Page 81 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

68 Classical methods

The Hamiltonian of f, is given by

H (u, v)= sup {vξ − f (u, ξ)} with v = f ξ (u, ξ) .
ξ
The associated Hamiltonian system is
⎧
u (x)= H v (u (x) ,v (x))
0
⎨
v (x)= −H u (u (x) ,v (x)) .
⎩ 0
The Hamiltonian system also has a first integral given by
d
0
[H (u (x) ,v (x))] = H u (u, v) u + H v (u, v) v ≡ 0.
0
dx
In physical terms we can say that if the Lagrangian f is independent of the
variable x (which is here the time), the Hamiltonian H is constant along the
trajectories.

2.4.1 Exercises
N
Exercise 2.4.1 Generalize Theorem2.10tothe case where u :[a, b] → R ,
N ≥ 1.
Exercise 2.4.2 Consider a mechanical system with N particles whose respective
3
masses are m i and positions at time t are u i (t)= (x i (t) ,y i (t) ,z i (t)) ∈ R ,
1 ≤ i ≤ N.Let
N N
1 X 0 2 1 X ¡ 02 02 02 ¢
T (u )= m i |u | = m i x + y + z i
0
i
i
i
2 2
i=1 i=1
be the kinetic energy and denote by U = U (t, u) the potential energy. Finally let
0
0
L (t, u, u )= T (u ) − U (t, u)
be the Lagrangian. Let also H be the associated Hamiltonian. With the help of
the preceding exercise show the following results.
(i) Write the Euler-Lagrange equations. Find the associated Hamiltonian
system.
(ii) Show that, along the trajectories (i.e. when v = L ξ (t, u, u )), the Hamil-
0
tonian can be written as (in mechanical terms it is the total energy of the system)
H (t, u, v)= T (u )+ U (t, u) .
0
p p 2
Exercise 2.4.3 Let f (x, u, ξ)= g (x, u) 1+ ξ . Write the associated Hamil-
tonian system and find a first integral of this system when g does not depend
explicitly on x.