Stanford University Department of Aeronautics and Astronautics

Introduction to Symmetry Analysis

Brian CantwellDepartment of Aeronautics and Astronautics

Stanford University

Chapter 4 -Classical Dynamics

Stanford University Department of Aeronautics and Astronautics

Consider a spring-mass system

H =1

2mdxdt

⎛⎝⎜

⎞⎠⎟

2

+12kx2 =T +V

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,

L =T −V

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 = r1 - r2.

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 ProblemLet

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