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Euler-Lagrange Equations of Second Kind
Phase Space, Generalized Coordinates, Constraints, Calculus of Variations and Euler-Lagrange Equations of Second Kind.
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Lagrangian Formalism
Lagrangian Formalism, Particle Systems, Friction and Dissipation.
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Symmetries and Conservation Laws
Generalized Momenta, Cyclic Coordinates, Noether's Theorem, Conservation of Energy.
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Stability and Bifurcation
Chaotic and Nonchaotic Dynamics, Sensitivity to Initial Conditions and Deterministic Chaos, Lyapunov Stability, Lyapunov's First Method, Lyapunov's Second Method, Bifurcations, Attractors.
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Euler-Lagrange Equations of First Kind
Lagrange Multiplier, Euler-Lagrange Equations of First Kind, SHAKE and RATTLE.
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Multi-body Problems and Rigid Bodies
Central Force, Two-body Problem, Effective Potential, Multi-body Problems, Center of Mass Theorem, Angular Momentum Theorem, Euler Angles, Lagrangian Equations of the Rigid Body.
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Linear and Nonlinear Oscillations
Oscillators with a Single Degree of Freedom, Transition to the Continuum, Linear and Non-linear Forces, Calculation of Perturbations, Harmonic Balance, Enforced Non-linear Oscillation, Self- and Parameter-excited Oscillation.
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Hamiltonian Formalism
Legendre Transformation, Hamiltonian mechanics, Poisson Brackets.
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Canonical Transformations and Invariances
Point Transformations, Canonical Transformations, Generators, Canonical Invariances of Poisson Brackets, Canonical Invariances of the Phase Volume.
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Liouville's Theorem
Phase Space Trajectories, Foundations of Statistical Mechanics, Liouville's Theorem and its Consequences.
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Discrete Lagrangian and Hamiltonian Formalisms
Symplectic Transformations, Symplecticity and Variational Integrators.
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Hamilton Jacobi Theory
Hamilton-Jacobi Formalism, Principal Function, Integrability, Level Set Method.
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Transition to Quantum Mechanics
Quantum Objects, Copenhagen Interpretation, Time-independent Schrödinger Equation, Time-dependent Schrödinger Equation, Single Configuration Ansatz, Time-dependent Self-consistent Field System, Ehrenfest’s Molecular Dynamics.
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Relativity
Space and Time, Galileo's principle, Einstein's Postulates, Lorentz Transformation, Time Dilation and Length Contraction, Minkowski Diagrams, Doppler Effect, Spacetime and Four-vectors, Relativistic Momentum, Mass and Energy, Photons.
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Fields
Classical View on Gravitation, Electrostatics, Magnetostatics, Electrodynamics, Maxwell's Equations, Gravitation in General Relativity, Quantum Fields.
