Re:Schrödinger equation emerges mathematically from classica...
by adrian_b on Saturday December 23, 2023 @03:13AM
(#38742544)
I have written above a more complete explanation.
Hamilton did not discover any "action". Like everybody since Maupertuis, a century earlier, Hamilton used "action" with the meaning "accumulated living force".
"Living force" is English for the Latin "vis viva", which is the old name of kinetic energy (the term "kinetic energy" was introduced only later, in 1854, by William Thomson, who in 1892 became Lord Kelvin; for a short time before 1854 "actual energy" was used instead of "kinetic energy" and opposed to "potential energy", as defined by Rankine in 1853). So "action", in the sense introduced by Maupertuis and used by everybody until the 20th century, meant integral of the kinetic energy (actually the living force was mv^2, so the double of the kinetic energy).
Hamilton has introduced a new physical quantity, never used by anyone before him, which he named just "function S". Unlike the principle of the minimum action, which is false except for certain restricted cases, Hamilton's variational principle about the "function S" is always true, including in relativistic mechanics and quantum mechanics.
Nowadays Hamilton's "function S" is usually called "Hamilton's action", because it has the same measurement unit like the traditional action, even if it is a different physical quantity. "Hamilton's" is frequently dropped, which does not cause much ambiguity, because now the traditional "action" is seldom mentioned.
Nevertheless, whenever history is discussed, a very clear distinction between "action" and Hamilton's "function S" must be maintained, otherwise it is impossible to understand the evolution of physics.
Hamilton has discovered his function S by starting from the system of equations of Lagrange and finding a way to deduce them from a simpler principle.
It is a little weird that even if Lagrange had discovered when young, together with Euler, the Euler-Lagrange equations for variational problems, many years later, when he has written his works about mechanics, he has never attempted to use any variational techniques in the formulation of his equations of dynamics (and he dismissed the principle of the minimum action as seldom applicable), so this relationship has been discovered only later, by Hamilton.
Hamilton did not discover any "action". Like everybody since Maupertuis, a century earlier, Hamilton used "action" with the meaning "accumulated living force".
"Living force" is English for the Latin "vis viva", which is the old name of kinetic energy (the term "kinetic energy" was introduced only later, in 1854, by William Thomson, who in 1892 became Lord Kelvin; for a short time before 1854 "actual energy" was used instead of "kinetic energy" and opposed to "potential energy", as defined by Rankine in 1853). So "action", in the sense introduced by Maupertuis and used by everybody until the 20th century, meant integral of the kinetic energy (actually the living force was mv^2, so the double of the kinetic energy).
Hamilton has introduced a new physical quantity, never used by anyone before him, which he named just "function S". Unlike the principle of the minimum action, which is false except for certain restricted cases, Hamilton's variational principle about the "function S" is always true, including in relativistic mechanics and quantum mechanics.
Nowadays Hamilton's "function S" is usually called "Hamilton's action", because it has the same measurement unit like the traditional action, even if it is a different physical quantity. "Hamilton's" is frequently dropped, which does not cause much ambiguity, because now the traditional "action" is seldom mentioned.
Nevertheless, whenever history is discussed, a very clear distinction between "action" and Hamilton's "function S" must be maintained, otherwise it is impossible to understand the evolution of physics.
Hamilton has discovered his function S by starting from the system of equations of Lagrange and finding a way to deduce them from a simpler principle.
It is a little weird that even if Lagrange had discovered when young, together with Euler, the Euler-Lagrange equations for variational problems, many years later, when he has written his works about mechanics, he has never attempted to use any variational techniques in the formulation of his equations of dynamics (and he dismissed the principle of the minimum action as seldom applicable), so this relationship has been discovered only later, by Hamilton.