Logistics
• Lecture notes allowed
• No numbers
• Same format as mid-terms. Four problems. I will
try to mix several concepts in one problem!
• Formulas from front page of Taylor will be given
• If you feel you need some formula, don’t hesitate
to ask!
How to prepare
• Review your lecture notes and make sure they are
complete
• Read the handouts that I posted
• Solve all homework
• Solve your mid-term tests
• Solutions are posted, but don’t look at them before
you solve the problem!
• Work out examples in textbook and lecture notes,
and look through end-of-chapter problems
• Don’t hesitate to contact me if you have any difficulties
Math
• Vectors, dot and cross product; Levi-Civita symbol
• Calculus
– Integrate by substitution of variable
– Differential in standard coordinate systems
• Vector calculus (formulas will be given as needed)
• Differential equations:
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Solve by separation of variables
Solve linear equations
Solve harmonic oscillator equation
Apply initial conditions
• Approximations, expansions, linearization
Linear problems
• Know the solution to the harmonic oscillator
problem! No dissipation, with dissipation, driven,
any initial conditions
• Know how to solve a general system of linear
ODE with constant coefficients!
Bird’s view of the classical
mechanics
Given a problem, now we know many ways to solve it
• Newton’s 2nd law: intuitive, but has
different form in different coordinates
• The only conservation laws that can be
applied “automatically” are energy,
momentum, or angular momentum
Lagrangian approach
• Based on variational principle; therefore
invariant with respect to transformation of
coordinates (q1, …qn)->(Q1, …,Qn)*
• Equations have the same form in any coordinate
system!
• In practice, you still start from natural
coordinates and L = K-U, and then make the
transformation of your choice
*Note:
velocities are not transformed independently, they follow the
transformation of q
Lagrangian approach: what kind
of problems can be solved?
• All linear problems without explicit time
dependence in coefficients. L contains only qiqj
and q˙ iq˙ j terms (quadratic). Then E-L equations
contain only linear terms.
– Linear in q˙ j term in the Lagrangian can be eliminated
as a total derivative df/dt
• T and U can be diagonalized simultaneously by
transformation to normal frequencies and normal
coordinates. Then the solution is straightforward.
Expect such problem on the final exam.
Coupled oscillations
• Normal frequencies and normal modes
• Normal coordinates: transformations in
both directions
• Inhomogeneous terms (e.g. constant
force)
• Typical problems: springs, pendulums
Lagrangian approach: what kind
of problems can be solved?
• Nonlinear problem if
– Only one degree of freedom and no time
dependence, i.e. energy is conserved
– Then equation E = T+U leads to the first-order
equation for x(t) that can always be solved in
quadratures
• All other nonlinear problems can be solved
only by reduction to a set of 1D problems
Nonlinear problems
• Need to have enough integrals of motion (equal to
the number of degrees of freedom)
• Within the Lagrangian approach, the only regular
way of finding I.o.M. is through cyclic coordinates.
Then the generalized momentum is conserved.
• Choosing the right system of coordinates is
critical! It should reflect the symmetry of the
potential U(qi)
• Examples: central force, EM field
Nonlinear problems
• Even when the problem is not solvable, there is a powerful
qualitative method: phase space (q, q˙) or (q, p)
• Motion is completely described by finding critical
(stationary) points q, q˙ = 0 and trajectories in their vicinity
• Motion in the vicinity of stationary points is described by
linearized equations! Know how to linearize!!
• Only four types of points on a 2D phase plane
• Know how to find them and the structure of phase
trajectories around them!
Hamiltonian dynamics
• Also based on the variational principle, same
invariance with respect to coordinate
transformations
– Know how to go from L to H! (next slide)
• Motion as a flow of “phase fluid” in phase space
• Powerful theorems and methods:
– Liouville’s theorem (conservation of phase volume)
– Poisson’s brackets: generate evolution of any function
f(p,q), may give new integrals of motion, allow you to
check if the transformation is canonical
Notable exceptions from H = K+U rule
• Non-inertial reference frame, for example
polar coordinates for a bead on a rotating
wire
• Charged particle in an electromagnetic
field
Canonical transformations
• The greatest advantage! Transform both q
and p. Much broader class than purely
coordinate transformations
• The main approach: transform to new
canonical coordinates in which the solution
to Hamilton’s equations is trivial
• Know how to find the generating function
and write the solution in old coordinates
Hamilton-Jacobi theory
• Most general way of finding integrals of
motion: by separation of variables in HJ
equation
• Again, identifying the right system of
coordinates is critical
• Method in search of problems: first identify
potentials and coordinates for which
variables can be separated, then see what
problems it can solve
Know the procedure!
• Choose the coordinates
• Know the Hamiltonian in standard
coordinate systems!
• Separate variables
• Write the solution
• Apply initial conditions
Rigid body rotation
• Don’t forget PHYS218 stuff!
– Center of mass
– Angular momentum and it’s split into COM
and “about COM” motion
– Kinetic energy and it’s separation into KE of
COM + KE of the motion relative to COM
– Rotation about fixed axis z. Moment of inertia.
Lz = Iz Ω. Polar coordinates.
Rigid body rotation
• L is not parallel to Ω. Tensor of inertia.
• Diagonalization of TI. Principal moments
and axes. Know how to do this!
• Euler’s equations
– Symmetric top
• Lagrangian and Hamiltonian in terms of
Euler’s angles
– Symmetric top
Non-separable (non-integrable)
systems
• Only approximate methods
• Adiabatic invariants
– Know how to find the actions and convert to
action-angle variables
– Know how to use adiabatic invariants to find
the time dependence of energy, amplitude,
and other physical quantities
• Method of averaging (not needed for
2014)
Road to chaos
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Nonlinearity
Non-integrability
Resonances
Destruction of separatrices
Resonances
• Nonlinear resonance and nonlinear
pendulum
• Back to phase space
• Instability and destruction of a separatrix
• Bifurcations
• Period-doubling bifurcation
• Properties of chaos
• From laminar to turbulent phase flow
Bifurcation diagram of a driven pendulum
θ(tn)
γ
Poincare section
Poincare section of the phase plane of a driven pendulum
http://www.physik3.gwdg.de
http://www.elmer.unibas.ch/pendulum/chaos.htm
wikipedia
Poincare maps on y,dy/dt plane for Henon-Heiles problem
C. Emanuelsson
Self-similarity
C. Emanuelsson
KAM tori for Henon-Heiles problem
C. Emanuelsson