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Added remark about the role of integration in Hamilton's stationary action

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edited Jan 12, 2021 at 22:33

About the phrasing of your question: it could be that mean to ask: before the introduction of QM, were there real applications of the Hamiltonian formalism in use, alongside the existing Lagrangian formalism?

Or it could be that you mean to ask: before QM, were there real applications of what we nowadays know as Hamilton's stationary action?

The answer below is for the second question.

Max Planck believed that Hamilton's stationary action constitutes something special.

Wikisource has a series of Lectures by Max Planck, delivered in 1909, the title of the seventh lecture is General Dynamics. Principle of Least Action.

The answer by contributor JHM offers the information that 'Jacobi appears to have simplified several redundancies in Hamilton's formalism'.

It would appear: while Hamilton's stationary action originated with Hamilton, it was later treatment that brought it into the form in which we know it today.

Derivation from first principles

For the case of a point mass there is the following derivation of Hamilton's stationary action from first principles. It's short. it would appear this is the shortest possible.

We have the work-energy theorem:

$$\int_{s_0}^s F ds = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2 $$

One implication of the work-energy theorem: along an entire trajectory the rate of change of kinetic energy must match the rate of change of potential energy.

$$d(E_k) = -d(E_p) \qquad (1) $$

(1) can be expressed as either a time derivative or a position derivative.

A trajectory is a true trajectory if the following is valid for every infinitisimally short subsection along the entire trajectory:

$$ \frac{d(E_k)}{dt} = -\frac{d(E_p)}{dt} \qquad (2) $$

$$ \frac{d(E_k)}{ds} = -\frac{d(E_p)}{ds} \qquad (3) $$

Again: (2) and (3) are two ways to express the same property.
(3) is the basis of Hamilton's stationary action:

Integration is a linear operation.
Therefore if for an entire trajectory (3) is satisfied then the following equation is automatically satisfied too:

$$ \frac{d(\int E_k dt)}{ds} = - \frac{d(\int E_p dt)}{ds} \qquad (4) $$

(4) expresses the stationary condition of Hamilton's stationary action.

(Many derivations start with Hamilton's stationary action, and derive (3) from (4). However, (3) is already a direct consequence of the work-energy theorem.)

(Many people wonder: in Hamilton's stationary action, what role is played by the integration? This derivation explains that. Integration is a linear operation, that is the key.)

Or it could be that you mean to ask: before QM, were there real applications of what we nowadays know as Hamilton's stationary action?

The answer below is for the second question.

Max Planck believed that Hamilton's stationary action constitutes something special.

Wikisource has a series of Lectures by Max Planck, delivered in 1909, the title of the seventh lecture is General Dynamics. Principle of Least Action.

The answer by contributor JHM offers the information that 'Jacobi appears to have simplified several redundancies in Hamilton's formalism'.

It would appear: while Hamilton's stationary action originated with Hamilton, it was later treatment that brought it into the form in which we know it today.

Derivation from first principles

For the case of a point mass there is the following derivation of Hamilton's stationary action from first principles. It's short. it would appear this is the shortest possible.

We have the work-energy theorem:

$$\int_{s_0}^s F ds = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2 $$

One implication of the work-energy theorem: along an entire trajectory the rate of change of kinetic energy must match the rate of change of potential energy.

$$d(E_k) = -d(E_p) \qquad (1) $$

(1) can be expressed as either a time derivative or a position derivative.

A trajectory is a true trajectory if the following is valid for every infinitisimally short subsection along the entire trajectory:

$$ \frac{d(E_k)}{dt} = -\frac{d(E_p)}{dt} \qquad (2) $$

$$ \frac{d(E_k)}{ds} = -\frac{d(E_p)}{ds} \qquad (3) $$

Again: (2) and (3) are two ways to express the same property.
(3) is the basis of Hamilton's stationary action:

Integration is a linear operation.
Therefore if for an entire trajectory (3) is satisfied then the following equation is automatically satisfied too:

$$ \frac{d(\int E_k dt)}{ds} = - \frac{d(\int E_p dt)}{ds} \qquad (4) $$

(4) expresses the stationary condition of Hamilton's stationary action.

(Many derivations start with Hamilton's stationary action, and derive (3) from (4). However, (3) is already a direct consequence of the work-energy theorem.)

Or it could be that you mean to ask: before QM, were there real applications of what we nowadays know as Hamilton's stationary action?

The answer below is for the second question.

Max Planck believed that Hamilton's stationary action constitutes something special.

Wikisource has a series of Lectures by Max Planck, delivered in 1909, the title of the seventh lecture is General Dynamics. Principle of Least Action.

The answer by contributor JHM offers the information that 'Jacobi appears to have simplified several redundancies in Hamilton's formalism'.

It would appear: while Hamilton's stationary action originated with Hamilton, it was later treatment that brought it into the form in which we know it today.

Derivation from first principles

For the case of a point mass there is the following derivation of Hamilton's stationary action from first principles. It's short. it would appear this is the shortest possible.

We have the work-energy theorem:

$$\int_{s_0}^s F ds = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2 $$

One implication of the work-energy theorem: along an entire trajectory the rate of change of kinetic energy must match the rate of change of potential energy.

$$d(E_k) = -d(E_p) \qquad (1) $$

(1) can be expressed as either a time derivative or a position derivative.

A trajectory is a true trajectory if the following is valid for every infinitisimally short subsection along the entire trajectory:

$$ \frac{d(E_k)}{dt} = -\frac{d(E_p)}{dt} \qquad (2) $$

$$ \frac{d(E_k)}{ds} = -\frac{d(E_p)}{ds} \qquad (3) $$

Again: (2) and (3) are two ways to express the same property.
(3) is the basis of Hamilton's stationary action:

Integration is a linear operation.
Therefore if for an entire trajectory (3) is satisfied then the following equation is automatically satisfied too:

$$ \frac{d(\int E_k dt)}{ds} = - \frac{d(\int E_p dt)}{ds} \qquad (4) $$

(4) expresses the stationary condition of Hamilton's stationary action.

(Many derivations start with Hamilton's stationary action, and derive (3) from (4). However, (3) is already a direct consequence of the work-energy theorem.)

(Many people wonder: in Hamilton's stationary action, what role is played by the integration? This derivation explains that. Integration is a linear operation, that is the key.)

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created Jan 12, 2021 at 22:15

Cleonis

Or it could be that you mean to ask: before QM, were there real applications of what we nowadays know as Hamilton's stationary action?

The answer below is for the second question.

Max Planck believed that Hamilton's stationary action constitutes something special.

Wikisource has a series of Lectures by Max Planck, delivered in 1909, the title of the seventh lecture is General Dynamics. Principle of Least Action.

The answer by contributor JHM offers the information that 'Jacobi appears to have simplified several redundancies in Hamilton's formalism'.

It would appear: while Hamilton's stationary action originated with Hamilton, it was later treatment that brought it into the form in which we know it today.

Derivation from first principles

For the case of a point mass there is the following derivation of Hamilton's stationary action from first principles. It's short. it would appear this is the shortest possible.

We have the work-energy theorem:

$$\int_{s_0}^s F ds = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2 $$

One implication of the work-energy theorem: along an entire trajectory the rate of change of kinetic energy must match the rate of change of potential energy.

$$d(E_k) = -d(E_p) \qquad (1) $$

(1) can be expressed as either a time derivative or a position derivative.

A trajectory is a true trajectory if the following is valid for every infinitisimally short subsection along the entire trajectory:

$$ \frac{d(E_k)}{dt} = -\frac{d(E_p)}{dt} \qquad (2) $$

$$ \frac{d(E_k)}{ds} = -\frac{d(E_p)}{ds} \qquad (3) $$

Again: (2) and (3) are two ways to express the same property.
(3) is the basis of Hamilton's stationary action:

Integration is a linear operation.
Therefore if for an entire trajectory (3) is satisfied then the following equation is automatically satisfied too:

$$ \frac{d(\int E_k dt)}{ds} = - \frac{d(\int E_p dt)}{ds} \qquad (4) $$

(4) expresses the stationary condition of Hamilton's stationary action.

(Many derivations start with Hamilton's stationary action, and derive (3) from (4). However, (3) is already a direct consequence of the work-energy theorem.)