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

Stanford University Department of Aeronautics and Astronautics Consider a spring-mass system Equation of motion Energy is conserved. The sum of kinetic energy, is called the Hamiltonian.

Stanford University Department of Aeronautics and Astronautics 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. There is a very general approach to problems of this type called Lagrangian dynamics. Usually the extremum is a minimum and this theory is often called the principle of least action. The kernel of the integral is called the Lagrangian. Typically,

Stanford University Department of Aeronautics and Astronautics Apply a small variation in the coodinates and velocities. Consider

Stanford University Department of Aeronautics and Astronautics At an extremum in S the first variation vanishes. Using Integrate by parts. At the end points the variation is zero.

Stanford University Department of Aeronautics and Astronautics The Lagrangian satisfies the Euler-Lagrange equations. Spring mass system The Euler-Lagrange equations generate

Stanford University Department of Aeronautics and Astronautics The Two-Body Problem The Lagrangian of the two-body system is

Stanford University Department of Aeronautics and Astronautics Set the origin of coordinates at the center-of-mass of the two points Insert (4.80) into (4.79). where r = r 1 - r 2.

Stanford University Department of Aeronautics and Astronautics In terms of the center-of-mass coordinates where the reduced mass is

Stanford University Department of Aeronautics and Astronautics Equations of motion generated by the Euler-Lagrange equations The Hamiltonian is

Stanford University Department of Aeronautics and Astronautics 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)

Stanford University Department of Aeronautics and Astronautics Use the Hamiltonian to determine the radius Integrate Determine the angle from conservation of angular momentum

Stanford University Department of Aeronautics and Astronautics The particle moving under the influence of the central field is constrained to move in an annular disk between two radii.

Stanford University Department of Aeronautics and Astronautics Kepler’s Two-Body Problem Let Lagrangian Generalized momenta

Stanford University Department of Aeronautics and Astronautics Gravitational constant Equations of motion In cylindrical coordinates

Stanford University Department of Aeronautics and Astronautics The two-body Kepler solution Relationship between the angle and radius or

Stanford University Department of Aeronautics and Astronautics Trajectory in Cartesian coordinates

Stanford University Department of Aeronautics and Astronautics Semi-major and semi-minor axes Apogee and perigee

Stanford University Department of Aeronautics and Astronautics Orbital period Area of the orbit Equate (4.108) and (4.109)

Stanford University Department of Aeronautics and Astronautics Invariant group of the governing equations