Calculus of variations MOC

Action functional

Let 𝑀 be an 𝑚-dimensional 𝐶𝛼-manifold. An action functional, also called a variational integral,¹ is a functional on some space 𝐹 of smooth maps from 𝑀 of the form

ℒ[𝑓]=∫𝑀𝐿[𝑓]

for some map into top forms

𝐿:𝐹→Ω𝑚(𝑀)

called the Lagrangian.

Further terminology

• A pair consisting of a space of fields and a lagrangian is a Lagrangian field theory.

• A Lagrangian which depends smoothly only on the base point 𝑥 ∈𝑀 and partial derivatives of 𝑓 up to order 𝑘 is called a Local Lagrangian of order 𝑘.

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Footnotes

1. 2004. Calculus of variations I, p. 3. ↩