MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian, and b) polar coordinates. Now in each case find the Hamiltonian H. Identify a symmetry of the system, and hence find a conserved quantity J. Show that J and H have vanishing Poisson bracket. Show that in case b), the radial and angular equations of motion separate, and write down the Hamiltonian of the radial motion. Describe the orbits of the system. Solutiona) We have T = m ẋ + ẏ + ż ) and Then = m ẋ + ẏ xẋ + yẏ) + k ), x + y V = k x + y. p x = mẋ + k x ẋ + xyẏ) x + y ), and similarly for p y. Inverting the linear relation between ẋ, ẏ) and p x, p y ), and substituting back to express E = T + V in terms of the canonical variables x, y, p x, p y ), we get, after some rearrangements, x m + k )x + y ) p x, p y ) + + k )y k xy k xy + k )x + y ) px p y ) +k x + y. b) In polars T = m ẋ + ẏ + ż )
and = m ṙ + k ) + r θ ), V = kr. Then p r = mṙ + k ), p θ = mr θ. Hence in terms of these canonical variables r, θ, p r, p θ ), we get: m + k ) p r + mr p θ + k r. Since H is independent of θ, J = p θ is a constant of the motion. Then {J, H} = {H, J} = H θ = 0. The same calculation can be done in any set of canonical variables - the Poisson bracket does not depend on how the canonical variables are chosen.. A sliding pendulum A mass M with horizontal coordinate X is free to slide along a smooth horizontal rod, which is taken to be the x-axis. A light rod of length l is attached to the first mass, and is free to swing in the x, z) plane; at the other end of this rod is a second mass m. Use the coordinate θ to denote the inclination of the swinging rod to the vertical. Gravity acts vertically downwards. Show that the Lagrangian for the system is LX, Ẋ, θ, θ) = M Ẋ + m Ẋ + lẋ θ cosθ) + l θ ) + lmg cosθ). Find the Hamiltonian of the system, and identify one other constant of the motion. Hence show that the equation for the motion of θ can
be derived from a Hamiltonian only involving θ and p θ, and constants of the motion. Show that if the system is started from rest, Ẋ t=0 = θ t=0 = 0, then the mass m moves along an ellipse. If θ is small, what is the frequency of the pendulum? Solution The kinetic energy is T = M Ẋ + m ẋ + ż ) = M Ẋ + m Ẋ + l cosθ) θ) + l sinθ) θ) while the potential, relative to the horizontal position, is V = mgl cosθ). Expanding, we find, setting L = T V : LX, Ẋ, θ, θ) = M Ẋ + m Ẋ + lẋ θ cosθ) + l θ ) + lmg cosθ). Now p X = MẊ + mẋ + ml cosθ) θ, p θ = ml θ + ml Ẋ cosθ). Substituting into E, which by Euler s theorem is T + V, we find l mm + m sin θ)) p ml X, p θ ) ml cosθ) ) ml cosθ) px M + m Since H is independent of X, we have that p X is conserved. If the system starts from rest it is and remains) zero. Now p X = 0 gives on integrating, M + m)x + ml sinθ) = K = constant. The coordinates of the bob of the pendulum are then K ml sinθ) x, z) = X + l sinθ), l cosθ)) = + l sinθ), l cosθ)) m + M K + Ml sinθ), l cosθ)), m + M 3 p θ ) lmg cosθ).
which is the parametric equation of an ellipse. If p X = 0, the motion of θ, p θ ) is given by the reduced Hamiltonian: l mm + m sin θ)) 0, p ml θ) ml cosθ) = M + m l mm + m sin θ)) p θ lmg cosθ). ) ) ml cosθ) 0 lmg cosθ) M + m p θ For small perturbations from equilibrium θ, p θ ), the equation of motion is approximately: θ = M + m l mm p θ, The frequency is then given by p θ = lmgθ. ω = gm + m) lm. 4
3. The diatomic molecule. Two atoms of masses m, m move freely in the plane, with a potential V x x ) between them. a) Write down the Lagrangian in Cartesian coordinates, and hence find the Hamiltonian. b) Rewrite the Lagrangian in new coordinates X, r), where X is the centre of mass, and x x ) = r. Find the Hamiltonian corresponding to this Lagrangian. Write down the constants of motion corresponding to the symmetries; and calculate the Poisson brackets of these quantities. Note it is essential to express all velocities in terms of the coordinates and their conjugate momenta. Identify 4 Poisson commuting constants of motion, and hence show that the system is integrable. Solution. L = m ẋ + m ẋ V x x ). m p + m p + V x x ).. Set x = X + m m +m r, and x = X m m +m r, so that we find on substituting and simplifying: L = m + m m m Ẋ + V r ). m + m )ṙ This leads to m + m ) P + m + m p r + V r ). m m Note that the centre of mass motion and the relative motion are completely decoupled in these coordinates. So the components of the total momentum P are conserved; this corresponds to the symmetry of translation in space X, Y ). Further the Hamiltonian H is conserved translation in t). Also the system is invariant under rotations so the quantity J = X P = XP Y Y P X is conserved; so is j = r p. 5
Note J does not Poisson commute with the components of P, though it does with P. A set of Poisson commuting conserved quantities is thus H, J, j and P. Since there are 4 of these, they are independent and the system has this many degrees of freedom, the system is integrable. 6