Explain Castigliano's theorems and their application in structural analysis.

Castigliano's first theorem: displacement equals partial derivative of strain energy with respect to force at that point (∂U/∂P = δ). Second theorem: force equals partial derivative of complementary energy with respect to displacement. For linear systems, both are equivalent. Application involves expressing U in terms of loads, differentiating, then evaluating integrals. Powerful for indeterminate structures, curved members, and combined loading cases where direct methods are complex.