0.2. Hamilton's equations 3

The basis of this concept is the Legendre transformation (see ARNOLD [A],

p.61f.) between tangent and cotangent bundles (see Section A.3)

TQ —+ T*Q,

(Q,0) '— • ( ? , P ) -

Then the time development described on TQ by the Lagrange function L =

L(q, q, t) (which we can and will assume to be convex in the second argument:

see, for example,

ARNOLD

([A], p. 65)) is replaced by the Hamiltonian

function H on phase space T*Q defined by

H(p, q, t) := pq - L(q, q, t) with p = — ,

where we have used the usual abbreviated symbols for the n-tuple

,

x

dL (dL dL\

P = (Pl,...,Pn), ^ = ( ^ , . . . ^ J , e t c .

The Lagrange equations (1) are here translated into Hamilton's equations

. dH . dH

( 2 ) « = ^ ' P = ~ ^ -

Because the total differential of H = H(p, q, t) (see Section A.4) gives

, „ dH , dH , dH ,

dH=^dp+-dq-dq+~Wdt

and by the definition H = pq — L(q, j, £), we also get

dH = qdp - —-dq - —dt.

oq ot

dL

Comparing (1) and p = —-, we get

oq

._dH_ 9I£__dL__. 9H _ dL

q~~d^' ~d^~~l^~~p' ~dt~~~dt'

Hamilton's equations (2) are now (when H is independent of t) a system

of ordinary differential equations, which, given a particular set of initial con-

ditions p°,q°, gives a unique curve 7* in phase space T*Q whose projection

7 onto the configuration space Q solves the original problem.

The Hamiltonian function is also written in the form

H = H(p,q,t) = (T) + V,

where V is the potential energy of the system and T is the kinetic energy

given in terms of the variables q and p.