Classical Mechanics LECTURE 28: HAMILTONIAN MECHANICS, NOETHER’S THEOREM Prof. N. Harnew University of Oxford HT 2017 1 OUTLINE : 28. HAMILTONIAN MECHANICS, NOETHER’S THEOREM 28.1 Hamilton mechanics 28.2 The physical significance of the Hamiltonian 28.3 Example: re-visit bead on rotating hoop 28.4 Noether’s theorem 2 28.1 Hamilton mechanics I Lagrangian mechanics : Allows us to find the equations of motion for a system in terms of an arbitrary set of generalized coordinates I Now extend the method due to Hamilton ! use of the conjugate (generalized) momenta p1; p2; ··· ; pn replace the generalized velocities q_ 1; q_ 2; ··· ; q_ n I This has advantages when some of conjugate momenta are constants of the motion and it is well suited to finding conserved quantities @L I From before, conjugate momentum : pk = @q_ k and E-L equation reads for coordinate k : p_ = @L k @qk d @L @L (since E-L is p_ k = ( ) = ) dt @q_ k @qk 3 The Hamiltonian, continued I An aside: use rules of I Lagrangian L = L(qk ; q_ k ; t) =) partial differentiation: dL @L P @L dqk @L dq_ k I dt = @t + k @q dt + @q_ dt k k I If f = f (x; y; z) @L P @L @L @ @ dy @ = + k ( q_ k + _ q¨k ) df f f f dz @t @qk @qk I dx = @x + @y dx + @z dx @L @L I Conjugate momentum definition : pk = , p_ k = @q_ k @qk dL @L P _ _ ¨ I Therefore dt = @t + k (pk qk + pk qk ) | {z } d _ dt (pk qk ) X d _ @L dH @L I dt (L − pk qk ) = @t dt = − @t k | {z } -H P I Define Hamiltonian H = k pk q_ k − L I If L does not depend explicitly on time, H is a constant of motion 4 28.2 The physical significance of the Hamiltonian P I From before : H = k pk q_ k − L @L @L I Where conjugate momentum : pk = , p_ k = @q_ k @qk 1 _ 2 _ 2 _ 2 I Take kinetic energy T = 2 m x + y + z 1 _ 2 _ 2 _ 2 I L = 2 m x + y + z − U(x; y; z) P _ 1 _ _ _ _ _ _ I H = k pk qk − L = 2 m (2x:x + 2y:y + 2z:z) − (T − U) = 2T − (T − U) = T + U = E ! total energy dH @L I From before dt = − @t dH ! If L does not depend explicitly on time dt = 0 ! energy is a constant of the motion I Can show by differentiation : Hamilton Equations ! @H = q_ ; @H = −p_ @pk k @qk k If a coordinate does not appear in the Hamiltonian it is cyclic or ignorable 5 28.3 Example: re-visit bead on rotating hoop First take the case of a free (undriven) system 1 2 _2 2 2 _2 I L = 2 m (R θ + R sin θ φ ) + mgR cos θ P @L I H = pk q_ k − L ; pk = k @q_ k @L 2 2 2 I p = = m R θ_ ; p = mR sin θ φ_ θ @θ_ φ 2 2 2 2 2 I H = m R θ_ + mR sin θ φ_ − L 1 2 _2 2 2 _2 = 2 m R θ + R sin θ φ −mgR cos θ ! H = T + U = E L does not depend explicitly on t , H; E conserved ! Hamiltonian gives the total energy Hamilton Equations : q_ = @H ; p_ = − @H k @pk k @qk _ @H ! pφ = − @φ = 0 (ignorable) 2 2 ! pφ = mR sin θ φ_ = Jz = constant of the motion 6 Example continued Now consider a DRIVEN system - hoop rotating at constant angular speed ! 1 2 _2 2 2 2 I L = 2 m (R θ + R ! sin θ) + mgR cos θ P @L I H = pk q_ k − L ; pk = k @q_ k @L 2 I p = = m R θ_ ; a single coordinate θ θ @θ_ 2 2 I H = m R θ_ − L 1 2 _2 2 2 2 = 2 m R θ − R ! sin θ −mgR cos θ dH @L I = − 1 2 _2 2 2 2 dt @t I E = 2 m (R θ + R ! sin θ) − mgR cos θ IH is a constant Hence E = H + mR2!2 sin2 θ of the motion, ! E (= T + U) 6= H E is not const. So what’s different ? In this case the hoop has been forced to rotate at an angular velocity !. External energy is being supplied to the system. 7 28.4 Noether’s theorem The theorem states : Whenever there is a continuous symmetry of the Lagrangian, there is an associated conservation law. I Symmetry means a transformation of the generalized coordinates qk and q_ k that leaves the value of the Lagrangian unchanged. I If a Lagrangian does not depend on a coordinate qk (ie. is cyclic) it is invariant (symmetric) under changes qk ! qk + δqk ; the @L corresponding generalized momentum pk = is conserved @q_k 1. For a Lagrangian that is symmetric under changes t ! t + δt, P @L the total energy H is conserved ! H = k @q_ q_ − L 2. For a Lagrangian that is symmetric under changes r ! r + δr, the linear momentum p is conserved 3. For a Lagrangian that is symmetric under small rotations of angle θ ! θ + δθ about an axis n^ such a rotation transforms the Cartesian coordinates by r ! r + δθ n^ × r , the conserved quantity is the component of the angular momentum J along the n^ axis 8.