Differentiation (making use of Eqn (11.2)) shows that dH/dt = d H/dt = -dL/dt and, hence, that if t does not appear explicitly in the Lagrangian, it does not appear in the Hamiltonian either, and H is a constant. In such a situation, called a 'conservative system', the Hamiltonian is shown readily to be the total energy in the system, i.e. H = T + V.

It also follows that dH d H

dpa d qa and these are Hamilton's canonical equations of motion for a system with N degrees of freedom. Whereas Lagrange's formulation results in N second-order differential equations, the Hamiltonian formulation delivers 2 N first-order equations. For the n-body problem, we can write = ri and p; = mi qi, and then the equations of motion (Eqn (9.13)) are simply pi = -ViV. These can be written in canonical form with the Hamiltonian n

Stimulated by Hamilton's work, Jacobi considered how to choose the coordinates for a particular problem so that the integration of Hamilton's canonical equations is as simple as possible. This leads to the concept of a canonical transformation, a transformation from variables pa, qa to Pa, Qa, that preserves the Hamiltonian nature of the equations. In other words, after applying the

Sometimes referred to as a 'contact transformation'. For a thorough treatment, see, for example, Whittaker (1937), pp. 292 ff. or Boccaletti and Pucacco (1996), pp. 76 ff.

transformation we obtain

for some function H*(Pa, Qa, t). A necessary and sufficient condition for a transformation to be canonical is that

£ pa dqa Pa dQa = dW, in which the time t is assumed constant and W is an arbitrary function of t and 2N of the 4N scalar variables pa, qa, Pa, and Qa. The new and old Hamiltonians are related via d W

Another approach to such transformations is via generating functions. Given afunction W1(qa, Qa, t) we can construct pa = d W1/d qa, Pa = —d W1/d Q a and then this defines a canonical transformation. Other possibilities are pa = d W2/dqa, Qa = d W2/3Pa with W2(qa, Pa, t), qa = -d W3 / dp a, Pa = — W3/S Q a with W3(pa, Qa, t), qa = -d W4 / dp a, Qa = d W4/3 Pa with W4(pa, Pa, t).

0 0

Post a comment