1 Survey of elementary principles Some history: 1600: Galileo Galilei 1564 – 1642 cf. section 7.0 ∼ Johannes Kepler 1571 – 1630 cf. section 3.7 1700: IsaacNewton 1643–1727 cf. section 1.1 ∼ 1750–1800: LeonhardEuler 1707–1783 cf. section 1.4 ∼ JeanLeRondd’Alembert 1717–1783 cf. section 1.4 Joseph-LouisLagrange 1736–1813 cf. section 1.4, 2.3 1850: CarlGustavJacobJacobi 1804–1851 cf. section 10.1 ∼ William Rowan Hamilton 1805 – 1865 cf. section 2.1 Joseph Liouville 1809 – 1882 cf. section 9.9 1900: Albert Einstein 1879 – 1955 cf. section 7.1 ∼ Emmy Amalie Noether 1882 – 1935 cf. section 8.2 1950: Vladimir Igorevich Arnold 1937 – 2010 cf. section 11.2 ≥ Alexandre Aleksandrovich Kirillov 1936 – BertramKostant 1928– Jean-MarieSouriau 1922–2012 JerroldEldonMarsden 1942–2010 Alan David Weinstein 1943 – FYGB08 – HT14 1 2014-11-24 1.1 Mechanics of a particle 1.1 Concepts: space, time kinematics, dynamics, statics coordinate system, reference frame, inertial frame, Galilean frame position, velocity, acceleration mass point, point mass inertial mass, gravitational mass, rest mass momentum, angular momentum force, torque, force field work, kinetic energy, conservative force, friction simply connected region, curl-free field potential energy, potential, total energy conservation law, conserved quantity, conserved charge Results: Newton’s second law conservation of momentum conservation of angular momentum conservation of total energy Formulas: (1.3) F~ = ~p˙ (1.12) W (P)= F~ d~s 12 P · (1.16) F~ (~r)= ~R V (~r) −∇ Elementary fact: Physics is independent of the choice of coordinate system. Warnings: j Do not mix up the notions ‘frame’ and ‘coordinate system’. j There are systems with F~ = ~ V but V time-dependent. −∇ FYGB08 – HT14 2 2014-11-24 1.2 Mechanics of a system of particles 1.2 Concepts: internal and external forces distance vector center of mass strong law of action and reaction mechanical equilibrium Results: Newton’s third law (weak law of action and reaction) for conservative forces depending only on distance center of mass motion conservation of total momentum conservation of total angular momentum conservation of total energy angular momentum as sum of c.m. term and term for relative motion kinetic energy as sum of c.m. term and term for relative motion Formulas: ˙ ~ ext ~ (1.19) ~pi = Fi + j Fji 1 ~ ~ (1.20 2 ) Fij = Fji P 2 − d ~ ~ ext (1.22) M dt2 R = F d ~ ~ ext (1.26) dt L = N ′ ~ (1.27) ~ri = ~ri + R (1.28) L~ = R~ M ~v + ~r′ ~p′ × i i × i 1 2 1 ′2 (1.31) T = 2 M v + 2 P i mi vi P FYGB08 – HT14 3 2014-11-24 1.3 Constraints 1.3 Concepts: dynamical and non-dynamical parts of a system constraint holonomic and non-holonomic constraints, semi-holonomic constraints skleronomic and rheonomic constraints independent dynamical variables, generalized coordinate, degree of freedom constraint force, applied force Results: presence of constraints = particle positions no longer independent ⇒ constraint forces are usually not known explicitly Formulas : (1.37) fκ = fκ(~r1, ~r2, ... ; t)=0 (1.38) ~ri = ~ri(q1, q2, ... , qN◦ ; t) q = q (~r , ~r , ... , ~r ; t) (holonomic constraints) − j j 1 2 N Examples: rigid body bead sliding on a wire disk rolling on a plane Warnings: j A generalized coordinate need not have dimension of length. FYGB08 – HT14 4 2014-11-24 1.4 D’Alembert’s principle and Euler--Lagrange equations 1.4 Concepts: virtual displacement, virtual work effective force generalized velocity generalized force Lagrangian total derivative Results: principle of virtual work, d’Alemberts principle ‘cancellation of dots’ rule for holonomic constraints (Euler-)Lagrange equations of motion Formulas: (1.43) F~ appl δ~r =0 i · i i X (1.45) (F~ appl ~p˙ ) δ~r =0 i − i · i i X ∂~r (1.49) Q = F~ i j i · ∂q i j X ∂~v ∂~r (1.51) i = i ∂q˙j ∂qj d ∂T ∂T (1.52) [ Qj] δqj =0 dt ∂q˙ − ∂q − j j j X d ∂T ∂T (1.53) = Qj dt ∂q˙j − ∂qj ∂V (1.54) Qj = ∂qj (1.56) L = T V − d ∂L ∂L (1.57) =0 dt ∂q˙j − ∂qj Warnings: j The notation F~ is often used for the applied rather than the total force. j Generically the kinetic energy depends also on the generalized coordinates, not only on the generalized velocities. FYGB08 – HT14 5 2014-11-24 1.5 Velocity dependent potentials and the dissipation function 1.5 Concepts: generalized potential, velocity dependent potential monogenic system Rayleigh’s dissipation function Results: — Formulas: ∂U d ∂U (1.58) Qj = + −∂qj dt ∂q˙j (1.62) U = eφ e A~ ~v e.m. − · (1.68) F~ fric = ~ −∇~vF ∂ (1.69) Qj = F −∂q˙j d ∂L ∂L ∂ (1.70) + F =0 dt ∂q˙j − ∂qj ∂q˙j Examples: charged particle in an electric and magnetic field frictional drag force on a sphere (Stokes’s law) 1.6 Simple applications of the Lagrangian formulation 1.6 Examples: particle in free space particle in free space, in cylindrical coordinates Atwood’s machine bead sliding on a uniformly rotating wire FYGB08 – HT14 6 2014-11-24 2 Variational Principles and Lagrange’s Equations 2.1 Hamilton’s principle 2.1 Concepts: integral principle functional, extremum of a functional configuration space, path in configuration space action, action integral Hamilton’s principle Results: — Formulas: t2 (2.1) S = S(t1, t2)= L dt t1 t2 Z (2.2) δS[q] δ L(q, q,˙ t) dt =0 ≡ Zt1 Warnings: j Two distinct meanings of “q”: as a coordinate of configuration space as a path q = q(t) in configuration space FYGB08 – HT14 7 2014-11-24 2.2 Calculus of variations 2.2 Concepts: functional, stationarity condition calculus of variations, independent and dependent variables path, neighboring paths, one-parameter family of paths boundary condition catenary, brachistochrone problem Results: — Formulas: x2 (2.3) J[y]= f(y, y,˙ x) dx Zx1 x2 ∂f d ∂f ∂y (2.9) ( ) dx =0 ∂y − dx ∂y˙ ∂α Zx1 α=0 Examples : shortest path between two points in a plane minimum area of a surface of revolution brachistochrone problem 2.3 Derivation of Lagrange’s equations from Hamilton’s principle 2.3 Results: Euler--Lagrange equations of motion derived from Hamilton’s principle – for monogenic systems with holonomic constraints FYGB08 – HT14 8 2014-11-24 2.4 Lagrange multipliers 2.4 Concepts: Lagrange multiplier semi-holonomic constraints Results: Formulas: t2 m (2.21) δ t1 (L + s=1 λs fs) dt =0 dR ∂L ∂LP m ∂fs (2.23) = Q˜j := λs dt ∂q˙j − ∂qj − s=1 ∂qj P Examples: Hoop rolling down an inclined plane Warnings: j Lagrange multipliers are directly applicable only in the case of holonomic systems Warning This section has been rewritten completely in newer versions of the book 2.5 Advantages of a variational principle formulation 2.5 Concepts: resistor, inductor, capacitor battery, electromotive force Results: electric-circuit analogues of mechanical quantities Formulas 1 : (2.42) j q¨j + jk q¨k + Rj q˙j + qj = j(t) L L Cj V k Xj=6 k Examples: battery in series with a resistance and an inductance inductance in series with a capacitance, as analogue of the simple harmonic oscillator FYGB08 – HT14 9 2014-11-24 2.6 Conservation laws and symmetry properties 2.6 Concepts: initial conditions first integral of equations of motion generalized momentum, canonical momentum, conjugate momentum cyclic coordinate, ignorable coordinate translational symmetry, rotational symmetry Results: conservation of the generalized momentum for a cyclic coordinate Formulas: ∂L (2.44) pj = ∂q˙j Examples: system of charged particles in an electromagnetic field uniform translation of a system uniform rotation of a system about a prescribed axis Warnings: j even when pj has dimension of momentum, it does not necessarily coincide with ordinary mechanical momentum FYGB08 – HT14 10 2014-11-24 3 The central force problem 3.1 Reduction to the equivalent one-body problem 3.1 Concepts: two-body problem interaction potential reduced mass Results: conservation of total momentum reduction of a two-body to a one-body problem Formulas: ′ m2 ′ m1 (3.2) ~r1 = ~r, ~r2 = ~r −m1 + m2 m1 + m2 ˙ (3.3) L = 1 M R~ 2 + 1 µ ~r˙ 2 U(~r, ~r,˙ ...) 2 2 − 1 1 1 (3.5) = + µ m1 m2 FYGB08 – HT14 11 2014-11-24 3.2 The equations of motion and first integrals 3.2 Concepts: central potential, central force spherical symmetry areal velocity planar polar coordinates quadrature Results: conservation of angular momentum, implying motion in a plane Kepler’s second law, resulting from angular momentum conservation energy conservation, yielding t = t(r) reduction of the central force problem to two quadratures Formulas: (3.8) ℓ = mr2 θ˙ dV ℓ2 (3.12) f(r) = m r¨ ≡ − dr − mr3 dr (3.17) dt = 2 ℓ2 (E V 2 ) m − − 2mr q ℓ dt (3.19) dθ = mr2 FYGB08 – HT14 12 2014-11-24 3.4 The virial theorem 3.4 Concepts: virial ideal gas, equipartition, pressure, Boltzmann constant Results: virial theorem for periodic motion ideal gas law Formulas: d (3.24) ~p ~r =2 T + F~ ~r dt i · i i · i i i X X 1 (3.26) T = F~ ~r −2 i · i 1 XdV (3.28) T = r 2 dr n +1 (3.29) T = V 2 FYGB08 – HT14 13 2014-11-24 3.3 The equivalent one-dimensional problem, and classification of orbits 3.3 Concepts: effective potential, angular momentum barrier bounded and unbounded motion turning point Results: qualitative form of possible orbits from graph for Veff circular orbits when the energy is minimal Formulas: ℓ2 (3.22) V = V + eff 2mr2 3.5 The differential equation for the orbit, and integrable power-law potentials 3.5 Concepts: orbit, orbit equation turning point elliptic functions Results: mirror symmetry of an orbit with at least one turning point orbits given by elementary functions for some power-law potentials Formulas: d ℓ d (3.32) = dt mr2 dθ d2u m d (3.34) + u = V ( 1 ) dθ2 −ℓ2 du u du (3.37) dθ = − 2mE 2mV 2 2 2 u ℓ − ℓ − q FYGB08 – HT14 14 2014-11-24 3.6 Conditions