Principle of least action

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Author: Dr. Chris G. Gray, Department of Physics University of Guelph

The principle of least action is the basic variational principle of particle and continuum systems. The true dynamical trajectories of a system are found by imagining all possible trajectories that the system could conceivably take, computing the action (a functional of the trajectory) for each of these trajectories, and selecting the one (or more) that makes the action "least" (actually stationary). The true trajectories are those that have least action.

1 Statements of Hamilton and Maupertuis Principles
2 History
3 Euler-Lagrange Equation
4 Restrictions to Holonomic and Nondissipative Systems
5 When Action is a Minimum
6 Relation of Hamilton and Maupertuis Principles
7 Generalizations
8 Practical Use of Action Principles
9 Relativistic Systems
10 Relation to Quantum Variational Principles
11 Continuum Mechanics and Field Theory
12 Conservation Laws
13 References (historical)
14 References
15 Further reading
16 See Also

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Statements of Hamilton and Maupertuis Principles

There are two major versions of the action, due to Hamilton and Maupertuis, and two corresponding action principles. The Hamilton principle is nowadays the most used. The Hamilton action $S$ is defined as an integral along an actual or virtual (trial) space-time trajectory $q(t)$ connecting two specified space-time events, initial event $A \equiv(q_A,t_A=0)$ and final event $B \equiv (q_B,t_B=T)$ ,

(1)

$S\; =\; \int _{0}^{T}L\, \left(q\; ,\; \dot{q}\right) \; d t\quad ,$

where $L\, \left(q\; ,\; \dot{q}\right)$ is the Lagrangian, and $\dot{q}\; =\; dq/d t$ . For most of what follows we will assume the simplest case where $L = K - V$ , where $K$ and $V$ are the kinetic and potential energies, respectively; an exception occurs in the relativistic section below. In general, q stands for the complete set of independent generalized coordinates, $q_1, q_2, \ldots$ , $q_f$ , where $f$ is the number of degrees of freedom. Any holonomic (coordinate) geometric constraints are assumed to be taken into account by the choice of the $q_1$ , $\ldots$ , $q_f$ . Nonholonomic (velocity) geometric constraints are excluded (see below). Hamilton's principle states that among all conceivable trajectories $q(t)$ that could connect the given end points $q_A$ and $q_B$ in the given time $T$ , the true trajectories are those that make $S$ stationary. As we shall see, if the trajectory is sufficiently short, the action $S$ is a minimum. In general, for long trajectories, $S$ is a saddle point (and is never a maximum). To emphasize the particular constraint on the varied trajectories, we write Hamilton's principle as

(2)

$\left(\delta S\right)_{T} \; =\; 0\quad ,$

where the constraint of fixed time $T$ is written explicitly, and the constraint of fixed end-positions $q_A$ and $q_B$ is left implicit. We will consider other variational principles below, but all will have fixed $q_A$ and $q_B$ (quantities other than $T$ will also be constrained) so we will always leave the constraint of fixed $q_A$ and $q_B$ implicit. It is clear from (1) that S is a functional of the trial trajectory q(t), and in (2) $\delta S$ denotes the first-order variation in S corresponding to the small variation $\delta q(t)$ in the trial trajectory.

The second major version of the action is Maupertuis' action $W$ , where

(3)

$W\; =\; \int _{q_{A} }^{q_{B} }pdq\; =\; \int _{0}^{T}2\, K\, d t\quad ,$

where the first (time-independent) form is the general definition, with $p\; =\; \partial L/\partial \dot{q}$ the canonical momentum, and pdq stands for $p_1dq_1 + p_2dq_2 + \ldots + p_fdq_f$ in general. The second (time-dependent) form for $W$ in (3) is valid for normal systems in which the kinetic energy $K$ is quadratic in the velocity components $\dot{q}_{1} \; ,\; \dot{q}_{2} \; ,\; \cdots \; ,\; \dot{q}_{f}$ . The Maupertuis action principle states that for true trajectories $W$ is stationary on trajectories with fixed end positions $q_A$ and $q_B$ and fixed energy $E$ . Following our earlier conventions, we write this principle as

(4)

$\left(\delta W\right)_{E} \; =\; 0\quad .$

Note that $E$ is fixed but $T$ is not in Maupertuis' principle (4), the reverse of the conditions in Hamilton's principle (2).

The two principles (2) and (4) are related by a Legendre transformation, as discussed below. Hamilton's principle is applicable to both conservative and nonconservative systems, with $L\, \left(q\; ,\; \dot{q}\; ,\; t\right)$ then explicitly time-dependent (e.g. due to a time-dependent potential), whereas the form (4) of Maupertuis' principle is restricted to conservative systems (it can be generalized – see Gray et al 2004).

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History

Various aspects of the history of action principles and variational principles in general are discussed in the historical references at the end of this article. Maupertuis' principle is older than Hamilton's principle by about a century (1744 vs 1834). Over the years, and even recently, a number of reformulations and generalizations of the two basic action principles have been given (see Gray et al 1996, 2004 for extensive discussions and references).

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Euler-Lagrange Equation

Solution of the variational problem posed by Hamilton's principle (2) yields the true trajectories $q(t)$ . Solution of Maupertuis' variational equation (4) using the time-dependent (second) form of $W$ in (3) also yields the the true trajectories, whereas using the time-independent (first) form of $W$ in (3) yields true orbits, i.e. spatial shape of the true paths. The solutions can be obtained directly from the variational principles (see below), or from the solution of the corresponding Euler-Lagrange differential equations which are equivalent to the variational principles. For Hamilton's principle, the Euler-Lagrange equation of motion (usually called simply Lagrange's equation) is (see, e.g., Brizard 2008, Goldstein et al 2002)

(5)

$\frac{d}{d t} \; \left(\frac{\partial L}{\partial \dot{q}} \right)\; -\; \frac{\partial L}{\partial q} \; =\; 0\quad ,$

where we have assumed one degree of freedom for simplicity of notation; if there is more than one, an equation of the form (5) holds for each of the $q_1$ , $\ldots$ , $q_f$ . The time-dependent version of Maupertuis' principle yields the same equation of motion for the space-time trajectories $q(t)$ . The time-independent version of Maupertuis' principle (often called Jacobi's principle) yields (Lanczos 1970, Landau and Lifshitz 1969) corresponding equations for the spatial paths (orbits).

Strictly speaking, because the action principles are formulated as boundary value problems (q is specified at two points $q_A$ and $q_B$ ) and not as initial value problems, there may be more than one solution: there can in fact be zero, one, two, ..., up to an infinite number of solutions in particular problems. For example, applying the Hamilton principle to the one dimensional harmonic oscillator and specifying q = 0 at t = 0 and at t = T (one period $2\pi/\omega$ ) gives an infinite number of solutions, i.e. $q(t) = A sin \omega t$ with one solution for each value of the amplitude A, which is arbitrary. The same system with the constraints q = 0 at t = 0 and q = A at t = T/4 has the unique solution $q(t) = A sin \omega t$ , and for the constraints q = 0 at t = 0 and at t = T/4 no solution exists. In practice, one usually has initial conditions in mind, and selects the appropriate solution of the corresponding boundary value problem, or imposes the initial conditions directly on the solution of the Euler-Lagrange equation of motion.

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Restrictions to Holonomic and Nondissipative Systems

The action principles (2) and (4) are restricted to holonomic systems, i.e. systems whose geometrical constraints (if any) involve only the coordinates and not the velocities. Simple examples of holonomic and nonholonomic systems are a particle confined to a spherical surface, and a wheel confined to rolling without slipping on a horizontal plane, respectively. Attempts to extend the action principles to nonholonomic systems have been controversial, and do not appear to have been successful (Papastavridis 2002).

In general, the action principles do not apply to dissipative systems, i.e. systems with frictional forces. However, for a few cases, Lagrangians for dissipative systems have been found, and Hamilton's principle then applies (for a brief review, see Gray et al 2004).

More generally, the question of whether a Lagrangian and corresponding action principle exist for a particular dynamical system, given the equation of motion and the nature of the forces acting on the system, is referred to as the "inverse problem of the calculus of variations" (Santilli 1978).

Figure 1: Space-time diagram for a family of true trajectories $x(t)$ for the quartic oscillator $[V(x) = (1/4)Cx^4]$ starting at $P(0,0)$ with $v_0 > 0$ . For this particular oscillator the kinetic focus occurs at a fraction 0.646 of the half-period $T_0/2$ , illustrated here for trajectory $0$ . The kinetic foci of all true trajectories of this family lie along the heavy gray line, the caustic, which is a hyperbolic curve for this oscillator. Squares indicate recrossing events of true trajectory $0$ with the other two true trajectories. (From Gray and Taylor 2007.)

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When Action is a Minimum

The action (either $S$ or $W$ ) is stationary for true trajectories; it is either a local minimum, or a saddle point (the action is larger for some trial trajectories and smaller for others, compared to the true trajectory action). Action is never a maximum (nonrelativistically). We discuss here only the case of the Hamilton action S for one-dimensional (1D) systems, and refer to Gray and Taylor (2007) for discussions of Maupertuis' action $W$ , and 2D etc. systems. For some 1D potentials $V(x)$ (those with $\partial^2V/ \partial x^2 \leq 0$ everywhere), e.g. $V(x) = 0$ , $V(x) = mg x$ , and $V(x) = -Cx^2$ , all true trajectories have minimum $S$ . For most potentials, however, only sufficiently short true trajectories have minimum action; the others have an action saddle point. "Sufficiently short" means that the final space-time event occurs before the so-called kinetic focus event of the trajectory. The latter is defined as the earliest event along the trajectory, following the initial event, where the second variation $\delta^2S = 0$ , for some trajectory variation. A more intuitive definition of kinetic focus can be given. As an example, consider a family of true trajectories for the quartic oscillator with $V(x) = (1/4) Cx^4$ , all starting at $P (x = 0$ at $t = 0)$ , and with initial velocity $v_0 > 0$ . Three trajectories of the family, denoted $0$ , $1$ , and $2$ , are shown in Fig. 1. These true trajectories intersect each other – note the squares in Fig. 1 showing intersections of trajectories $1$ and $2$ with trajectory $0$ . The kinetic focus event $Q_0$ of the true trajectory $0$ , with starting event $P$ , is the event closest to $P$ at which a second true trajectory, with slightly different initial velocity at $P$ , intersects trajectory $0$ , in the limit for which the two trajectories coalesce as their initial velocities at $P$ are made equal. Based on this definition a simple prescription for finding the kinetic focus can be given (Gray and Taylor 2007), and for a quartic oscillator trajectory starting at P(0,0) the kinetic focus Q occurs at time $t_Q = 0.646(T/2)$ , where T is the period, as shown in Fig.1 for trajectory 0.

The other trajectories shown in Fig. 1 have their own kinetic foci, i.e. $Q_1$ for trajectory $1$ and $Q_2$ for trajectory $2$ . The locus of all the kinetic foci of the family is called the caustic (it is an envelope), and is shown as the heavy gray line in Fig. 1.

Thus, for trajectory $0$ in Fig. 1, if the trajectory terminates before kinetic focus $Q_0$ , the action $S$ is a minimum; if the trajectory terminates beyond $Q_0$ , the action is a saddle point.

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Relation of Hamilton and Maupertuis Principles

The Hamilton and Maupertuis principles are related by a Legendre transformation (Gray et al 1996, 2004). Recall first that the Lagrangian $L \left(q\; ,\; \dot{q}\right)$ and Hamiltonian $H(q, p)$ are so-related, i.e.

(6)

$H \left(q\; ,\; p\right)\; =\; p \dot{q}\; -\; L \left(q\; ,\; \dot{q}\right)\quad .$

If we integrate (6) with respect to $t$ along an arbitrary virtual or trial trajectory between two points $q_A$ and $q_B$ , and use the definitions (1) and (3) of $S$ and $W$ we get $\bar{E}T = W - S$ , or

(7)

$S\; =\; W\; -\; \bar{E}\; T\quad ,$

where $\bar{E}\; \equiv \; \int _{0}^{T}d t\; H/T$ is the mean energy along the trial trajectory. (Along a true trajectory, with $\bar{E}= E =$ const, (7) reduces to the well-known relation (Goldstein et al 2002) $S=W-ET$ .) From the Legendre transformation relation (7) between $S$ and $W$ , for conservative systems one can derive Hamilton's principle from Maupertuis' principle, and vice-versa (Gray et al, 1996, 2004). The two action principles are thus equivalent for conservative systems, and related by a Legendre transformation whereby one changes between energy and time as independent variables.

The existence in mechanics of two actions and two corresponding variational principles with a Legendre transformation between them is analogous to the situation in thermodynamics (Gray et al 2004). There, as established by Gibbs, one introduces two free energies related by a Legendre transformation, i.e. the Helmholtz and Gibbs free energies, with each free energy satisfying a variational principle which determines the thermal equilibrium state of the system.

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Generalizations

If we vary the trial trajectory $q(t)$ in (7), with no variation in end positions $q_A$ and $q_B$ , the corresponding variations $\delta S$ , $\delta W$ , $\delta \bar{E}$ and $\delta T$ are seen to be related by

(8)

$\delta S\; +\; \bar{E}\; \delta \; T\; =\; \delta \; W\; -\; T \; \delta \; \bar{E} .$

Next one can show (Gray et al 1996) that the two sides of (8) separately vanish for variations around a true trajectory. The left side of (8) then gives $\delta S + E \delta T = 0$ , since $\bar{E} = E$ (a constant) on a true trajectory, which is called the unconstrained Hamiltonian principle. This can be written in standard form $\delta S = \lambda \delta T$ , where $\lambda$ is a constant Lagrange multiplier, here determined as $\lambda = -E$ (negative of energy of the true trajectory). If one constrains $T$ to be fixed for all trial trajectories, then $\delta T = 0$ and we have ( $\delta S)_T = 0$ , the usual Hamilton principle. If instead we constrain $S$ to be fixed we get ( $\delta T)_S = 0$ , the so-called reciprocal Hamilton principle.

The right side of (8) gives $\delta W - T \delta \bar{E} = 0$ , which is called the unconstrained Maupertuis principle, which can also be written as $\delta W = \lambda \delta \bar{E}$ where $\lambda = T$ (duration of true trajectory) is a constant Lagrange multiplier. If one constrains $\bar{E}$ to be fixed for the trial trajectories, one gets ( $\delta W)_\bar{E} = 0$ , which is a generalization of Maupertuis' principle (4); we see that the constraint of fixed energy in (4) can be relaxed to one of fixed mean energy. If instead we constrain W to be fixed, we get ( $\delta \bar{E})_W = 0$ , which is called the reciprocal Maupertuis principle. In these generalizations of Maupertuis' principle, conservation of energy is a consequence of the principle for time-invariant systems (just as it is for Hamilton's principle), whereas conservation of energy is an assumption of the original Maupertuis principle.

In all the variational principles discussed here, we have held the end-points $q_A$ and $q_B$ fixed. It is possible to derive additional generalized principles (Gray et al 2004) which allow variations $\delta q_A$ and $\delta q_B$ in the end-points.

As we shall see, these alternative formulations of the action principles, particularly the reciprocal Maupertuis principle, have advantages when using action principles to solve practical problems, and also in making the connection to quantum variational principles. We note that reciprocal variational principles are common in geometry and in thermodynamics (see Gray et al 2004 for discussion and references), but their use in mechanics is relatively recent.

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Practical Use of Action Principles

Just as in quantum mechanics, variational principles can be used directly to solve a dynamics problem, without employing the equations of motion. This is termed the the direct variational or Rayleigh-Ritz method. The solution may be exact (in simple cases) or essentially exact (using numerical methods), or approximate and analytic (using a restricted and simple set of trial trajectories). We illustrate the approximation method with a simple pedagogical example and refer the reader elsewhere for more complicated examples dealing with research problems (Gray et al 1996, 2004). Consider a one-dimensional quartic oscillator, with Hamiltonian

(9)

$H\; =\; \frac{p^{2} }{2 m} \; +\; \frac{1}{4} \; C\; x^{4} \quad .$

Unlike a harmonic oscillator, the frequency $\omega$ will depend on the amplitude or energy of motion. We wish to estimate this dependence. As a trial trajectory we take

(10)

$x(t) = A \sin \omega t$ ,

where the amplitude A is regarded as known (given W and $\omega$ -see eq.(12)) and where we treat $\omega$ as a variational parameter; we will vary $\omega$ such that an action principle is satisfied. For definiteness, we use the reciprocal Maupertuis principle $(\delta \bar{E})_W = 0$ discussed in the previous section, but the other action principles can be employed similarly. From the definitions, we find the mean energy $\bar{E}$ and action $W$ over a cycle of the trial trajectory (10) to be

(11)

$\bar{E}\; =\; \frac{\omega }{4 \pi} \; W\; +\; C\; \frac{3\; W^{2} }{32 \pi ^{2} m^2 \omega ^{2} } \quad ,$

(12)

$W\; =\; \pi \; \omega \; m\; A^{2} \quad .$

Treating $\omega$ as a variational parameter in (11) and applying $\left(\partial \bar{E}/\partial \omega \right)_{W} \; =\; 0$ gives

(13)

$\omega \; =\; \left(\frac{3\; C\; W}{4\; \pi \; m^{2} } \right)^{1/3} \quad .$

Substituting (13) in (11) gives for $\bar{E}$

(14)

$\bar{E}\; =\; \frac{1}{2} \; \left(\frac{C}{m^{2} } \right)^{1/3} \left(\frac{3\; W}{4 \pi } \right)^{4/3} \quad ,$

which can be combined with (13) to give

(15)

$\omega \left(\bar{E}\right)\; =\; \left(\frac{2\; C\; \bar{E}}{m^{2} } \right)^{1/4} \quad ,$

i.e. a variational estimate of the frequency as a function of the energy. The approximation (15) is accurate to 0.75% and can be improved systematically by including terms $B \sin{3\omega t}$ , $D \sin{5\omega t}$ , etc., in the trial trajectory $x(t)$ .

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Relativistic Systems

The Hamilton and Maupertuis principles, and the generalizations discussed above, can be put in Lorentz invariant form (Gray et al 2004). As an example of the relativistic Hamilton principle, consider a particle of mass m and charge e in an external electromagnetic field with a potential having contravariant components $A^\alpha = (A_0, A_i) \equiv (\phi, A_i)$ , and covariant components $A_\alpha = \left(A_{0} ,\; -\; A_{i} \right) \equiv (\phi, - A_i)$ , where $\phi$ and $A_i$ (for $i = 1, 2, 3$ ) are the usual scalar and vector potentials. A Lorentz invariant form for the Hamilton action for this system is (Jackson 1999, Landau and Lifshitz 1962, Lanczos 1970)

(16)

$S\; =\; m\; \int d s\; +\; e\; \int A_{\alpha } \; d x^{\alpha } \quad .$

The signs of the Lagrangian and corresponding action can be chosen arbitrarily; here we choose the sign of Lanczos (1970) in (16). The four-dimensional path in (16) runs from the initial space-time point $x_{A} \; =\; \left(x_{A}^{0} \; ,\; x_{A}^{1} \; ,\; x_{A}^{2} \; ,\; x_{A}^{3} \right)$ to the final space-time point $x_{B} \; =\; \left(x_{B}^{0} \; ,\; x_{B}^{1} \; ,\; x_{B}^{2} \; ,\; x_{B}^{3} \right)$ , with corresponding proper times $s_A$ and $s_B$ . Here $ds$ is the infinitesimal interval of the path (or of the proper time), $ds^2 = dx_\alpha dx^\alpha = g_{\alpha \beta} dx^\alpha dx^\beta = dx_{0}^{2} \; -\; dx_{i}^{2}$ , the metric has signature ( $+$ , $-$ , $-$ , $-$ ), and we use the summation convention and take $c$ (speed of light ) $= 1$ . S itself is not gauge invariant, but a gauge transformation $A_\alpha \rightarrow A_\alpha + {\partial f}/{\partial x^\alpha}$ (for arbitrary $f$ ) adds only constant boundary points terms to $S$ , so that $\delta S$ is unchanged. The Hamilton principle is thus gauge invariant.

If we introduce a parameter $\tau$ along the four-dimensional path (a valid choice is proper time s along the true or any virtual path), we can write $S$ in standard form, $S = \int L d \tau$ , where $L = m [v_\alpha v^\alpha ]^{1/2} + e A_\alpha v^\alpha$ is the Lagrangian and $v^\alpha = dx^\alpha /d \tau$ . The Euler-Lagrange equation yields the covariant Lorentz equation of motion

(17)

$m\; \frac{d\, v_{\alpha } }{d\, s} \; =\; e\; F_{\alpha \beta } \; v^{\beta } \quad ,$

where $F_{\alpha \beta} = \partial_\alpha A_\beta - \partial_\beta A_\alpha$ , with $\partial_\alpha = \partial / \partial x^\alpha$ , and we have chosen the parameter $\tau = s$ , the true path proper time. Specific examples, such as an electron in a uniform magnetic field, are discussed in the references (Gray et al 2004, Jackson 1999). As discussed briefly below, the equations for the field (Maxwell equations) can also be derived from an action principle.

Action principles are important also in general relativity. Note from (16) that for a special relativistic free particle the action principle $\delta S = \delta \int ds = 0$ can be interpreted as a "principle of stationary proper time" (Rohrlich 1965), or more colloquially as a "principle of maximal aging" (Taylor and Wheeler 1992). The proper time is stationary, here a maximum, for the true trajectory (which is straight in a Lorentz frame) compared to the proper time for all virtual trajectories. The principle of maximal aging is also valid, for "short" trajectories, in general relativity for the motion of a particle in a gravitational field (Taylor and Wheeler 2000). For "long" true trajectories (defined above in the fifth section) the proper time is a saddle point (Misner et al 1973, Wald 1984). In general relativity the Einstein gravitational field equations can also be derived from an action principle, using the so-called Einstein-Hilbert action (Landau and Lifshitz 1962, Misner et al 1973). The latter is perhaps the best example of a case where new physics was derived from action principles, since Einstein and Hilbert were both motivated by action principles, at least partly, in establishing the field equations. There was a near miss in the case of wave mechanics, as described in the next section.

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Relation to Quantum Variational Principles

We discuss here only the Schrödinger time-independent quantum variational principle; for discussion and references to various quantum time-dependent principles, we refer to Gray et al (2004). As is well known (e.g. Merzbacher 1998), the time-independent Schrödinger equation

(18)

$\hat{H}\; \psi _{n} \; =\; E_{n} \; \psi _{n}$

for the stationary states $\psi_n$ , with energies $E_n$ , is equivalent to the variational principle

(19)

$\left(\delta \; \frac{\left\langle \psi \; \left|\, \hat{H}\, \right|\; \psi \right\rangle }{\left\langle \psi \; |\; \psi \right\rangle } \right)_{n} \; =\; 0\quad ,$

where $\hat{H}$ is the Hamiltonian operator corresponding to the classical Hamiltonian $H(q, p)$ . The subscript in (19), quantum number $n$ , indicates a constrained variation of $\psi$ such that $\psi_n$ is the particular stationary solution selected; for example, to obtain the ground state, one could restrict the search to nodeless trial functions $\psi$ . As mentioned earlier, (19) is the basis of a very useful approximation scheme in quantum mechanics (Epstein 1974), analogous to the direct use of classical action principles to solve classical dynamics problems (see above).

The reader will notice the similarity of (19) to one of the classical variational principles discussed above, i.e. the reciprocal Maupertuis principle applied to the case of stationary motions:

(20)

$\left(\delta \bar{E}\right)_{W} \; =\; 0\quad .$

The classical mean energy $\bar{E}\; \equiv \; \int _{0}^{T}d t\; H/T$ is clearly analogous to the quantum mean energy $\left\langle \psi \; \left|\, \hat{H}\, \right|\; \psi \right\rangle /\left\langle \psi \; |\; \psi \right\rangle$ . The constraints ( $W$ in (20), n in (19)) are also analogous because at large quantum numbers we have for stationary bound motions $W_n \sim nh$ (Bohr-Sommerfeld), where $h$ is Planck's constant. Thus fixed $n$ and fixed $W$ are equivalent, at least for large quantum numbers.

The above heuristic arguments can be tightened up. First, (20) can be derived (in simple cases) in the classical limit $(h \to 0)$ from (19) (Gray et al 1996). Conversely, one can "derive" quantum mechanics (i.e. (19)) by applying quantization rules to (20) (Gray et al 1999). Schrödinger, in his first paper on wave mechanics (Schrödinger 1926a), tried to derive the quantum variational principle from a classical variational principle. Unfortunately he did not have available the formulation (20) of the classical action principle, and abandoned this route to quantum mechanics. Very quickly, in his second paper (Schrödinger 1926b), he found the route which is now in the text books.

A semiclassical variational principle can be based on the reciprocal Maupertuis principle (20) (Gray et al 2004). Thus, for bound states, one first determines the classical energy as a function of the action W by solving (20) as described earlier (e.g. see eq.(14) for the quartic oscillator), and then imposes the Bohr-Sommerfeld quantization condition (or one of its refinements) on action W. This gives the allowed energies semiclassically as a function of the quantum number.

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Continuum Mechanics and Field Theory

Action principles can be applied to field-like quantities $\phi (x, t)$ , both classically (Goldstein et al 2002, Landau and Lifshitz 1962, Soper 1976, Burgess 2002, Jackson 1999, Morse and Feshbach 1953, Brizard 2008) and quantum-mechanically (Dyson 2007, Wentzel 1949). The systems can be nonrelativistic or relativistic. We have already mentioned above the application of action principles to the electromagnetic and gravitational fields, and to the Schrödinger wave function. These methods are also widely applied in classical continuum mechanics, e.g., to strings, membranes, elastic solids and fluids (Yourgrau and Mandelstam 1968, Lanczos 1970, Reddy 2002).

We illustrate with one simple example, the one-dimensional vibrating string, following Brizard (2008). The nonrelativistic classical equation of motion for the transverse displacement $\phi(x, t)$ is

(21)

$\rho \; \frac{\partial ^{2} \; \phi }{\partial \; t^{2} } \; -\; \tau \; \frac{\partial ^{2} \; \phi }{\partial \; x^{2} } \; =\; 0\quad ,$

where $\rho$ is the density and $\tau$ the tension. Eq. (21) is the well known wave equation. It is assumed that $\phi (x, t)$ is zero at the two ends, $x = 0$ and $x = \ell$ , and that $\phi (x, t)$ is given at two times, $t = 0$ and $t = T$ . One easily verifies that the equation of motion (21) follows from the action principle $\delta S = 0$ , with the given constraints, where

(22)

$S\; =\; \int _{0}^{T}d t\; \int _{0}^{\ell}d x\; \mathcal{L}\; \left(\phi \; ,\; \partial _{t} \phi \; ,\; \partial _{x} \phi \right)\quad ,$

with

(23)

$\mathcal{L}\; \left(\phi \; ,\; \partial _{t} \phi \; ,\; \partial _{x} \phi \right) = \frac{1}{2} \; \rho \left(\frac{\partial \phi }{\partial t} \right)^{2} \; -\; \frac{1}{2} \; \tau \left(\frac{\partial \phi }{\partial x} \right)^{2}$

the Lagrangian density $\left(\int_0^{\ell} dx\; \mathcal{L}\; =\; L \right)$ . Because of the simple quadratic Lagrangian density (23), the variation of (22) can readily be done directly; alternatively, we can use the Euler-Lagrange equation for 1D fields $\phi(x, t)$ , a generalization of (5),

(24)

$\frac{\partial }{\partial t} \; \left(\frac{\partial \mathcal{L}} {\partial \left(\partial _{t} \phi \right)} \right)\; +\; \frac{\partial }{\partial x} \; \left(\frac{\partial \mathcal{L}}{\partial \left(\partial _{x} \phi \right)} \right)\; -\; \frac{\partial \mathcal{L}}{\partial \phi } \; =\; 0\quad ,$

which also gives (21).

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Conservation Laws

Conservation laws are a consequence of symmetries of the Lagrangian or action. For example, conservation of energy follows from invariance under time translation. The link between symmetries and conservation laws holds for particle and continuum systems (Noether's theorem). The conservation laws can be derived either from the Lagrangian and equations of motion (Goldstein et al 2002), or directly from the action and the variational principle (Goldstein et al 2002, Lanczos 1970, Oliver 1994, Schwinger et al 1998). Since Noether's theorem is to be discussed elsewhere in Scholarpedia, we refer to that article for details.

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References (historical)

Goldstine, H.H. (1980). A History of the Calculus of Variations from the 17th Through the 19th Century, Springer, New York.
Hankins, T.L. (1980). Sir William Rowan Hamilton, Johns Hopkins U.P., Baltimore.
Lanczos, C. (1970). The Variational Principles of Mechanics, 4th edition, University of Toronto Press, Toronto.
Terrall, M. (2002). The Man who Flattened the Earth, University of Chicago Press, Chicago. (biography of Maupertuis)
Todhunter, I. (1861). A History of the Progress of the Calculus of Variations During the Nineteenth Century, Cambridge U.P., Cambridge.
Yourgrau, W. and S. Mandelstam (1968). Variational Principles in Dynamics and Quantum Theory, 3rd edition, Saunders, Philadelphia.

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References

Brizard, A.J. (2008). An Introduction to Lagrangian Mechanics, World Scientific, Singapore.
Burgess, M. (2002). Classical Covariant Fields, Cambridge U.P., Cambridge.
Dyson, F. (2007). Advanced Quantum Mechanics, World Scientific, Singapore.
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[edit]

Suggested by:	Dr. Edwin F. Taylor, Massachusetts Institute of Technology
Invited by:	Dr. Riccardo Guida, Institut de Physique Théorique; CEA, IPhT; CNRS; Gif-sur-Yvette, France
Action editor:	Dr. Riccardo Guida, Institut de Physique Théorique; CEA, IPhT; CNRS; Gif-sur-Yvette, France
Assistant editor:	Mr. Leo Trottier, PhD student; University of California, San Diego

Principle of least action

From Scholarpedia

Contents

Statements of Hamilton and Maupertuis Principles

History

Euler-Lagrange Equation

Restrictions to Holonomic and Nondissipative Systems

When Action is a Minimum

Relation of Hamilton and Maupertuis Principles

Generalizations

Practical Use of Action Principles

Relativistic Systems

Relation to Quantum Variational Principles

Continuum Mechanics and Field Theory

Conservation Laws

References (historical)

References

Further reading

See Also

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