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Introduction to Symmetry AnalysisPowerPoint Presentation

Introduction to Symmetry Analysis

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Introduction to Symmetry Analysis

Chapter 4 -Classical Dynamics

Brian Cantwell

Department of Aeronautics and Astronautics

Stanford University

Equation of motion

Energy is conserved. The sum of kinetic energy,

is called the Hamiltonian.

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.

Consider called Lagrangian dynamics.

Apply a small variation in the coodinates and velocities.

At an extremum in S the first variation vanishes. called Lagrangian dynamics.

Using

Integrate by parts.

At the end points the variation is zero.

The Lagrangian satisfies the Euler-Lagrange equations. called Lagrangian dynamics.

Spring mass system

The Euler-Lagrange equations generate

The Two-Body Problem called Lagrangian dynamics.

The Lagrangian of the two-body system is

Set the origin of coordinates at the center-of-mass of the two points

Insert (4.80) into (4.79).

where r = r1 - r2.

In terms of the center-of-mass coordinates two points

where the reduced mass is

Equations of motion generated by the Euler-Lagrange equations

The Hamiltonian is

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)

Use the Hamiltonian to determine the radius is convenient to express the position of the particle in terms of cylindrical coordinates.

Integrate

Determine the angle from conservation of angular momentum

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

Kepler’s Two-Body Problem is constrained to move in an annular disk between two radii.

Let

Lagrangian

Generalized momenta

Gravitational constant is constrained to move in an annular disk between two radii.

Equations of motion

In cylindrical coordinates

The two-body Kepler solution is constrained to move in an annular disk between two radii.

Relationship between the angle and radius

or

Trajectory in Cartesian coordinates is constrained to move in an annular disk between two radii.

Semi-major and semi-minor axes is constrained to move in an annular disk between two radii.

Apogee and perigee

Orbital period is constrained to move in an annular disk between two radii.

Area of the orbit

Equate (4.108) and (4.109)

Invariant group of the governing equations is constrained to move in an annular disk between two radii.

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