Direct Energy, 2018a

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CONTENTS
13.3 Derivation of the Lagrangian ..................296
13.4 Deriving the Thomas Fermi Equation ............305
13.5 From Thomas Fermi Theory to Density Functional Theory . 308
13.6 Problems .............................309
14 Lie Analysis 311
14.1 Introduction ...........................311
14.1.1 Assumptions and Notation ...............312
14.2 Types of Symmetries ......................313
14.2.1 Discrete versus Continuous ...............313
14.2.2 Regular versus Dynamical . . . ............314
14.2.3 Geometrical versus Nongeometrical ..........314
14.3 Continuous Symmetries and Innitesimal Generators ....315
14.3.1 Denition of Innitesimal Generator .........315
14.3.2 Innitesimal Generators of the Wave Equation ....316
14.3.3 Concepts of Group Theory ..............320
14.4 Derivation of the Innitesimal Generators ..........322
14.4.1 Procedure to Find Innitesimal Generators ......322
14.4.2 Thomas Fermi Equation Example ...........323
14.4.3 Line Equation Example . . . .............326
14.5 Invariants ............................330
14.5.1 Importance of Invariants ................330
14.5.2 Noether's Theorem ...................330
14.5.3 Derivation of Noether's Theorem ...........331
14.5.4 Line Equation Invariants Example ...........333
14.5.5 Pendulum Equation Invariants Example .......334
14.6 Summary ............................336
14.7 Problems .............................337
Appendices 341
A. Variable List . ...........................341
B. Select Units of Measure ......................349
C. Overloaded Terminology .....................351
D. Specic Energies ..........................353
References 355
Index 370
About the Book 374
About the Author 374