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# Analytical Dynamics - Lagrangian and 3D Rigid Body Dynamics

## Syllabus

Kinematics and Dynamics of a Single Particle | Lecture 1 of a Course.
Planar kinematics and kinetics of a particle.
Rotating and translating frames, linear momentum and angular momentum and their rates of change.
Demonstrations of the transport theorem, Matlab demo for mass sliding on parabola.
Tetherball dynamics, conservation of angular momentum and central forces.
Multi-particle system, center of mass, total linear momentum | center of mass motion | superparticle.
Multi-particle system: center-of-mass frame, angular momentum, energy, and applications.
Two particle 2D example, rigid body of particles and its kinematics.
Moment of inertia tensor/matrix for a rigid body, principal axis frame.
Newton-Euler equations for a rigid body | center of mass & inertia tensor calculation worked example.
Rotational dynamics about an arbitrary reference point, planar rigid body motion, car jump example.
3D rigid body kinematics, rotation matrices & Euler angles, Euler principal axis & angle of rotation.
Rigid body kinematic differential equation for Euler angles and rotation matrix.
Free Rigid Body Dynamics | Stability About Principal Axes | Qualitative Analysis of Spinning Objects.
Torque-free motion of a symmetric rigid body, kinetic energy of a rigid body | caber toss analysis.
Free rigid body phase space; spin stabilization of frisbees.
Lagrangian mechanics introduction | generalized coordinates, constraints, and degrees of freedom.
D’Alembert’s Principle of Virtual Work | active forces and workless constraint forces.
Lagrange's equations from D’Alembert’s principle | several worked examples.
Lagrange’s equations with conservative and non-conservative forces | phase space introduction.
Phase portraits via potential energy | bifurcations | constraint forces via Lagrange multipliers.
Lagrange multipliers and constraint forces | nonholonomic constraints | downhill race various shapes.
Constants of motion, ignorable coordinates and Routh procedure | spherical pendulum eqns derived.
Chaos in mechanical systems, Routh procedure, ignorable coordinates & symmetries | Noether's theorem.
Friction and phase portraits | Coulomb friction | cone of friction | falling broom | spinning top.
Rolling coin, bicycles, fish, Chaplygin swimmer | small oscillations about equilibrium.
Normal modes of mechanical systems.
Quasivelocities & dynamic equations | Kane's method, Kane's equations, avoiding Lagrange multipliers.
Coupled rigid bodies, impulsive dynamics, applications| trap jaw ants, leaping lizards, falling cat.

### Taught by

Ross Dynamics Lab

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