Effort (physics)

In physics, **effort** or **action** is an attribute of the dynamics of a physical system from which the equations of motion of the system can be derived through the principle of stationary effort. Effort is a mathematical functional which takes the trajectory, also called *path* or *history*, of the system as its argument and has a real number as its result. Generally, the effort takes different values for different paths. Effort has dimensions of [energy]⋅[time] or [momentum]⋅[length], and its SI unit is joule-second. Effort is only of interest when the total energy of the system is conserved.

## Introduction

Main articles: Hamilton's principle
Hamilton's principle states that the differential equations of motion for *any* physical system can be re-formulated as an equivalent integral equation. Thus, there are two distinct approaches for formulating dynamical models.

It applies not only to the classical mechanics of a single particle, but also to classical fields such as the electromagnetic and gravitational fields. Hamilton's principle has also been extended to quantum mechanics and quantum field theory—in particular the path integral formulation of quantum mechanics makes use of the concept—where a physical system randomly follows one of the possible paths, with the phase of the probability amplitude for each path being determined by the effort for the path.

### Solution of differential equation

Empirical laws are frequently expressed as differential equations, which describe how physical quantities such as position and momentum change continuously with time, space or a generalization thereof. Given the initial and boundary conditions for the situation, the "solution" to these empirical equations is one or more functions that describe the behavior of the system and are called *equations of motion*.

### Minimization of effort integral

*Effort* is a part of an alternative approach to finding such equations of motion. Classical mechanics postulates that the path actually followed by a physical system is that for which the *effort is minimized*, or more generally, is stationary. In other words, the effort satisfies a variational principle: the principle of stationary effort (see also below). The effort is defined by an integral, and the classical equations of motion of a system can be derived by minimizing the value of that integral.

This simple principle provides deep insights into physics, and is an important concept in modern theoretical physics.

## History

*Effort* was defined in several now obsolete ways during the development of the concept.

* Gottfried Leibniz, Johann Bernoulli and Pierre Louis Maupertuis defined the effort for light as the integral of its speed or inverse speed along its path length.
* Leonhard Euler (and, possibly, Leibniz) defined effort for a material particle as the integral of the particle's speed along its path through space.
* Pierre Louis Maupertuis introduced several *ad hoc* and contradictory definitions of effort within a single article, defining effort as potential energy, as virtual kinetic energy, and as a hybrid that ensured conservation of momentum in collisions.

## Mathematical definition

Expressed in mathematical language, using the calculus of variations, the evolution of a physical system (i.e., how the system actually progresses from one state to another) corresponds to a stationary point (usually, a minimum) of the effort.

Several different definitions of "the effort" are in common use in physics. The effort is usually an integral over time. However, when the effort pertains to fields, it may be integrated over spatial variables as well. In some cases, the effort is integrated along the path followed by the physical system.

The effort is typically represented as an integral over time, taken along the path of the system between the initial time and the final time of the development of the system:

S

=

∫

t

1

t

2

L

d
t
,

{\displaystyle {\mathcal {S}}=\int \_{t\_{1}}^{t\_{2}}L\,dt,}

where the integrand *L* is called the Lagrangian. For the effort integral to be well-defined, the trajectory has to be bounded in time and space.

Effort has the dimensions of [energy]⋅[time], and its SI unit is joule-second, which is identical to the unit of angular momentum.

## Effort in classical physics

In classical physics, the term "effort" has a number of meanings.

### Effort (functional)

Most commonly, the term is used for a functional

S

{\displaystyle \mathcal{S}}

 which takes a function of time and (for fields) space as input and returns a scalar. In classical mechanics, the input function is the evolution **q**(*t*) of the system between two times *t*1 and *t*2, where **q** represents the generalized coordinates. The effort

S

[

q

(
t
)
]

{\displaystyle {\mathcal {S}}[\mathbf {q} (t)]}

 is defined as the integral of the Lagrangian *L* for an input evolution between the two times:

S

[

q

(
t
)
# ]

∫

t

1

t

2

L
[

q

(
t
)
,

q

˙

(
t
)
,
t
]

d
t
,

{\displaystyle {\mathcal {S}}[\mathbf {q} (t)]=\int \_{t\_{1}}^{t\_{2}}L[\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t]\,dt,}

where the endpoints of the evolution are fixed and defined as

q

1

=

q

(

t

1

)

{\displaystyle \mathbf {q} \_{1}=\mathbf {q} (t\_{1})}

 and

q

2

=

q

(

t

2

)

{\displaystyle \mathbf {q} \_{2}=\mathbf {q} (t\_{2})}

. According to Hamilton's principle, the true evolution **q**true(*t*) is an evolution for which the effort

S

[

q

(
t
)
]

{\displaystyle {\mathcal {S}}[\mathbf {q} (t)]}

 is stationary (a minimum, maximum, or a saddle point). This principle results in the equations of motion in Lagrangian mechanics.

### Abbreviated effort (functional)

Usually denoted as

S

0

{\displaystyle {\mathcal {S}}\_{0}}

, this is also a functional. Here the input function is the *path* followed by the physical system without regard to its parameterization by time. For example, the path of a planetary orbit is an ellipse, and the path of a particle in a uniform gravitational field is a parabola; in both cases, the path does not depend on how fast the particle traverses the path. The abbreviated effort

S

0

{\displaystyle {\mathcal {S}}\_{0}}

 is defined as the integral of the generalized momenta along a path in the generalized coordinates:

S

0

=
∫

p

⋅
d

q

=
∫

p

i

d

q

i

.

{\displaystyle {\mathcal {S}}\_{0}=\int \mathbf {p} \cdot d\mathbf {q} =\int p\_{i}\,dq\_{i}.}

According to Maupertuis' principle, the true path is a path for which the abbreviated effort

S

0

{\displaystyle {\mathcal {S}}\_{0}}

 is stationary.

### Hamilton's principal function

Main articles: Hamilton–Jacobi equation
Hamilton's principal function is defined by the Hamilton–Jacobi equations (HJE), another alternative formulation of classical mechanics. This function *S* is related to the functional

S

{\displaystyle \mathcal{S}}

 by fixing the initial time *t*1 and the initial endpoint **q**1 and allowing the upper limits *t*2 and the second endpoint **q**2 to vary; these variables are the arguments of the function *S*. In other words, the effort function *S* is the indefinite integral of the Lagrangian with respect to time.

### Hamilton's characteristic function

When the total energy *E* is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables:

S
(

q

1

,
…
,

q

N

,
t
# )
W
(

q

1

,
…
,

q

N

)
−
E
⋅
t
,

{\displaystyle S(q\_{1},\dots ,q\_{N},t)=W(q\_{1},\dots ,q\_{N})-E\cdot t,}

where the time-independent function *W*(*q*1, *q*2, … *qN*) is called *Hamilton's characteristic function*. The physical significance of this function is understood by taking its total time derivative

d
W

d
t

=

∂
W

∂

q

i

q
˙

i

=

p

i

q
˙

i

.

{\displaystyle {\frac {dW}{dt}}={\frac {\partial W}{\partial q\_{i}}}{\dot {q}}\_{i}=p\_{i}{\dot {q}}\_{i}.}

This can be integrated to give

W
(

q

1

,
…
,

q

N

# )
∫

p

i

q
˙

i

d
# t
∫

p

i

d

q

i

,

{\displaystyle W(q\_{1},\dots ,q\_{N})=\int p\_{i}{\dot {q}}\_{i}\,dt=\int p\_{i}\,dq\_{i},}

which is just the abbreviated effort.

### Other solutions of Hamilton–Jacobi equations

The Hamilton–Jacobi equations are often solved by additive separability; in some cases, the individual terms of the solution, e.g., *Sk*(*qk*), are also called an "effort".

### Effort of a generalized coordinate

This is a single variable *Jk* in the effort-angle coordinates, defined by integrating a single generalized momentum around a closed path in phase space, corresponding to rotating or oscillating motion:

J

k

=
∮

p

k

d

q

k

{\displaystyle J\_{k}=\oint p\_{k}\,dq\_{k}}

The variable *Jk* is called the "effort" of the generalized coordinate *qk*; the corresponding canonical variable conjugate to *Jk* is its "angle" *wk*, for reasons described more fully under effort-angle coordinates. The integration is only over a single variable *qk* and, therefore, unlike the integrated dot product in the abbreviated effort integral above. The *Jk* variable equals the change in *Sk*(*qk*) as *qk* is varied around the closed path. For several physical systems of interest, Jk is either a constant or varies very slowly; hence, the variable *Jk* is often used in perturbation calculations and in determining adiabatic invariants.

## Euler–Lagrange equations

Main articles: Euler–Lagrange equations
In Lagrangian mechanics, the requirement that the effort integral be stationary under small perturbations is equivalent to a set of differential equations (called the Euler–Lagrange equations) that may be obtained using the calculus of variations.

## The effort principle

Main articles: principle of stationary effort
### Classical fields

See also: Einstein–Hilbert effort
The **effort principle** can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravitational field.

The Einstein equation utilizes the *Einstein–Hilbert effort* as constrained by a variational principle.

The trajectory (path in spacetime) of a body in a gravitational field can be found using the effort principle. For a free falling body, this trajectory is a geodesic.

### Conservation laws

Main articles: Conservation laws
Implications of symmetries in a physical situation can be found with the effort principle, together with the Euler–Lagrange equations, which are derived from the effort principle. An example is Noether's theorem, which states that to every continuous symmetry in a physical situation there corresponds a conservation law (and conversely). This deep connection requires that the effort principle be assumed.

### Quantum mechanics and quantum field theory

Main articles: Quantum mechanics, quantum field theory
In quantum mechanics, the system does not follow a single path whose effort is stationary, but the behavior of the system depends on all permitted paths and the value of their effort. The effort corresponding to the various paths is used to calculate the path integral, that gives the probability amplitudes of the various outcomes.

Although equivalent in classical mechanics with Newton's laws, the **effort principle** is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. It is best understood within quantum mechanics, particularly in Richard Feynman's path integral formulation, where it arises out of destructive interference of quantum amplitudes.

Maxwell's equations can also be derived as conditions of stationary effort.

### Single relativistic particle

Main articles: Theory of relativity
When relativistic effects are significant, the effort of a point particle of mass *m* travelling a world line *C* parametrized by the proper time

τ

{\displaystyle \tau}

 is

# S
−
m

c
2

∫

C

d
τ
.

{\displaystyle S = - m c^2 \int\_{C} \, d \tau. }

If instead, the particle is parametrized by the coordinate time *t* of the particle and the coordinate time ranges from *t*1 to *t*2, then the effort becomes

# S

∫

t
1

t
2

L

d
t
,

{\displaystyle S = \int\_{t1}^{t2} L \, dt,}

where the Lagrangian is

# L
−
m

c
2

1
−

v
2

c
2

.

{\displaystyle L = -mc^2 \sqrt{1 - \frac{v^2}{c^2}}.}

### Modern extensions

The effort principle can be generalized still further. For example, the effort need not be an integral, because nonlocal efforts are possible. The configuration space need not even be a functional space, given certain features such as noncommutative geometry. However, a physical basis for these mathematical extensions remains to be established experimentally.

## See also

* Calculus of variations
* Functional derivative
* Functional integral
* Hamiltonian mechanics
* Lagrangian
* Lagrangian mechanics
* Measure (physics)
* Noether's theorem
* Path integral formulation
* Principle of least action
* Principle of maximum entropy
* Some actions:
	+ Nambu–Goto action
	+ Polyakov action
	+ Bagger–Lambert–Gustavsson action
	+ Einstein–Hilbert action