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# Variational Principles of Mechanics

## AAE60700

3

January 19, 2021

### Learning Objective:

Graduate students in science and engineering will benefit from a strong background in the more abstract and intellectually satisfying areas of dynamical theory. The course will concentrate on the discoveries of Hamilton and Jacobi which have resulted in variational principles of dynamics having unusual elegance and beauty. After reviewing the Special Theory of Relativity, General Relativity will be introduced. Tensor algebra will be developed and Einstein's three crucial tests (precession of Mercury's orbit, gravitational redshift, and deflection of light) will be derived.

### Description:

Variational principles of mechanics are derived from the calculus of variations. It is shown that Lagrange's equations can be derived from Hamilton's principle, in which the canonical integral (containing the Lagrangian) is extremized. Similarly, a variational principle is used to derive the motion of freely falling bodies in General Relativity. These variational principles give unifying insight into dynamics, optics, optimization, General Relativity, and other topics. Familiarity with General Relativity is becoming more and more relevant to aerospace engineers because of its importance in the Global Positioning System, high-precision spacecraft trajectory propagation, and in new tests of General Relativity that are being conducted in space missions.
Lecture Notes Example

### Topics Covered:

Generalized coordinates, configuration space, nonholonomic constraints, kinetic energy and Riemmanian geometry, transformation equations, the two postulates of Special Relativity, Lorentz transformations. Calculus of variations, extremum problems, Euler-Lagrange equations, variation with auxiliary conditions and nonholonomic conditions. The twin paradox, simultaneity in Special Relativity, the invariant interval, world line, Einstein's train. Hamilton's principle and nonholonomic equations of constraint. Addition of velocities in Special Relativity, the velocity parameter, generalized Doppler effect, and relativistic dynamics. Einstein's famous mass-energy equation. The momentum-energy four-vector. Conservative systems, Jacobi integral, ignorable coordinates, the Routhian function, and Liouville systems. Accelerated systems in Special Relativity, relativistic rockets, and the limits of space travel. Derivation of Hamilton's equations for holonomic and nonholonomic systems. Special Relativity in four-vectors. A brief history of General Relativity, equivalence principle, deflection of light, precession of the perihelion of Mercury's orbit, gravitational red shift, Einstein's elevator. Non-contemporaneous variations, principle of least action, Fermat's principle of optics, Kepler's problem.

### Prerequisites:

Graduate standing in engineering, science, or mathematics.

### Applied / Theory:

25 / 75

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### Web Content:

Syllabus, lecture notes, and homework assignments.

### Homework:

75% of the grade will be based on the student's journal which will be turned in approximately every two weeks for grading. Electronic submission of homework.

### Projects:

Required - May be job related. 25% of the grade will be based on a written term paper to be turned in near the end of the semester.

No exams.

### Textbooks:

Official textbook information is now listed in the Schedule of Classes. NOTE: Textbook information is subject to be changed at any time at the discretion of the faculty member. If you have questions or concerns please contact the academic department.
Required: The Variational Principles of Mechanics (Dover Books on Physics), 4th Revised Edition, by Cornelius Lanczos, Dover Publications ISBN:9780486650678

### Computer Requirements:

ProEd Minimum Computer Requirements

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