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.
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