Mathematical Methods
5 Variational calculus
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Notes
1
Introduction and Prerequisites
2
Linear vector spaces
3
Operators, Eigenvectors and Eigenvalues
4
Green functions
5
Variational calculus
A
Contour Integration
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Chapter 5
Variational calculus
5.1
Functionals and stationary points
5.2
Stationary points
5.3
Special cases with examples: first integrals
5.3.1
Functional of first derivative only
5.3.2
No explicit dependence on
x
5.4
Generalisations
5.4.1
Variable end points
5.4.2
One endpoint free
5.4.3
More than one function: Hamilton’s principle
5.4.4
More dimensions: field equations
5.4.5
Higher derivatives
5.5
Constrained variational problems
5.5.1
Lagrange’s undetermined multipliers
5.5.2
Generalisation to functionals
5.5.3
Eigenvalue problems
5.5.4
The Rayleigh-Ritz method
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