Principle of Least Action and Lagrange's Equations of Mechanics - Basics of Calculus of Variations

Explore the Principle of Least Action and Lagrange's Equations of Mechanics in this comprehensive lecture from an advanced dynamics course. Delve into variational principles of mechanics, including the Principle of Least Action and its equivalence to Lagrange's equations. Learn how to derive Newton's equations using a physicsy approach, then dive into calculus of variations techniques to obtain Euler-Lagrange equations. Examine practical applications like the Brachistochrone problem and cubic spline curves for data fitting. Gain a deeper understanding of canonical transformations, Hamiltonian systems, and the mathematical foundations of classical mechanics in this 59-minute video presented by Dr. Shane Ross, a Virginia Tech professor with a Caltech PhD.

Canonical transformations come from generating functions via variational principles.
Principal of least action.
Initial approach to understanding how principle of least action leads to Newton's equations.
Euler-Lagrange equations: More general, calculus of variations approach to principle of critical action, leading to Euler-Lagrange equations (Lagrange's equations in mechanics context).
Euler-Lagrange equations, example uses.
Brachistochrone problem.
Cubic spline curves (data fitting).