Introduction to Symmetry Analysis

1. Introduction to Symmetry Analysis Chapter 4 -Classical Dynamics Brian Cantwell Department of Aeronautics and Astronautics Stanford University

2. Consider a spring-mass system Equation of motion Energy is conserved. The sum of kinetic energy, is called the Hamiltonian.

3. There is a very general approach to problems of this type called Lagrangian dynamics. Dynamical systems that conserve energy follow a path in phase space that corresponds to an extremum in a certain integral of the coordinates and velocities called the action integral. The kernel of the integral is called the Lagrangian. Typically, Usually the extremum is a minimum and this theory is often called the principle of least action.

4. Consider Apply a small variation in the coodinates and velocities.

5. At an extremum in S the first variation vanishes. Using Integrate by parts. At the end points the variation is zero.

6. The Lagrangian satisfies the Euler-Lagrange equations. Spring mass system The Euler-Lagrange equations generate

7. The Two-Body Problem The Lagrangian of the two-body system is

8. Set the origin of coordinates at the center-of-mass of the two points Insert (4.80) into (4.79). where r = r1 - r2.

9. In terms of the center-of-mass coordinates where the reduced mass is

10. Equations of motion generated by the Euler-Lagrange equations The Hamiltonian is

11. The motion of the particle takes place in a plane and so it is convenient to express the position of the particle in terms of cylindrical coordinates. The Hamiltonian is the total energy which is conserved The equations of motion in cylindrical coordinates simplify to Angular momentum is conserved (Kepler’s Second Law)

12. Use the Hamiltonian to determine the radius Integrate Determine the angle from conservation of angular momentum

13. The particle moving under the influence of the central field is constrained to move in an annular disk between two radii.

14. Kepler’s Two-Body Problem Let Lagrangian Generalized momenta

15. Gravitational constant Equations of motion In cylindrical coordinates

16. The two-body Kepler solution Relationship between the angle and radius or

17. Trajectory in Cartesian coordinates

18. Semi-major and semi-minor axes Apogee and perigee

19. Orbital period Area of the orbit Equate (4.108) and (4.109)

20. Invariant group of the governing equations