, Symplectomorphisms (Lecture 5)    • Canonical Transformations & the Symplectic...   ▶️ Next: Generating Functions for Canonical Transformations — Examples & the Big Picture (Lecture 7)    • Generating Function of a Canonical Transfo...   ▶️ This course builds on prior knowledge of Lagrangian systems:    • Lagrangian Mechanics & 3D Rigid Body Dynam...   ▶️ Continuation course: Center Manifolds, Normal Forms & Bifurcations    • Local Bifurcation Theory: Center Manifolds...   🔗 Related courses: Lagrangian & Rigid Body Dynamics:    • Lagrangian Mechanics & 3D Rigid Body Dynam...   Hamiltonian Dynamics:    • Hamiltonian Mechanics: Full Graduate Cours...   Local Bifurcation Theory:    • Local Bifurcation Theory: Center Manifolds...   Three-Body Problem:    • Three-Body Problem: Trajectory Design & Lo...   Nonlinear Dynamics & Chaos:    • Nonlinear Dynamics & Chaos — Full Course F...   Recorded: July 2021 #Hamiltonian #EulerLagrange #PrincipleLeastAction #LeastAction #Brachistochrone #HamiltonsPrinciple #CalculusOfVariations #Lagrangian #CanonicalTransformation #SymplecticGeometry #NonlinearDynamics #ClassicalMechanics #GraduateLevel