Physical Dynamics (SPA5304) Lecture Plan 2018 The numbers on the left margin are approximate lecture numbers. Items in gray are not covered this year 1 Advanced Review of Newtonian Mechanics 1.1 One Particle 1 Particle, frame, trajectory, velocity, acceleration, Newton s principle of determinacy, number of degrees of freedom, generalised coordinates. Newton s second law (ṗ = F), conservation of momentum. 2 Angular momentum, time evolution of L ( L = r F), conservation of angular momentum. Consequences of conservation of angular momentum: planar orbits, constant area velocity for the motion of two particles subject to central force (e.g. gravitational force, for which this is called Kepler s 2nd law). 3 Work of a force along a certain path P(1, 2), kinetic energy. Proof that W [P(1, 2)] = T 2 T 1, where T := (m/2)ṙ 2. Conservative systems (F = V (r)), potential energy. A force F is conservative iff the work of F along any path P(1, 2) only depends on the initial and final positions but not on the particular path. Examples of potentials. Free falling particle, gravitational potential. Harmonic oscillator. Energy conservation, E(r, ṙ) := T + V. Explicit check that Ė = 0. 4 Examples of conservative systems. Freely falling particle, application of energy conservation: E init = E final. Harmonic oscilator. Qualitative analysis of motion in one dimension from the potential: The motion with a given energy E is limited to the region where V E. Turning points, equilibrium positions, stability. The pendulum. Derivation of the equations of motion in three different ways: 1. Using Newton s equations; 2. using energy conservation; 3. using the time evolution of L. 1.2 Many Particles 5 Internal and external forces. Newton s third law (also called weak law of action and reaction): F ij = F ji. Centre of mass R, total momentum P. Proof that Ṗ = F, where F is the sum of all external forces (assuming the weak law of action and reaction). 1
Total angular momentum L, proof that L = i r i F (e) i (assuming the strong law of action and reaction: F ij = ˆr ij F ij ). Proof that L = R P + L, where L := i r i p i. Proof that L = i r i F (e) i. 6 Work done by the forces acting on a system of n particles. Kinetic energy, proof that T = T COM + T where T COM := (1/2) i miṙ2 and T := (1/2) i m iṙ 2 i. 7 Conservative forces. Proof that if the interaction depends only on the distance, then the force is conservative. Internal and external potential energy, total energy E = i T i + i V i + 1 2 i,j V ij. Energy conservation. Definition of rigid body, no work done by the internal forces. 2 Lagrangian Mechanics 8 Functionals. Calculus of variations: extremal curves of a functional, Euler-Lagrange equations. Many variables. Coordinate independence (proof). Examples: Curve of minimal length in euclidean space, brachistochrone (2018: covered in EXC 3). Applications to mechanics: Hamilton s principle of least action. Action, Lagrangian and Euler-Lagrange equations. 9 Lagrangians differing by the total derivative of a function of the coordinates give rise to the same motion. Generalised coordinates, generalised momenta, generalised forces. Cyclic coordinates, conserved quantities. Example: a particle moving in a plane, Kinetic energy and angular 10 momentum in plane polar coordinates. Planar pendulum, using Newtonian and Lagrangian approaches. Constraints. Holonomic constraints. Configuration manifold. Non-holonomic constraints. Examples (a disk rolling without slipping). Holonomic constraints and constraint forces. Constraint forces do no work. Variational calculus with constraints. The role of generalised coordinates. General procedure for solving problems with constraints. 11 Examples: Atwood s machine (time-independent holonomic constraint), a bead sliding on a uniformly rotating wire (time-dependent holonomic constraint). 12 Curvilinear coordinates: spherical polar and cylindrical coordinates. Kinetic energy and angular momentum in these coordinates. 3 Central force problem Reduction to one-body problem. Centre-of-mass R and relative motion r. Decomposition of T into center-of-mass and relative parts. Reduced mass µ := m 1 m 2 /(m 1 + m 2 ). Decoupling of the centre-of-mass motion. 2
Central force. Conservation of angular momentum. Planar motion. Proof that L = r µṙ. Reduction to the equivalent one-dimensional problem, effective potential. 13 Derivation of the orbits for the Kepler potential. Cartesian equation of the orbits. Conical sections. Kepler s laws of planetary motion. 14 Relation between the motion of r and that of sun and earth. Bertrand s theorem. Example: Spherical pendulum (2018: done in EXC). 4 Symmetries and conservation laws Symmetries. Noether s theorem. Conserved charges. Expression of the conserved quantity in the case of a Lagrangian exactly invariant (δl = 0) under a symemtry transformation. Translation invariance momentum conservation, rotational invariance angular momentum conservation. 15 Expression of the conserved quantity when δl = d (δf ). Time translation invariance dt energy conservation. Definition of Hamiltonian, H := p q L. Proof that dh/dt = L/ t. Why and when H =energy. 5 Rigid Bodies Inertial frame S I (origin O) versus body-fixed frame S II (origin O ). Identification of motion of the rigid body with that of S II. Number of degrees of freedom of a rigid body: 3 (translational) + 3 (rotational) = 6. 16 Fundamental formula of rigid kinematics: ρ = Ṙ + ω r where ρ = OP, R = OO, r = O P, ω: angular velocity. Kinetic energy of a rigid body: (A) if O is fixed, T = 1 2 ω I O ω, (2) if O is the COM, T = 1 2 MṘ2 + 1 2 ω I COMω, where I is the intertia tensor. Continuum version of I. 17 General properties of the inertia tensor: symmetry, additivity. Principal axes, principal moments, principal axis system of a rigid body. Asymmetric, symmetric, and spherical tops. Examples of inertia tensors: rotator, homogeneous rigid rod, sphere, cuboid. examples in EXC and HW. More Displaced axis theorem. Moment of inertia with respect to a fixed axis ˆn, Iˆn. Kinetic energy of a rigid body rotating about a fixed axis, T = 1 2 ω2 Iˆn. Formula for Iˆn, Iˆn = i m id 2 i, where d i is the distance of a generic point P i in the rigid body from the axis. 18 Physical pendulum, small oscillations. Parallel axis (Huygens-Steiner) theorem. (Reading week) 3
19 Angular momentum of a rigid body with respect to O: L O = R MṘ+I COMω, where R is the centre of mass coordinate relative to O and I G is the inertia tensor with respect to the centre of mass. Proof that L COM = I COM ω, where L COM is the angular momentum of the rigid body with respect to the centre of mass. Re-expressing the kinetic energy of a rigid body as T = 1 2 MṘ2 + 1 2 ω L COM. Re-expressing the kinetic energy of a rigid body with a fixed point O as T = 1 2 ω L O. 6 Spinning Tops Rotating frame. Derivation of Euler s equations. 20 Solution of Euler s equations in the case of a free symmetric top (I 1 = I 2 ). Angular momentum. Chandler s wobble. Asymmetric top. Solution in principle. Study of the stability of the rotation near one of the principal axes for the asymmetric top (the tennis racket theorem ). Definition of the Euler angles (ϕ, θ, ψ). Expression of the components of the angular velocity in the body fixed-frame (ω 1, ω 2, ω 3 ) in terms of the angular velocities ϕ, θ, ψ: ω 1 = ϕ sin θ sin ψ + θ cos ψ, ω 2 = ϕ sin θ cos ψ θ sin ψ, ω 3 = ϕ cos θ + ψ. Expression for the kinetic energy of a symmetric top (I 1 = I 2 ): T = (I 1 /2)( ϕ 2 sin 2 θ + θ 2 ) + (I 3 /2)( ϕ cos θ + ψ) 2. Revisiting free symmetric spinning top. Lagrangian in terms of Euler angles. Conserved quantities. Absence of nutation ( θ = 0). Precession of the top symmetry axis about the direction of L. Coplanarity of (1) the top symmetry axis, (2) ω, and (3) the angular momentum L. Expressions for the angular velocities: ϕ = L I 1 (precession of the top symmetry axis about the direction of L) and ψ = L cos θ ( 1 I 1 1 I 3 ) (spinning of the top about its symmetry axis), and equation for the inclination θ of the top symmetry axis, E = + cos2 θ I 3 ). L 2 2 ( sin2 θ I 1 Revisiting the free symmetric top, making contact with Euler s equations. Ω = ψ, Ω =. Relation between viewpoints of the intertial and body-fixed frames. Feynman s ω 3 I 3 I 1 I 1 plate: for small θ, wobbling is twice as fast as spinning, ϕ 2 ψ. Lagrangian for the symmetric top with a fixed point in a gravity field (Lagrange s top). Study of the case L I 3 L II 3. One-dimensional effective potential for the inclination θ of the top axis with respect to the vertical ẑ I -direction. Qualitative analysis of the precession and nutation of the top symmetry axis. The case L I 3 = L II 3. Sleeping top (θ = 0), stability of the solution for the case L 2 3 > 4 Mgd I 1, where d is the distance of the centre of mass from the fixed point. 4
7 Small Oscillations 21 Motivations. Definition of equilibrium position. One-dimensional case, conditions for the stability of the equilibrium position, Lagrangian of the small oscillations, frequency of small oscillations. 22 Example: bead on a wire. Multi-dimensional case. Conditions for the stability of the equilibrium position, Lagrangian of the small oscillations. Secular (or characteristic) equation det(ω 2 T V) = 0. Normal frequencies and normal modes. 23,24 Examples: double pendulum, linear tri-atomic molecule, pendulum with moving suspension point. 8 Hamiltonian Mechanics 25 Legendre transformation. Hamiltonian and Hamilton s equations. Phase space. 26 Examples. harmonic oscillator, particle in a potential, planar pendulum. Many degrees of freedom. Cyclic coordinates. Examples: particle in a potential. 27 Poisson brackets. Time evolution of a physical observable A = A(q, p, t): A = {A, H} + A/ t. Time evolution of the Hamiltonian, Ḣ = H/ t = L/ t. Properties of the Poisson bracket. {A, BC} = {A, B}C + B{A, C}. Jacobi identity: {A, {B, C}} + {B, {C, A}} + {C, {A, B}} = 0. Proof that if A and B are two conserved quantities ({A, H} = {B, H} = 0), then {A, B} is also conserved, i.e. {{A, B}, H} = 0 (Poisson s theorem). Relation to the commutator in quantum mechanics. 28 Lagrangian and Hamiltonian for a charged particle in an electromagnetic field. Lorentz force. Gauge transformations, gauge-invariance of the action. 29 Motion of a charged particle in a constant, uniform magnetic field. A = 1 (B r). 2 Noether charges associated to translations and rotations around the direction of the magnetic field. Solution to the equations of motion. Liouville s theorem. 9 Motion in a Non-Inertial Frame 32 Derivation of the equations of motion in a non-inertial frame. Fictitious forces: centrifugal force, Coriolis force (and more). Foucault s pendulum. 5
Problem class 30,31 Problem class. 6