Didactic Notes on General Strategy

Didactic Notes on General Strategy

Essential reference

 * Feynman's lecture on physics
 * C.-N. Yang's lecture on college physics
 * R.-K. Su's lecture on Quantum mechanics

Work and Mechanical Energy
The center of the class is the concept of Energy and how a process quantity, Work, is related to the state quantity, Kinetic Energy.
 * Introduction to energy, by showing its definition from various sources, giving examples how the expression "negative energy" is daily used.
 * Conclude that it is a conversed scalar quantity (eg. mass and money), it describes the properties of a given state (c.f. a process).
 * The definition of work, pointing out key elements, module of forces, displacement, angles (0, 90, 180). Give a simple numerical example, calculate the work done by individual force.
 * In relation with previous knowledge, give an example of the work done by the counterforce, and an example of work done by friction force.
 * Work done by a variant force, integral, graphical explanation
 * Example, work done by a string, calculated by the area, emphasize that work is related to the process itself
 * Introduce the theorem of work and kinematic energy by mentioning that force causes the variation of velocity, velocity causes the variation of position, and what kind of variation does work cause
 * Using an example of constant accelerating to show the theorem, but say that the result is general.
 * Kinetic energy is a physical quantity of a given state, give the same simple example to calculate the work by the rest force by using the theorem, mentioning in the case of variant force, the calculation of the work becomes very complicated

Poynting Vector and Energy Conservation
The center of the class is Conservation Laws which is talked about through out the course
 * Review Maxwell Eqs, mention that the cause of the introduction of displacement current is not the experimental observation but charge conservation, emphasize that conservation law always plays an important role in physics.
 * Topic today is also about conservation law of energy, which is connected with Maxwell eq.
 * From Lorentz force derived the expression of work done by EM field per unit time
 * Emphasize Maxwell eq. implies charge conservation, show it explicitly
 * Explain that energy conservation should be able to write down similarly, write energy conservation without current simply by replacing charge density and current density with energy density and energy current density
 * Explain when electric current exists, energy will be transformed into other forms through the work done by EM field, complete the conservation eq, emphasizing that explicit form of energy current density is unknown at the moment
 * Write down explicitly the expression of the energy density of EM field, does partial derivative with respect to time, show explicitly the expression of Poynting vector
 * Emphysize its physical content for a second time
 * Give examples that radiation transport energy
 * Mention that radiation also transport momentum and momentum is also conserved. Give samples such as radiometer, solar sail and comet's tail.
 * Mention that in the derivation, Poynting is actually determined only up to a Curl of any vector, the ambiguity can be solved by deriving the conservation law using Lagrangian formalism. It is more generally since it is connected with symmetry.
 * Calculate the Poynting vector using the example of plane wave, emphasize in this case it has a intuitive physical interpretation.
 * Calculate the Poynting vector for the case of charging a capacitance, showing why the energy flow is entering from the side of the capacity instead of through the cable, emphasizing intuition does not alway work.
 * Show in the example of the field created by a permanent magnet and a static charge, Poynting vector is non-vanishing, explain why.

Important reference: Feynman's lecture, Jackson's classical electrodynamics

Geometrical Optics

 * General division of optics: geometrical optics, physical optics, modern optics.
 * Geometrical optics: basic assumptions (Fermat's principle, light ray, paraxial approximation), reflection and refraction, lens
 * Physical optics: theoretical foundations (Maxwell's equations, Kirchhoff diffraction equation, Huygens–Fresnel equation), recover geometrical results, superposition and interference, polarization
 * Modern optics: theoretical foundations (quantum mechanics, quantum eletrodynamics), laser
 * Theoretical foundation of geometrical optics: light ray (eg. shadow), independent propagation, reversibility) validity (wave length, eg. no geometrical acoustics)
 * Historical review, the failure of "particle interpretation of light", the velocity of light is bigger in the water by momentum conservation, in contradiction to data
 * Reflection and refraction, eg. equation for curved refracting surface
 * Lens, real/virtual magnified/diminishes erect/inverted image, eg. derive lensmaker's equation by using two successive refracting surfaces
 * Demonstrations, glass diverges laser, use laser to show total reflection

Relativistic Fluid Dynamics

 * The basic notion of fluid: continuous medium with any element satisfies EoM (eg. Euler's equation) EoS and equation of continuity.
 * The element in question should be macroscopically small comparing to the size of the system but microscopically big to contain enough degree of freedoms (to apply concept of thermodynamics).
 * In comparison to what we have learned in colleague physics, the approach is more programmatic and mathematical (eg. the notion that fluid cannot support tangential tension is due to isotropy)
 * Instead of using $$v(x,y,z,t)$$, $$\rho(x,y,z,t)$$ and $$P(x,y,z,t)$$, one uses the energy momentum tensor, a covariant quantity to address the properties of the system.
 * For ideal fluid we change the EoM from Euler's equation to its relativistic version, rewrite the equation of continuity in a covariant form and maintain the EoS.
 * Review the energy momentum tensor. The physical contents of $$T^{munu}$$ is the $$\mu$$-th component of 4-momentum $$p$$ that goes across a unit surface area perpendicular to $$x^\nu$$. In other words, the flow of $$p^\mu$$.
 * A simpler understanding can be achieved by think of $$n^\mu$$, the number flow, which measures the number of particles that goes across the a unit surface area perpendicular to $$x^\mu$$
 * Since the particle number is a scalar, we can say that the flow of a scalar is a vector, and the flow of a vector is a tensor.
 * A tricky point to notice is that though we are already accustomed to the particle flow across a spatial surface, it is new to a certain point that the number of particle (worldlines) goes across a time-like surface actually corresponds to the particle density.
 * Therefore $$T^{00}$$ is the density of $$p^0$$, the energy density. $$T^{01}$$ is the energy goes across a unit surface perpendicular to $$x^1$$, $$T^{12}$$ is the $$p^1=p^x$$ that goes across a surface perpendicular to $$x^2$$. For the last case, the surface area has the dimension of a time interval multiplies by a 2d spatial surface, therefore it has the dimension as well as physical content of the pressure.
 * Now we show the energy momentum tensor reduces to that of a diagonal matrix for an ideal fluid in it CM frame. This is done by using the isotopy as well as Landau's definition of CM.
 * To show the problem more intuitively, we introduce the definition of the energy-momentum tensor in Weinberg's book, and show as an exercise that for ideal gas, one has exactly $$T^{00}=\epsilon$$, $$T^{11}=P$$, the l.h.s. are obtained by using the definition of the energy-momentum tensor, while the r.h.s. are obtained by using thermodynamics of ideal gas.
 * Finally, we write down the EOM for ideal fluid in term of $$\epsilon$$ and $$P$$, and decompose it parallel to and perpendicular to the four velocity $$u^\mu$$.
 * We obtain the explicit form of those equations.
 * We show that the part parallel to the four velocity corresponds to the conservation of entropy flow, while the part perpendicular to the four velocity corresponds to the non-relativistic Euler equation while take adequate approximations.