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## Concluding remarks about Phys 410

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In this course, we have …
The physics of small oscillations about stable equilibrium pointsDriven damped oscillations, resonance, Fourier series
Re-visited Newtonian mechanics at a slightly higher levelMomentum, angular momentum, work, kinetic energy, conservative forces, potential energy
Learned about inertial and non-inertial reference frames,and how they affect the equations of motion (Coriolis, centrifugal)
Lagrangianand Hamiltonian mechanicsgeneralized coordinates, constraints, calculus of variations, Lagrange’s andHamilton’s equations
Detailed examination of some interesting forces:drag force (both linear and quadratic in v)Lorentz forceRocket motionCentral forceCoriolisand Centrifugal forces
Considered orbits for inverse-square-law forces, and scattering
Developed a quantitative and precise description ofclassical mechanical systems
Relativistic Mechanics: Kinematics and DynamicsTime dilation, length contraction, Lorentz transformation, Lorentz invariancefour-vectors (momentum, force)
Considerednonlinear mechanics:Driven damped pendulum:attractors, harmonics, sub-harmonics,perioddoubling bifurcations,Feigenbaumnumber, sensitivitytoinitialconditions, theLyapunovexponent, period-doubling cascade, chaos,bifurcation diagrams, state-space orbits,andthePoincarésection
Rotation of rigid bodies about an arbitrary axisInertia tensor, principal moments, principal axes
The motion of coupled oscillatorsNormal modes, normal coordinates
A Graduate Classical Mechanics class covers essentially the same topics…
Galilean invariance
Some “Take-Away” Skills for Phys 410
Recognize and recall the general solution …
Identify constraints, choose appropriate generalized coordinates, write down theLagrangian, find the conjugate momenta, write down the Hamiltonian. Solve them.Exploit ‘ignorable coordinates’ and the associated conservation laws
Transform a 2-body problem to the CM + relative coordinates, solve each problem systematically
Know how to write down vector quantities in terms of components in various coordinatesystems (Cartesian, spherical, cylindrical), and take dot products, cross products, etc.
Recognize ‘small oscillations’ situations and attack them systematically (normal modes,coords)
Recognize and exploit conservation laws
Recognize the presence of nonlinearity and utilize nonlinear dynamics concepts to understandthe motion
Understand and utilize the power of simple principles to explain complex physical phenomenaEinstein’s postulates of special relativity
Understand that details matter in nonlinear dynamics, but systematic analysis isstill useful

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