Download International Physics Olympiad (IPhO) Formulas Cheat Sheet and more Cheat Sheet Complex Numbers Theory in PDF only on Docsity!
Formulas for IPhO
Version: July 4, 2018
I Mathematics
1. Taylor series (truncate for approximations):
F ( x ) = F ( x 0 ) +
F ( n )( x 0 )( x − x 0 ) n/n! Special case — linear approximation: F ( x ) ≈ F ( x 0 ) + F ′( x 0 )( x − x 0 ) Some examples for | x | 1: sin x ≈ x, cos x ≈ 1 − x^2 / 2 , ex^ ≈ 1 + x ln(1 + x ) ≈ x, (1 + x ) n^ ≈ 1 + nx
2. Perturbation method: find the solution itera-
tively using the solution to the ”non-perturbed” (directly solvable) problem as the 0th^ approxi- mation; corrections for the next approximation are calculated on the basis on the previous one.
3. Solution of the linear differential equation
with constant coefficients ay ′′^ + by ′^ + cy = 0: y = A exp( λ 1 x ) + B exp( λ 2 x ) , where λ 1 , 2 is the solution of the characteristic equation aλ^2 + bλ + c = 0 if λ 1 6 = λ 2. If the solution of the characteristic equation is com- plex, while a , b and c are real numbers, then λ 1 , 2 = γ ± iω and y = Ceγx^ sin( ωx + ϕ 0 ).
4. Complex numbers
z = a + b i = | z | e i ϕ, ¯ z = a − i b = | z | e −i ϕ
| z |^2 = z z ¯ = a^2 + b^2 , ϕ = arg z = arcsin
b | z | Re z = ( z + ¯ z ) / 2 , Im z = ( z − z ¯) / 2 | z 1 z 2 | = | z 1 || z 2 | , arg z 1 z 2 = arg z 1 + arg z 2 e i ϕ^ = cos ϕ + i sin ϕ cos ϕ = e
i ϕ (^) + e −i ϕ 2 ,^ sin^ ϕ^ =^
e i ϕ^ − e −i ϕ 2 i
5. Cross and dot products of vectors are dis-
tributive: a ( b + c ) = ab + ac. ~a · ~b = ~b · ~a = axbx + ay by +... = ab cos ϕ | ~a × ~b | = ab sin ϕ ; ~a × ~b = − ~b × ~a ⊥ ~a,~b ~a × ~b = ( ay bz − by az ) ~ex + ( az bx − bz ax ) ~ey +... ~a × [ ~b × ~c ] = ~b ( ~a · ~c ) − ~c ( ~a · ~b ). Mixed prod. (volume of parallelep. def. by 3 vec.): ( ~a,~b, ~c ) ≡ ( ~a · [ ~b × ~c ]) = ([ ~a × ~b ] · ~c ) = ( ~b, ~c, ~a ).
6. Cosine and sine laws:
c^2 = a^2 + b^2 − 2 ab cos ϕ a/ sin α = b/ sin β = 2 R
7. sin( α ± β ) = sin α cos β ± cos α sin β
cos( α ± β ) = cos α cos β ∓ sin α sin β tan( α ± β ) = (tan α + tan β ) / (1 ∓ tan α tan β ) cos^2 α = 1+cos 2 2 α , sin^2 α = 1 −cos 2 2 α cos α cos β = cos( α + β )+cos( 2 α − β ),... cos α + cos β = 2(cos α + 2 β + cos α − 2 β ),....
8. An angle inscribed in a circle is half of the
central angle that subtends the same arc on the circle. Conclusions: hypotenuse of a right triangle is the diameter of its circumcircle; if the angles of a quadrilateral are supplementary, it is a cyclic quadrilateral.
9. √ Surface area of a triangle = 12 aha = pr =
p ( p − a )( p − b )( p − c ) = abc/ 4 R.
10. Triangle’s centroid: intersection point of
medians, divides medians to 2:1.
11 ∗. Vector approach to geometry problems.
12. Taking derivatives:
( f g )′^ = f g ′^ + f ′ g, f [ g ( x )]′^ = f ′[ g ( x )] g ′ (sin x )′^ = cos x, (cos x )′^ = − sin x ( ex )′^ = ex, (ln x )′^ = 1 /x, ( xn )′^ = nxn −^1 (arctan x )′^ = 1 / (1 + x^2 ) (arcsin x )′^ = −(arccos x )′^ = 1 /
1 − x^2
13. Integration: the formulas are the same as
for derivatives, but with swapped left-hand-side
and rhs. (inverse operation!), e.g.∫
xn d x = xn +1 / ( n + 1).
Special case of the substitution method:∫
f ( ax + b )d x = F ( ax + b ) /a.
14. Conic sections: a 11 x^2 + 2 a 12 xy + a 22 y^2 +
a 1 x + a 2 y + a 0 = 0 with a 11 = a 22 — circle; with a 11 ·( a 11 a 22 − a^212 ) > 0 — ellipse,... < 0 — hyperbola, with a 11 a 22 = a^212 — parabola. El- lipse: l 1 + l 2 = 2 a , α 1 = α 2 , A = πab ; hy- perbola: l 1 − l 2 = 2 a , α 1 + α 2 = 0; parabola: l + h = const, α 1 = α 2.
15. Numerical methods. Newton’s iterative
method for finding roots f ( x ) = 0: xn +1 = xn − f ( xn ) /f ′( xn ).
Trapezoidal rule for approximate integration:∫
b
a
f ( x )d x ≈
b − a 2 n
[ f ( x 0 ) + 2 f ( x 1 ) +...
+2 f ( xn − 1 ) + f ( xn )]
16. Derivatives and integrals of vectors: differ-
entiate/integrate each component; alternatively differentiate by applying the triangle rule for the difference of two infinitesimally close vectors.
II General recommendations
1. Check all formulas for veracity: a) examine
dimensions; b) test simple special cases (two parameters are equal, one param. tends to 0 or ∞); c) verify the plausibility of solution’s qualitative behaviour.
2. If there is an extraordinary coincidence in
the problem text (e.g. two things are equal) then the key to the solution might be there.
3. Read carefully the recommendations in the
problem’s text. Pay attention to the problem’s formulation — insignificant details may carry vital information. If you have solved for some time unsuccessfully, then read the text again — perhaps you misunderstood the problem.
4. Postpone long and time-consuming math-
ematical calculations to the very end (when everything else is done) while writing down all the initial equations which need to be simplified.
5. If the problem seems to be hopelessly diffi-
cult, it has usually a very simple solution (and a simple answer). This is valid only for Olympiad problems, which are definitely solvable.
6. In experiments a) sketch the experimental
scheme even if you don’t have time for measure- ments; b) think, how to increase the precision of the results; c) write down (as a table) all your direct measurements.
III Kinematics
1. For a point or for a translational motion of
a rigid body (integral → area under a graph):
~v =
d ~x d t
, ~x =
~v d t ( x =
vx d t etc.)
~a =
d ~v d t
d^2 ~x d t^2
, ~v =
~a d t
t =
v − x^1 d x =
a − x^1 d vx, x =
vx ax
d vx If a = Const_._ , then previous integrals can be found easily, e.g. x = v 0 t + at^2 / 2 = ( v^2 − v^20 ) / 2 a.
2. Rotational motion — analogous to the trans-
lational one: ω = d ϕ/ d t , ε = d ω/ d t ; ~a = ~τ d v/ d t + ~nv^2 /R
3. Curvilinear motion — same as point 1, but
vectors are to be replaced by linear velocities, accelerations and path lengths.
4. Motion of a rigid body. a) vA cos α =
vB cos β ; ~vA , ~vB — velocities of pts. A and B ; α , β — angles formed by ~vA , ~vB with line AB. b) The instantaneous center of rotation ( 6 = center of curvature of material pt. trajectories!) can be found as the intersection pt. of perpendic- ulars to ~vA and ~vB , or (if ~vA, ~vB ⊥ AB ) as the intersection pt. of AB with the line connecting endpoints of ~vA and ~vB.
5. Non-inertial reference frames:
~v 2 = ~v 0 + ~v 1 , ~a 2 = ~a 0 + ~a 1 + ω^2 R~ + ~aCor Note: ~aCor ⊥ ~v 1 , ~ω ; ~aCor = 0 if ~v 1 = 0.
6 ∗. Ballistic problem: reachable region
y ≤ v 02 / (2 g ) − gx^2 / 2 v^20_._ For an optimal ballistic trajectory, initial and final velocities are perpendicular.
7. For finding fastest paths, Fermat’s and Huy-
gens’s principles can be used.
8. To find a vector (velocity, acceleration), it
is enough to find its direction and a projection to a single (possibly inclined) axes.
IV Mechanics
1. For a 2D equilibrium of a rigid body: 2
eqns. for force, 1 eq. for torque. 1 (2) eq. for force can be substituted with 1 (2) for torque. Torque is often better — “boring” forces can be eliminated by a proper choice of origin. If forces are applied only to 2 points, the (net) force application lines coincide; for 3 points, the 3 lines meet at a single point.
2. Normal force and friction force can be com-
bined into a single force, applied to the contact point under angle arctan μ with respect to the normal force.
3. Newton’s 2 nd^ law for transl. and rot. motion:
F^ ~ = m~a, M~ = I~ε ( M~ = ~r × F~ ). For 2D geometry M~ and ~ε are essentially scalars and M = F l = Ftr , where l is the arm of a force.
4. Generalized coordinates. Let the system’s
state be defined by a single parameter ξ and its time derivative ξ ˙ so that the pot. energy Π = Π( ξ ) and kin. en. K = μ ξ ˙^2 / 2; then μ ξ ¨ = −dΠ( ξ ) / d ξ. (Hence for transl. motion: force is the derivative of pot. en.)
5. If the system consists of mass points mi :
~rc =
mi~ri/
mj , P~ =
mi~vi ~L =
mi~ri × ~vi, K =
miv i^2 / 2
Iz =
mi ( x^2 i + y^2 i ) =
( x^2 + y^2 )d m.
6. In a frame where the mass center’s velocity is ~vc (index c denotes quantities rel. to the mass center): ~L = ~Lc + M Σ R~c × ~vc, K = Kc + M Σ v^2 c / 2 P^ ~ = P~c + M Σ ~vc 7. Steiner’s theorem is analogous ( b — distance of the mass center from rot. axis): I = Ic + mb^2. 8. With P~ and ~L from pt. 5, Newton’s 2 nd^ law: F^ ~ Σ = d P /~ d t, M~ Σ = d ~L/ d t 9 ∗. Additionally to pt. 5, the mom. of inertia rel. to the z -axis through the mass center can be found as Iz 0 =
i,j mimj^ [( xi^ −^ xj^ )
(^2) + ( yi −
yj )^2 ] / 2 M Σ.
10. ∑ Mom. of inertia rel. to the origin θ = mi~r^2 i is useful for calculating Iz of 2D bodies or bodies with central symmetry using 2 θ = Ix + Iy + Iz. 11. Physical pendulum with a reduced length ˜ l :
ω^2 ( l ) = g/ ( l + I/ml ) , ω ( l ) = ω (˜ l − l ) =
g/ ˜ l, ˜ l = l + I/ml
12. Coefficients for the momenta of inertia: cylinder 12 , solid sphere 25 , thin spherical shell 2 3 , rod^
1 12 (rel. to endpoint^
1 3 ), square^
1
13. Often applicable conservation laws: energy (elastic bodies, no friction), momentum (no net external force; can hold only along one axis), angular momentum (no net ext. torque, e.g. the arms of ext. forces are 0 (can be written rel. to 2 or 3 pts., then substitutes conservation of lin. mom.). 14. Additional forces in non-inertial frames of ref.: inertial force − m~a , centrifugal force mω^2 R~ and Coriolis force∗^2 m~v × Ω ~ (better to avoid it; being ⊥ to the velocity, it does not create any work). 15. Tilted coordinates: for a motion on an inclined plane, it is often practical to align axes along and ⊥ to the plane; gravit. acceleration has then both x - and y - components. Axes may also be oblique (not ⊥ to each other), but then with ~v = vx~ex + vy~ey , vx 6 = to the x -projection of ~v. 16. Collision of 2 bodies: conserved are a) net momentum, b) net angular mom., c) angular
mom. of one of the bodies with respect to the impact point , d) total energy (for elastic colli- sions); in case of friction, kin. en. is conserved only along the axis ⊥ to the friction force. Also: e) if the sliding stops during the impact, the final velocities of the contact points will have equal projections to the contact plane; f) if slid- ing doesn’t stop, the momentum delivered from one body to the other forms angle arctan μ with the normal of the contact plane.
17. Every motion of a rigid body can be repre- sented as a rotation around the instantaneous center of rotation C (in terms of velocities of the body points). NB! Distance of a body point P from C 6 = to the radius of curvature of the trajectory of P. 18. Tension in a string: for a massive hang- ing string, tension’s horizontal component is constant and vertical changes according to the string’s mass underneath. Pressure force (per unit length) of a string resting on a smooth surface is determined by its radius of curvature and tension: N = T /R. Analogy: surface ten- sion pressure p = 2 σ/R ; to derive, study the pressure force along the diameter. 19. Liquid surface takes equipot. shape (ne- glecting σ ); in incompr. liquid, p = p 0 − w , w —vol. dens. of pot. en. (for a gas, see X-6). 20. Bernoulli law for incompr. fluid:
p +
ρv^2 + ρϕ = const; in homog. field, the gravit. potential ϕ = gh. For gas of specific heat cp [J/kg], 1 2
v^2 + cpT = const_._
21 ∗. Momentum continuity by straight stream- lines: p + ρv^2 = const. 22 ∗. Adiabatic invariant: if the relative change of the parameters of an oscillating system is small during one period, the area of the loop drawn on the phase plane (ie. in p - x coordi- nates) is conserved with a very high accuracy.
23. For studying stability use a) principle of minimum potential energy or b) principle of small virtual displacement. 24 ∗. Virial theorem for finite movement: a) If F ∝ | ~r |, then 〈 K 〉 = 〈Π〉 (time averages); b) If F ∝ | ~r |−^2 , then 2 〈 K 〉 = − 〈Π〉. 25. Tsiolkovsky rocket equation ∆ v = u ln Mm.
V Oscillations and waves
1. Damped oscillator: ¨ x + 2 γ x ˙ + ω 02 x = 0 ( γ < ω 0 ). Solution of this equation is (cf. I.2.): x = x 0 e − γt^ sin( t
ω^20 − γ^2 − ϕ 0 ).
2. Eq. of motion for a system of coupled oscil- lators: ¨ xi =
j aij^ xj^.
3. A system of N coupled oscillators has N different eigenmodes when all the oscillators oscillate with the same frequency ωi , xj = xj 0 sin ( ωit + ϕij ), and N eigenfrequencies ωi (which can be multiple, ωi = ωj ). General solu- tion (with 2 N integration constants Xi and φi ) is a superposition of all the eigenmotions : xj =
i
Xixj 0 sin( ωit + ϕij + φi )
4. If a system described with a generalized coordinate ξ (cf IV-2) and K = μ ξ ˙^2 / 2 has an equilibrium state at ξ = 0, for small oscilla- tions Π( ξ ) ≈ κξ^2 / 2 [where κ = Π′′(0)] so that ω^2 = κ/μ. 5. The phase of a wave at pt. x, t is ϕ = kx − ωt + ϕ 0 , where k = 2 π/λ is a wave vec- tor. The value at x, t is a 0 cos ϕ = < a 0 eiϕ. The phase velocity is vf = νλ = ω/k and group velocity vg = dω/dk. 6. For linear waves (electromagn. w., small- amplit. sound- and water w.) any pulse can be considered as a superpos. of sinusoidal waves; a standing w. is the sum of two identical counter- propagating w.: ei ( kx − ωt )^ + ei (− kx − ωt )^ = 2 e − iωt^ cos kx. 7. Speed of sound in a gas cs =
( ∂p/∂ρ )adiab =
γp/ρ = ¯ v
γ/ 3_._
8. √ Speed of sound in elastic material cs = E/ρ. 9. Sp. of waves in shallow ( h λ ) water: v =
gh ; in a string: v =
T /ρ lin.
10. Doppler’s effect: ν = ν 0 1+ 1 − vu ‖‖^ /c /css. 11. Huygens’ principle: wavefront can be con- structed step by step, placing an imaginary wave source in every point of previous wave front. Results are curves separated by distance ∆ x = cs ∆ t , where ∆ t is time step and cs is the velocity in given point. Waves travel perpendic- ular to wavefront.
VI Geometrical optics. Photometry.
1. Fermat’s principle: waves path from point A to point B is such that the wave travels the least time. 2. Snell’s law: sin α 1 / sin α 2 = n 2 /n 1 = v 1 /v 2_._ 3. If refraction index changes continuously, then we imaginarily divide the media into lay- ers of constant n and apply Snell’s law. Light ray can travel along a layer of constant n , if the requirement of total internal reflection is marginally satisfied, n ′^ = n/r (where r is the curvature radius). 4. If refraction index depends only on z , the photon’s mom. px , py , and en. are conserved: kx, ky = Const ., | ~k | /n = Const_._ 5. The thin lens equation (pay attention to signs): 1 /a + 1 /b = 1 /f ≡ D. 6. Newton’s eq. ( x 1 , x 2 — distances of the image and the object from the focal planes): x 1 x 2 = f^2. 7. Parallax method for finding the position of an image: find such a pos. for a pencil’s tip that it wouldn’t shift with resp. to the image when moving perpendicularly the position of your eye. 8. Geometrical constructions for finding the paths of light rays through lenses: a) ray passing the lens center does not refract; b) ray ‖ to the optical axis passes through the focus; c) after refr., initially ‖ rays meet at the focal plane; d) image of a plane is a plane; these two planes meet at the plane of the lens. 9. Luminous flux Φ [unit: lumen (lm)] mea- sures the energy of light (emitted, passing a contour, etc), weighted according to the sensi- tivity of an eye. Luminous intensity [candela (cd)] is the luminous flux (emitted by a source) per solid angle: I = Φ / Ω. Illuminance [lux (lx)] is the luminous flux (falling onto a surface) per unit area: E = Φ /S. 10. Gauss theorem for luminous flux: the flux through a closed surface surrounding the point
18. Forces acting on a dipole: F = ( E ~d~e )′, F = ( B ~d~μ )′; interaction between 2 dipoles: F ∝ r −^4. 19. Point charge as a magn. dipole: dμ ∝ Φ ∝ v^2 ⊥ /B is an adiab. inv (see IV-20). 20. Electric and magnetic images: grounded (superconducting for magnets) planes act as mir- rors. Field of a grounded (or isolated) sphere can be found as a field of one (or two) fictive charge(s) inside the sphere. The field in a planar waveguide (slit between metallic plates) can be obtained as a superposition of electromagnetic plane waves. 21. Ball’s (cylinder’s) polarization in homo- geneous (electric) field: superpos. of homoge- neously charged (+ ρ and − ρ ) balls (cylinders), d ∝ E. 22. Eddy currents: power dissipation density ∼ B^2 v^2 /ρ ; momentum given during a single pass: F τ ∼ B^2 a^3 d/ρ (where d — thickness; a — size). 23. Inside a superconductor and for fast pro- cesses inside a conductor B = 0 and thus I = 0 (current flows in surface layer — skin effect). 24. Charge in homog. magnetic field B~ = B~ez moves along a cycloid with drift speed v = E/B = F/eB ; generalized mom. is conserved p ′ x = mvx − Byq, p ′ y = mvy + Bxq, as well as gen. angular mom. L ′^ = L + 12 Bqr^2. 25. MHD generator ( a — length along the direction of E~ ): E = vBa, r = ρa/bc. 26. Hysteresis: S-shaped curve (loop) in B - H -coordinates (for a coil with core also B - I - coord.): the loop area gives the thermal energy dissipation density per one cycle). 27. Fields in matter: D~ = εε 0 E~ = ε 0 E~ + P~ , where P~ is dielectric polarization vector (vol- ume density of dipole moment); H~ = B/μμ~ 0 = B/μ^ ~ 0 − J~ , where J~ is magnetization vector (vol- ume density of magnetic moment). 28. In an interface between two substances Et , Dn (= εEt ), Ht (= Bt/μ ) and Bn are continu- ous. 29. Energy density: W = 12 ( εε 0 E^2 + B^2 /μμ 0 ). 30. For μ 1, fieldlines of B are attracted to the ferromagnetic (acts as a potential hole, cf. pt. 28). 31. Current density ~j = ne~v = σ E~ = E/ρ~. 32. Lenz’s law: system responds so as to op- pose to changes.
X Thermodynamics
1. pV = mμ RT 2. Internal energy of one mole U = 2 i RT. 3. Volume of one mole at standard cond. is 22,4 l. 4. Adiabatic processes: slow as compared to sound speed, no heat exchange: pV γ^ = Const_._ (and T V γ −^1 = Const_._ ). 5. γ = cp/cv = ( i + 2) /i. 6. Boltzmann’s distribution: ρ = ρ 0 e − μgh/RT^ = ρ 0 e − U/kT^. 7. Maxwell’s distribution (how many molecules have speed v ) ∝ e − mv
(^2) / 2 kT .
8. Atm. pressure: if ∆ p p , then ∆ p = ρg ∆ h. 9. p = 13 mn ¯ v^2 = nkT , ¯ v =
3 kT /m , ν = vnS.
10. Carnot’s cycle: 2 adiabats, 2 isotherms. η = ( T 1 − T 2 ) /T 1 ; derive using S - T -coordinates. 11. Heat pump, inverse Carnot: η = (^) T 1 T −^1 T 2. 12. Entropy: dS = dQ/T. 13. I law of thermodynamics: δU = δQ + δA 14. II law of thermodynamics: ∆ S ≥ 0 (and η real ≤ η Carnot). 15. Gas work (look also p. 10) A =
p d V, adiabatic: A =
i 2
∆( pV )
16. Dalton’s law: p =
pi.
17. Boiling: pressure of saturated vapour pv = p 0 ; at the interface betw. 2 liquids: pv 1 + pv 2 = p 0. 18. Heat flux P = kS ∆ T /l ( k — thermal conductivity); analogy to DC circuits ( P corre- sponds to I , ∆ T to U , k to 1 /ρ ). 19. Heat capacity: Q =
c ( T )d T. Solids: for low temperatures, c ∝ T^3 ; for high T , c = 3 N k , where N — number of ions in crystal lattice.
20. Surface tension: U = Sσ, F = lσ, p = 2 σ/R. 21. Stefan-Boltzmann law (gray body): P = εσT^4. 22. Wien’s law: f max = AkB T /h ( A ≈ 2_._ 8), λ max = hc/A ′ kB T ( A ′^ ≈ 5)
XI Quantum mechanics
1. ~p = ¯ h~k (| ~p | = h/λ ), E = ¯ hω = hν. 2. Interference: as in wave optics. 3. Uncertainty (as a math. theorem): ∆ p ∆ x ≥
¯ h 2
, ∆ E ∆ t ≥
¯ h 2
, ∆ ω ∆ t ≥
For qualitative estimates by non-smooth shapes, h serves better (∆ p ∆ x ≈ h etc).
4. Spectra: hν = En − Em ; width of spectral lines is related to lifetime: Γ τ ≈ ¯ h. 5. Oscillator’s (eg. molecule) en. levels (with eigenfrequency ν 0 ): En = ( n + 12 ) hν 0. For many eigenfrequencies: E =
i hniνi.
6. Tunnelling effect: barrier Γ with width l is easily penetrable, if Γ τ ≈ ¯ h , where τ = l/
Γ /m.
7. Bohr’s model: En ∝ − 1 /n^2. In a (clas- sically calculated) circular orbit, there is an integer number of wavelengths λ = h/mv. 8. Compton effect — if photon is scattered from an electron, photon’s ∆ λ = λC (1 − cos θ ). 9. Photoeffect: A + mv^2 / 2 = hν ( A - work of exit for electrons). I - U -graph: photocurrent starts at the counter-voltage U = −( hν − A ) /e , saturates for large forward voltages. 10. Stefan-Boltzmann: P = σT^4.
XII Kepler laws
1. F = GM m/r^2 , Π = − GM m/r. 2. Gravitational interaction of 2 point masses (Kepler’s I law): trajectory of each of them is an ellipse, parabola or hyperbola, with a focus at the center of mass of the system. Derive from R.-L. v. (pt 9). 3. Kepler’s II law (conserv. of angular mom.): for a point mass in a central force field, radius vector covers equal areas in equal times. 4. Kepler’s III law: for two point masses at elliptic orbits in r −^2 -force field, revolution peri- ods relate as the longer semiaxes to the power of 32 : T (^) 12 /T (^) 22 = a^31 /a^32_._ 5. Full energy ( K + Π) of a body in a gravity field: E = − GM m/ 2 a. 6. For small ellipticities ε = d/a 1, trajec- tories can be considered as having a circular shapes, with shifted foci. 7. Properties of an ellipse: l 1 + l 2 = 2 a ( l 1 , l 2 — distances to the foci), α 1 = α 2 (light from one focus is reflected to the other), S = πab. 8. A circle and an ellipse with a focus at the circle’s center can touch each other only at the longer axis. 9 ∗. Runge-Lenz vector (the ellipticity vector):
~ε =
~L × ~v GM m
XIII Theory of relativity
1. Lorentz transforms (rotation of 4D space-time of Minkowski geometry), γ = 1 /
1 − v^2 /c^2 : x ′^ = γ ( x − vt ) , y ′^ = y, t ′^ = γ ( t − vx/c^2 ) p ′ x = γ ( px − mv ) , m ′^ = γ ( m − pxv/c^2 )
2. Length of 4-vector: s^2 = c^2 t^2 − x^2 − y^2 − z^2 m^20 c^2 = m^2 c^2 − p^2 x − p^2 y − p^2 z 3. Adding velocities: w = ( u + v ) / (1 + uv/c^2 ). 4. Doppler effect: ν ′^ = ν 0
(1 − v/c ) / (1 + v/c ).
5. Minkowski space can be made Euclidean if time is imaginary ( t → ict ). Then, for rot. angle ϕ , tan ϕ = v/ic. Express sin ϕ , and cos ϕ via tan ϕ , and apply the Euclidean geometry formulae. 6. Shortening of length: l ′^ = l 0 /γ. 7. Lengthening of time: t ′^ = t 0 γ. 8. Simultaneity is relative, ∆ t = − γv ∆ x/c^2. 9. F~ = d~p/dt [= (^) dtd ( m~v ), where m = m 0 γ ]. 10. Ultrarelativistic approximation: v ≈ c , p ≈ mc ,
1 − v^2 /c^2 ≈
2(1 − v/c ). 11 ∗. Lorentz tr. for E - B : B~ ′|| = B~ ||, E~ ′|| = E~ ||,
E^ ~ ⊥′ = γ ( E~ ⊥ + ~v × B~ ⊥) , B~ ⊥′ = γ ( B~ ⊥ − ~v × E~ ⊥ c^2
