Emmy Noether's paper "Invariant Variation Problems" explores the relationship between invariance and variation in calculus of variations, combining methods from the calculus of variations and Lie's group theory. The paper discusses how invariants under continuous groups of transformations can be characterized by specific relationships between the Lagrange expressions and their derivatives. It presents two main theorems: the first states that if an integral is invariant under a finite continuous group, then certain linear combinations of the Lagrange expressions become divergences, and conversely, invariance follows from this. The second theorem states that if an integral is invariant under an infinite continuous group, then there are specific identity relationships between the Lagrange expressions and their derivatives up to a certain order, and conversely, invariance follows from these relationships. The paper also discusses the implications of these theorems for the theory of relativity, showing how they can be used to derive conservation laws and first integrals. It provides examples, such as the Weierstrass parametric representation and the general theory of relativity, to illustrate these concepts. The paper concludes by connecting these results to a Hilbertian assertion about the failure of proper conservation laws in general relativity, arguing that such failure is a characteristic feature of the general theory of relativity. The paper emphasizes the importance of group theory in understanding invariance and conservation laws in variational problems.Emmy Noether's paper "Invariant Variation Problems" explores the relationship between invariance and variation in calculus of variations, combining methods from the calculus of variations and Lie's group theory. The paper discusses how invariants under continuous groups of transformations can be characterized by specific relationships between the Lagrange expressions and their derivatives. It presents two main theorems: the first states that if an integral is invariant under a finite continuous group, then certain linear combinations of the Lagrange expressions become divergences, and conversely, invariance follows from this. The second theorem states that if an integral is invariant under an infinite continuous group, then there are specific identity relationships between the Lagrange expressions and their derivatives up to a certain order, and conversely, invariance follows from these relationships. The paper also discusses the implications of these theorems for the theory of relativity, showing how they can be used to derive conservation laws and first integrals. It provides examples, such as the Weierstrass parametric representation and the general theory of relativity, to illustrate these concepts. The paper concludes by connecting these results to a Hilbertian assertion about the failure of proper conservation laws in general relativity, arguing that such failure is a characteristic feature of the general theory of relativity. The paper emphasizes the importance of group theory in understanding invariance and conservation laws in variational problems.