ASO: Calculus of Variations (2025-26)
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- Lecturer: Profile: Paul Dellar
Course information
Course term: Trinity
Course lecture information: 8 lectures
Course overview:
The calculus of variations concerns problems in which one wishes to find the minima or extrema of some quantity over a system that has functional degrees of freedom. The typical example is an integral that assigns a real number to a function, such as the arc length of a curve. Many important problems across pure and applied mathematics and physics can be expressed in variational form. They range from finding the shape of a soap bubble, a surface that minimizes its surface area, to finding the configuration that minimises the energy of a piece of elastic material. Hamilton's principle of least action is a powerful and coordinate-independent way to formulate the laws of classical mechanics. This course shows that the solutions of such variational problems satisfy systems of differential equations, the Euler-Lagrange equations. Moreover, symmetries of a variational problem lead to first integrals of the Euler-Lagrange equations, such as a conserved energy in classical mechanics.
Learning outcomes:
Students will be able to formulate variational problems, derive their Euler-Langrange equations, and identify conserved quantities.
Course synopsis:
The basic variational problem and the Euler-Lagrange equation. Examples, including the brachistochrone and axisymmetric soap films.
Extension to several dependent variables. Hamilton's principle for free particles and particles subject to holonomic constraints. Equivalence with Newton's second law. Geodesics on surfaces. Extension to several independent variables.
Examples including Laplace's equation. Lagrange multipliers and variations subject to constraint. Eigenvalue problems for Sturm-Liouville equations. The Rayleigh-Ritz method for approximating eigenvalues.
Extension to several dependent variables. Hamilton's principle for free particles and particles subject to holonomic constraints. Equivalence with Newton's second law. Geodesics on surfaces. Extension to several independent variables.
Examples including Laplace's equation. Lagrange multipliers and variations subject to constraint. Eigenvalue problems for Sturm-Liouville equations. The Rayleigh-Ritz method for approximating eigenvalues.