Startling simplicity P (6) iδui = δL − ( !) d P 1 @L 2 @L (1) k @L (k−1) − dx 1 (1) δui + 1 (2) δui + ··· + 1 (k) δui @ui @ui @ui ( !) d2 P 2 @L 3 @L (1) k @L (k−2) + dx2 2 (2) δui + 2 (3) δui + ··· + 2 (k) δui @ui @ui @ui k dk P k @L ··· + (−1) dxk k (k) δui : @ui On its face, an equation Euler or Lagrange could have written. Direct consequence of additivity, chain rule, and integration by parts. In several independent variables those lines become a sum: @A @A Div(A) = 1 + ··· + n : @x1 @xn Noether’s Aj are not functions. They are differential operators. d The last three lines are a derivative dx . Direct consequence of additivity, chain rule, and integration by parts. In several independent variables those lines become a sum: @A @A Div(A) = 1 + ··· + n : @x1 @xn Noether’s Aj are not functions. They are differential operators. d The last three lines are a derivative dx . On its face, an equation Euler or Lagrange could have written. In several independent variables those lines become a sum: @A @A Div(A) = 1 + ··· + n : @x1 @xn Noether’s Aj are not functions. They are differential operators. d The last three lines are a derivative dx . On its face, an equation Euler or Lagrange could have written. Direct consequence of additivity, chain rule, and integration by parts. Noether’s Aj are not functions. They are differential operators. d The last three lines are a derivative dx . On its face, an equation Euler or Lagrange could have written. Direct consequence of additivity, chain rule, and integration by parts. In several independent variables those lines become a sum: @A @A Div(A) = 1 + ··· + n : @x1 @xn d The last three lines are a derivative dx . On its face, an equation Euler or Lagrange could have written. Direct consequence of additivity, chain rule, and integration by parts. In several independent variables those lines become a sum: @A @A Div(A) = 1 + ··· + n : @x1 @xn Noether’s Aj are not functions. They are differential operators. 1918: not symmetry, but transforming variables: x1;:::; xn ! y1;:::; yn. 1918: defines a divergence as a kind of “expression” – for good reasons we will see. She (like Klein) put “conservation laws” in quote marks. Thus Noether’s originality, generality, simplicity. Noether thought in formulas (always). circa. 1930 1918: defines a divergence as a kind of “expression” – for good reasons we will see. She (like Klein) put “conservation laws” in quote marks. Thus Noether’s originality, generality, simplicity. Noether thought in formulas (always). 1918: not symmetry, but transforming variables: x1;:::; xn ! y1;:::; yn. circa. 1930 for good reasons we will see. She (like Klein) put “conservation laws” in quote marks. Thus Noether’s originality, generality, simplicity. Noether thought in formulas (always). 1918: not symmetry, but transforming variables: x1;:::; xn ! y1;:::; yn. 1918: defines a divergence as a kind of “expression” – circa. 1930 She (like Klein) put “conservation laws” in quote marks. Thus Noether’s originality, generality, simplicity. Noether thought in formulas (always). 1918: not symmetry, but transforming variables: x1;:::; xn ! y1;:::; yn. 1918: defines a divergence as a kind of “expression” – for good reasons we will see. circa. 1930 Thus Noether’s originality, generality, simplicity. Noether thought in formulas (always). 1918: not symmetry, but transforming variables: x1;:::; xn ! y1;:::; yn. 1918: defines a divergence as a kind of “expression” – for good reasons we will see. circa. 1930 She (like Klein) put “conservation laws” in quote marks. Noether thought in formulas (always). 1918: not symmetry, but transforming variables: x1;:::; xn ! y1;:::; yn. 1918: defines a divergence as a kind of “expression” – for good reasons we will see. circa. 1930 She (like Klein) put “conservation laws” in quote marks. Thus Noether’s originality, generality, simplicity. Euler and Lagrange knew roughly: Symmetry $ conservation. Hamilton’s whole method of “ignorable coordinates.” Noether cites complicated derivations of SR and Newtonian mechanics from symmetries; extremely specific on-going GR. Her breakthrough: Utter generality. It is all simply: A combination of the methods of the formal calculus of vari- ations with those of Lie’s group theory. (Noether 1918) Merely put the two together: the result lies on the page before you. (Vast generality ! unforeseen applications. Not Noether’s concern.) Background. Noether cites complicated derivations of SR and Newtonian mechanics from symmetries; extremely specific on-going GR. Her breakthrough: Utter generality. It is all simply: A combination of the methods of the formal calculus of vari- ations with those of Lie’s group theory. (Noether 1918) Merely put the two together: the result lies on the page before you. (Vast generality ! unforeseen applications. Not Noether’s concern.) Background. Euler and Lagrange knew roughly: Symmetry $ conservation. Hamilton’s whole method of “ignorable coordinates.” Her breakthrough: Utter generality. It is all simply: A combination of the methods of the formal calculus of vari- ations with those of Lie’s group theory. (Noether 1918) Merely put the two together: the result lies on the page before you. (Vast generality ! unforeseen applications. Not Noether’s concern.) Background. Euler and Lagrange knew roughly: Symmetry $ conservation. Hamilton’s whole method of “ignorable coordinates.” Noether cites complicated derivations of SR and Newtonian mechanics from symmetries; extremely specific on-going GR. Utter generality. It is all simply: A combination of the methods of the formal calculus of vari- ations with those of Lie’s group theory. (Noether 1918) Merely put the two together: the result lies on the page before you. (Vast generality ! unforeseen applications. Not Noether’s concern.) Background. Euler and Lagrange knew roughly: Symmetry $ conservation. Hamilton’s whole method of “ignorable coordinates.” Noether cites complicated derivations of SR and Newtonian mechanics from symmetries; extremely specific on-going GR. Her breakthrough: It is all simply: A combination of the methods of the formal calculus of vari- ations with those of Lie’s group theory. (Noether 1918) Merely put the two together: the result lies on the page before you. (Vast generality ! unforeseen applications. Not Noether’s concern.) Background. Euler and Lagrange knew roughly: Symmetry $ conservation. Hamilton’s whole method of “ignorable coordinates.” Noether cites complicated derivations of SR and Newtonian mechanics from symmetries; extremely specific on-going GR. Her breakthrough: Utter generality. A combination of the methods of the formal calculus of vari- ations with those of Lie’s group theory. (Noether 1918) Merely put the two together: the result lies on the page before you. (Vast generality ! unforeseen applications. Not Noether’s concern.) Background. Euler and Lagrange knew roughly: Symmetry $ conservation. Hamilton’s whole method of “ignorable coordinates.” Noether cites complicated derivations of SR and Newtonian mechanics from symmetries; extremely specific on-going GR. Her breakthrough: Utter generality. It is all simply: Merely put the two together: the result lies on the page before you. (Vast generality ! unforeseen applications. Not Noether’s concern.) Background. Euler and Lagrange knew roughly: Symmetry $ conservation. Hamilton’s whole method of “ignorable coordinates.” Noether cites complicated derivations of SR and Newtonian mechanics from symmetries; extremely specific on-going GR. Her breakthrough: Utter generality. It is all simply: A combination of the methods of the formal calculus of vari- ations with those of Lie’s group theory. (Noether 1918) the result lies on the page before you. (Vast generality ! unforeseen applications. Not Noether’s concern.) Background. Euler and Lagrange knew roughly: Symmetry $ conservation. Hamilton’s whole method of “ignorable coordinates.” Noether cites complicated derivations of SR and Newtonian mechanics from symmetries; extremely specific on-going GR. Her breakthrough: Utter generality. It is all simply: A combination of the methods of the formal calculus of vari- ations with those of Lie’s group theory. (Noether 1918) Merely put the two together: (Vast generality ! unforeseen applications. Not Noether’s concern.) Background. Euler and Lagrange knew roughly: Symmetry $ conservation. Hamilton’s whole method of “ignorable coordinates.” Noether cites complicated derivations of SR and Newtonian mechanics from symmetries; extremely specific on-going GR. Her breakthrough: Utter generality. It is all simply: A combination of the methods of the formal calculus of vari- ations with those of Lie’s group theory. (Noether 1918) Merely put the two together: the result lies on the page before you. Not Noether’s concern.) Background. Euler and Lagrange knew roughly: Symmetry $ conservation. Hamilton’s whole method of “ignorable coordinates.” Noether cites complicated derivations of SR and Newtonian mechanics from symmetries; extremely specific on-going GR. Her breakthrough: Utter generality. It is all simply: A combination of the methods of the formal calculus of vari- ations with those of Lie’s group theory. (Noether 1918) Merely put the two together: the result lies on the page before you. (Vast generality ! unforeseen applications. Background. Euler and Lagrange knew roughly: Symmetry $ conservation. Hamilton’s whole method of “ignorable coordinates.” Noether cites complicated derivations of SR and Newtonian mechanics from symmetries; extremely specific on-going GR. Her breakthrough: Utter generality. It