Direct Energy, 2018a

11 CALCULUS OF VARIATIONS 255
The path x 1 (t) describes a case where the mass travels at a constant speed.
The path x 2 (t) describes a case where the mass accelerates whenthe restraint
is removed, and the path x 3 (t) describes a case where the mass rst
accelerates thenslows. The actionof each path canbe calculated using Eq.
11.36. For example purposes, the values of m =1kg and K = π 2 J m 2 are
used. The path x 1 (t) has S =0.355, the path x 2 (t) has S =0.364, and the
physical path x 3 (t) has zero action S =0.
We can derive the path that minimizes the action and that is found in
nature using the Euler-Lagrange equation.
∂L
∂x − d dt
∂L
) =0 (11.37)
∂ ( dx
dt
The rst term is the generalized potential. The second term is the time
derivative of the generalized momentum. The equationof motionis found
by putting these pieces together.
Kx + m d2 x
=0 (11.38)
dt2 The rst term of the equationof motionis − ∣ −→ ∣ ∣∣.
F spring The second term
represents the accelerationof the mass. We have just found the equationof
motion, and it is a statement of Newton's second law, force is mass times
acceleration. It is also a statement of conservation of force on the mass.
Equation11.38 is a second order linear dierential equationwith constant
coecients. It is the famous wave equation, and its solution is well
known
(√ ) (√ )
K K
x(t) =c 0 cos
m t + c 1 sin
m t (11.39)
where c 0 and c 1 are constants determined by the initial conditions. If we
securely attach the mass to the spring, as opposed to letting the mass get
kicked away, it will oscillate as described by the path x(t).
Energy is conserved in this system. To verify conservation of energy,
we can show that the total energy does not vary with time. The total
energy is given by the Hamiltonian of Eq. 11.31. In this example, both
the Hamiltonian and the Lagrangian do not explicitly depend on time,
∂H
=0and ∂L =0. Instead, they only depend on changes intime. For
∂t
∂t
this reason, we say both the total energy and the Lagrangian have time
translation symmetry, or we say they are time invariant. The spring and
mass behave the same today, a week from today, and a year from today.