sign in sign up
<<
Home , Relative velocity

Lagrangian : Lecture 14

Galilean Transformations and Phase

Galilean Transformations In the lecture 12 we saw that the Hamiltonian generates translations in , the total linear generates translations in space and I hope you took my suggestion to show that the total generates rotations of space. The latter two generators have vanishing Poisson brackets with the Hamiltonian, are explicitly time- independent, and hence are conserved quantities. In this lecture we complete our discussion of the principle of relativity by considering the dependence on boosts — changes between inertial frames moving at constant relative . Rather than trying to derive the generator of boosts from their transformation properties, we the other way: we construct a conserved quantity and then work out what transformation it corresponds to. We start with the quantity 1 X X = m q . M j x j j P This is the position of the centre of (where M = j m j). It is not conserved, except in the case that we are in the centre-of-mass frame. In general, it moves at constant velocity 1 X 1 X  X˙ = m q˙ = p = x , M j x j M x j M j j where x is the total linear momentum in the x direction. We use this to construct a quantity that is conserved:   X B = M X − Xt˙ = (m j qx j − px jt) . j This is a calculation, based on the current position and momentum of the centre of mass and the current time, of (M ) the position the centre of mass would have had at time t = 0. This is a conserved (time-independent) quantity, despite its expression being explicitly time dependent. To calculate the transformation that this corresponds to, we can take the of B with the qs j and momenta ps j: ! X X ∂qs j ∂B ∂qs j ∂B q → q + [q , B] = q + −  , s j s j s j s j ∂q ∂p ∂p ∂q u k uk uk uk uk ! X X ∂ps j ∂B ∂ps j ∂B p → p + [p , B] = p + −  . s j s j s j s j ∂q ∂p ∂p ∂q u k uk uk uk uk We have ∂qs j ∂qs j ∂ps j ∂ps j = δsuδ jk, = 0, = 0, = δsuδ jk, ∂quk ∂puk ∂quk ∂puk and ∂B ∂ X = (m q − t p ) = −δ t, ∂p ∂p j x j x j xs uk uk j ∂B ∂ X = (m q − t p ) = m δ . ∂q ∂q j x j x j k xs uk uk j Therefore qs j → qs j − δxs  t , which gives

qx j → qx j −  t ,

qy j → qy j ,

qz j → qz j .

Similarly, px j → px j − m j  . With  = v, these are the Galilean transformations between two frames moving with relative velocity v in the x direction. Thus B, the position of the centre of mass at time 0, is the generator of boosts in the x direction. We can use the conservation of B, which follows from the principle of relativity, to prove another property of the Hamiltonian. But, because B has explicit time dependence, this follows a slightly different pattern to the previous cases we have considered. We still have dB ∂B = [B, H] + = 0 , dt ∂t but since ∂B X = − p , ∂t x j j we can conclude that [B, H] = x. Thus the change in Hamiltonian when we boost by an infinitesimal velocity∗ −dX˙ is

dH = −[H, B] dX˙ = [B, H] dX˙ = x dX˙ = M X˙ dX˙ , ∗The minus sign is so that we are boosting in the direction of increasing the centre-of-mass velocity, X˙ → X˙ + dX˙. where X˙ is the velocity of the centre of mass. Therefore, starting from the centre-of- mass frame, in which X˙ = 0 and the Hamiltonian is Hc.o.m, and boosting in steps of dX˙ to a frame in which the centre of mass has velocity V, we obtain 1 H = H + MV2 . c.o.m 2 Therefore, knowing only the principle of relativity, which requires the internal of a system to be independent of the velocity of the inertial frame in which it is measured, we have proved that the kinetic of an object of mass M moving at V is 1 2 2 MV .

Phase Space The state of a system with N degrees of freedom is completely specified by the values of (qi, pi) at some time, t. We can think of these 2N values as coordinates of a point in a 2N-dimensional “phase space”. The motion of the system in phase space is determined by Hamilton’s equations. Liouville’s theorem states that an ensemble of systems that obey Hamilton’s equations behave in phase space like an incompressible fluid.