Emmy Noether's paper "Invariante Variationsprobleme" (1918) explores the relationship between variational problems and continuous groups, as introduced by Sophus Lie. The paper combines the methods of the calculus of variations with Lie's group theory to derive theorems that describe the invariance of integrals under transformations from a continuous group. Noether's main results are: 1. **Theorem I**: If an integral \( I \) is invariant under a finite continuous group \( \mathfrak{G}_\rho \), then \( \rho \) linearly independent combinations of the Lagrange expressions become divergences. Conversely, if \( \rho \) linearly independent combinations of the Lagrange expressions become divergences, then \( I \) is invariant under \( \mathfrak{G}_\rho \). 2. **Theorem II**: If an integral \( I \) is invariant under an infinite continuous group \( \mathfrak{G}_{\infty, \rho} \) where the arbitrary functions occur up to the \( \sigma \)-th derivative, then there exist \( \rho \) identity relationships between the Lagrange expressions and their derivatives up to the \( \sigma \)-th order. Conversely, if these relationships hold, then \( I \) is invariant under \( \mathfrak{G}_{\infty, \rho} \). Noether also discusses the converse of these theorems, showing that if the Lagrange expressions are relative invariant under a group, then the integral \( I \) is invariant. She provides examples to illustrate these theorems, including the Weierstrass parametric representation and the general theory of relativity. The paper concludes by discussing the implications of these theorems for the existence of first integrals in variational problems and the laws of conservation of energy.Emmy Noether's paper "Invariante Variationsprobleme" (1918) explores the relationship between variational problems and continuous groups, as introduced by Sophus Lie. The paper combines the methods of the calculus of variations with Lie's group theory to derive theorems that describe the invariance of integrals under transformations from a continuous group. Noether's main results are: 1. **Theorem I**: If an integral \( I \) is invariant under a finite continuous group \( \mathfrak{G}_\rho \), then \( \rho \) linearly independent combinations of the Lagrange expressions become divergences. Conversely, if \( \rho \) linearly independent combinations of the Lagrange expressions become divergences, then \( I \) is invariant under \( \mathfrak{G}_\rho \). 2. **Theorem II**: If an integral \( I \) is invariant under an infinite continuous group \( \mathfrak{G}_{\infty, \rho} \) where the arbitrary functions occur up to the \( \sigma \)-th derivative, then there exist \( \rho \) identity relationships between the Lagrange expressions and their derivatives up to the \( \sigma \)-th order. Conversely, if these relationships hold, then \( I \) is invariant under \( \mathfrak{G}_{\infty, \rho} \). Noether also discusses the converse of these theorems, showing that if the Lagrange expressions are relative invariant under a group, then the integral \( I \) is invariant. She provides examples to illustrate these theorems, including the Weierstrass parametric representation and the general theory of relativity. The paper concludes by discussing the implications of these theorems for the existence of first integrals in variational problems and the laws of conservation of energy.