From Hamilton's principle to Lagrange's equation (The calculus of variations [Cou2, pp.737-767]).
Hamilton's principle is important because it gives the correct form of equation of motion- Lagrange's equations. However, almost all physics books fail to interpret Hamilton's principle correctly.
In order to capture the spirit of Hamilton's principle, many physicists try to formulate their argument as carefully as possible while deriving Lagrange's equations. However, none of the following gimmicks succeeds in capturing the spirit.
Virtual displacement [Go2, p.39, l.-8] at a fixed time t [Go2, p.16, l.-12].
q can be a parameter other than time [Rob, p.138, l.14].
The principle of least action [Lan1, p.2, l.5] ( The nomenclature is incorrect and misleading because we are looking for the extremum rather than the minimum of action).
Distinguish boundary conditions from initial conditions [Rob, p.138, l.-17-l.-7].
Allow the final time to vary while the initial and final coordinates
remain fixed [Lan1, p.140, l.-7].
[Go2, p.45, l.-6-p.46, l.4].
[Born, p.734, (73)] (The meaning of the statement is not clear).
For some reason [Rob, p.139, l.5] (There must be a reason, but the author fails to pinpoint it).
Distinguish a short segment from a long segment [Lan1, p.2, l.-4].
[Go2, p.18, l.-8-l.-7].
Remark.
The correct formulation of Hamilton's principle is given in [Cou2, vol.2, p.758, l.21-l.27].
From Lagrange's equation to Hamilton's principle.
To drastically reduce the domain of admissible [Cou, vol.1, p.168, l.-13]
paths and reveal what role [Go2, p.438, l.-5-p.442, l.15] α_{i}'s
play in Lagrange's equation, we must assume that the equation of motion satisfies certain differential equations although we do not know their exact form. This assumption enables us to obtain the integration constants from a theoretical solution. Simply put, the "variation" in Hamilton's variational principle
essentially means varying around the integrals of motion [Go2, p.16, l.-12].
Thus, if the infinitesimal parameter α's in [Go2,p.38, l.9] were to be defined as (β_{1}-α_{1},...,β_{n}-α_{n}), where α_{i}'s are defined as in [Go2, p.439, l.-11] and β_{i}'s vary around the integration constants α_{i}'s,
then we could proceed in two directions:
Since α_{i}'s are the constants of motion,
¶S/¶α_{i}=0, where S is defined as in [Go2, p.439, (10-4)]. Thus we have derived Hamilton's principle from Lagrange's equations [Go2, p.442, (10-13)].
By [Go2, pp.43-45, §2-3] (Note. α should be defined as perturbed integration constants instead), we may derive Lagrange's equations from Hamilton's principle.
Remark. The integration constants α_{i}'s can be considered parameters for perturbation, while a path is a dependent variable. We may vary parameters β_{i}'s around α_{i}'s arbitrarily, but we can not vary the dependent variable-path completely arbitrarily as in [Lan1, p.2, l.-10] because a path is determined by the parameters. In other words, the q_{i}(t)+δq_{i})(t)'s still have to satisfy Lagrange's equations even though we allow β_{i}'s to vary around α_{i}'s arbitrarily.
[Lan1, p.127, l.7], [Lan1, p.127, (39.6)] and [Lan1, p.128, (39.8)] all come
from [Lan1, p.126, (39.1)] by the change of coordinates [1]. This is because
the Lagrangian is the unique [Lan1, p.4, l.-13-l.-4] entity derived from Hamilton’s principle
[Lan1, p.2, l.7] to characterize the equation of motion [Lan1, p.3, (2.6)].
Although the operation such as ¶L/¶v=p [Lan1, p.128, (39.10)] is absolutely legitimate in the theory of differential equations, it
is still abstract. Therefore, we would like to check whether the derived
results agree with common sense in physics [Lan1, p.128, l.-3-p.129, l.3].
The neglected term is second order in dy [Jack,
p.44, l.15].
Proof. Let a be a constraint parameter of y varying around 0. Then dy = (¶y/¶a)da.
¶/¶x (dy) =
¶/¶x (¶y/¶a)da
= ¶/¶a(¶y/¶x)da.
In fact, the neglected term is second order in da.
I[y] is a stationary minimum under the condition
given in [Jack, p.44, l.21-l.22].
Proof.
Let a be a constraint parameter of y
varying around 0. If a changes from 0 to da, dI = I[y+dy]-I[y]
is nonnegative when the neglected term in [Jack, p.44, (1.64)] is considered. Therefore, y
reaches a minimum when a = 0.