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Unformatted text preview: Classical Dynamics and Fluids P 45 R UTHERFORD SCATTERING Classical Dynamics and Fluids P 46 R UTHERFORD SCATTERING II Classical Dynamics and Fluids P 47 H YPERBOLIC O RBITS IN R EPULSIVE P OTENTIAL — A NOTHER W AY O C P impact parameter semilatus rectum semimajor axis f f Path for repulsive force b a r r 8 • We can simply use the results for the attractive potential case and let φ exceed φ ∞ . The radius r is then negative and the particle traces out the repulsive branch, getting closest to O at φ = π , so that r ( π ) = a (1 + e ) . • This works because the potential energy A r is negative when r < and so represents a repulsive potential. • This approach has the considerable advantage that no sign changes are needed, but it has something of a “through the looking glass” character to it. Classical Dynamics and Fluids P 48 T HE T HREEB ODY P ROBLEM • Use a computer. . . • Some hierarchical systems can be stable indefinitely e.g. Sun, Earth and the Moon. • A general 3body encounter can be very complicated, but a general feature emerges. • If 3 bodies are allowed to attract each other from a distance (a), they will speed up and interact strongly (b). Eventually the interaction is likely to form a close binary (negative gravitational binding energy) releasing kinetic energy, which may be enough for the bodies to escape to infinity (c). • This mechanism is responsible for “evaporation” of stars from star clusters. (maybe also invalidates the virial theorem?) • The planet Pluto has a close companion Charon, and has an eccentric orbit which takes it inside Neptune’s orbit. A 3body collision amongst Neptune’s moons is the most likely cause. Classical Dynamics and Fluids P 49 R IGID B ODY D YNAMICS • A rigid body is a special case of the manyparticle system we have already studied, in which all the interparticle distances are fixed. • The location of all the particles in the body can therefore be described by 6 coordinates: the position R of the centre of mass, and 3 angles ( θ,φ,χ ) (the Euler angles), which describe the attitude of the body with respect to the spatial ( x,y,z )axes. We will define these angles later. • More importantly, the velocity of any particle in the body is determined by the velocity v of the CoM and a single angular velocity ω . • The dynamics of the rigid body is then determined by its total mass and the inertia tensor that relates the angular momentum J to the angular velocity ω . • This inertia tensor is the generalisation of the moment of inertia of a body rotating about a fixed axis. • For a body spinning about a fixed axis (say ˆ e z ), the moment of inertia I = X m ( x 2 + y 2 ) relates the angular momentum J to the angular velocity ω via J = Iω ....
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This note was uploaded on 05/09/2009 for the course DAMTP NST 1B Phy taught by Professor Sfgull during the Spring '07 term at Cambridge.
 Spring '07
 SFGull

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