94 The Lagrange-Hamilton Method
To obtain another k equation, define the Hamiltonian
k
H(p, q) = Y<PM ~ L(l> <?)> (^2 -^3 -^45 )where, according to (2.3.43), each qj is a function of p and q. Then direct
calculations show that
dqL_d_H_ dpl_ d_H_
dt - dp^ dt - dq,' •?-i"--'fc- ^6Ab>Equations (2.3.46) are a system of 2k first-order ODEs, known as
Hamilton's equations.
EXERCISE 2.3.5.B Derive equations (2.3-46) Hint: the first follows directly by
computing dH/dpj and using (2.3.42), (2.3-43), and qj = dqj/dt; for the second,
note that dL/dqj = —dH/dqj and use (2.3-44)-
EXERCISE 2.3.6F Show that if (2.3.31) is used instead of (2.3.27), then the
second equation in (2.3-46) becomes
We conclude the section by showing that, in a typical mechanical prob-
lem, when the vector q is the position and V is the potential energy, the
Hamiltonian is the total energy of the system.
Theorem 2.3.1 Consider a system of n point masses in M^3. Assume
that the position vectors rj and the potential V of the system do not depend
on q and depend only on q. Then the Hamiltonian of the system is equal
to the total energy:
Proof. By assumption,
Then
H(p,q)=£K + V. (2.3.48)ri = ri(q), i = l,...,n. (2.3.49)ElH- (2-3-50)
*d<b