Principle of least action
The principle of least action is the basic variational principle of particle and continuum systems. In Hamilton's formulation, a true dynamical trajectory of a system between an initial and final configuration in a specified time is 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 one that makes the action locally stationary (traditionally called "least"). True trajectories are those that have least action.
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 any actual or virtual (conceivable or trial) space-time trajectory \(q(t)\) connecting two specified space-time events, initial event \(A \equiv(q_A,t_A=0)\) and final event \(B \equiv (q_B,t_B=T)\ ,\) \[\tag{1} S\; =\; \int _{0}^{T}L\, \left(q\; ,\; \dot{q}\right) \; d t\quad , \]
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 (see Section 4). 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. In Hamilton's principle the conceivable or trial trajectories are not constrained to satisfy energy conservation, unlike the case for Maupertuis' principle discussed later in this section (see also Section 7). Energy conservation will emerge as a consequence of the Hamilton principle for time-invariant systems (Section 12). More than one true trajectory may satisfy the given constraints of fixed end-positions and travel time (see Section 3). As we shall see in Section 5, if the trajectory is sufficiently short, the action \(S\) is a local minimum for a true trajectory. In general, for long trajectories \(S\) is a saddle point (and is never a maximum) for a true trajectory. To emphasize a particular constraint on the varied trajectories, we write Hamilton's principle as \[\tag{2} \left(\delta S\right)_{T} \; =\; 0\quad , \]
where the constraint of fixed travel 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. Some smoothness restrictions are also often imposed on the trial trajectories. 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 Hamilton principle means that the variation of the action \( \delta S \) vanishes for any small trajectory variation \( \delta q(t) \) around a true trajectory consistent with the given constraints.
The second major version of the action is Maupertuis' action \(W\ ,\) where \[\tag{3} W\; =\; \int _{q_{A} }^{q_{B} }pdq\; =\; \int _{0}^{T}2\, K\, d t\quad , \]
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 principle states that for true trajectories \(W\) is stationary on trial trajectories with fixed end positions \(q_A\) and \(q_B\) and fixed energy \(E = K+V\ .\) Following our earlier conventions, we write this principle as \[\tag{4} \left(\delta W\right)_{E} \; =\; 0\quad . \]
Note that \(E\) is fixed but \(T\) is not in Maupertuis' principle (4), the reverse of the conditions in Hamilton's principle (2).
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 (in multidimensions) true orbits, i.e. spatial shape of the true paths. In the latter case, in two and more dimensions, the action W can be rewritten as an integral involving the arc length along the orbit (Jacobi's form), and the problem then resembles a geodesic or reciprocal isoperimetric problem (Lanczos 1970). In all cases the solutions can be obtained directly from the variational principles (see Section 8), or from the solution of the corresponding Euler-Lagrange differential equations (see Section 3) which are equivalent to the variational principles.
Hamilton's principle is applicable to both conservative and nonconservative systems, with the Lagrangian \(L\, \left(q\; ,\; \dot{q}\; ,\; t\right)\) explicitly time-dependent in the latter case (e.g. due to a time-dependent potential V(q,t)), whereas the form (4) of Maupertuis' principle is restricted to conservative systems (it can be generalized – see Gray et al 2004). For conservative systems the two principles (2) and (4) are related by a Legendre transformation, as discussed in Section 6.
An appealing feature of the action principles is their brevity and elegance in expressing the laws of motion. They are valid for any choice of coordinates (i.e., they are covariant), and readily yield conservation laws from symmetries of the system (Section 12). They generate covariant equations of motion (Section 3), but they also supply an alternative and direct route to finding true trajectories which bypasses equations of motion; this route can be implemented analytically as an approximation scheme (Section 8), or numerically to give essentially exact trajectories (Beck et al 1989, Basile and Gray 1992, Marsden and West 2001). Action principles transcend classical particle and rigid body mechanics and extend naturally to other branches of physics such as continuum mechanics (Section 11), relativistic mechanics (Section 9), quantum mechanics (Section 10), and field theory (Section 11), and thus play a unifying role. They have occasionally assisted in developing new physics (see comment at the end of Section 9). The Hamilton and Maupertuis principles are not applicable, however, if the system is nonholonomic, and usually not if the system is dissipative (Section 4).
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 1996a, 2004 for extensive discussions and references). In Section 7 we discuss several of the more recent generalizations.
Euler-Lagrange Equations
Using standard calculus of variations techniques one can carry out the first-order variation of the action, set the result to zero as in (2) or (4), and thereby derive differential equations for the true trajectory, called the Euler-Lagrange equations, which are equivalent to the variational principles. For Hamilton's principle, the corresponding Euler-Lagrange equation of motion (often called simply Lagrange's equation) is (see, e.g., Brizard 2008, Goldstein et al 2002) \[\tag{5} \frac{d}{d t} \; \left(\frac{\partial L}{\partial \dot{q}_{\alpha}} \right)\; -\; \frac{\partial L}{\partial q_{\alpha}} \; =\; 0\quad , \]
where \(\alpha = 1,2,...,f\ .\) As with the action principles, eqs.(5) are covariant (i.e. valid for any choice of the coordinates \( q_\alpha \)), and can be written out explicitly as coupled second-order differential equations for the \( q_\alpha \)'s. For particle systems these equations reduce to the standard Newton equations of motion if one chooses Cartesian coordinates in an inertial frame. 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 yields (Lanczos 1970, Landau and Lifshitz 1969) corresponding differential equations for the true spatial paths (orbits).
As a simple example, consider the Hamilton principle for the one-dimensional harmonic oscillator with coordinate x. The Lagrangian is \(L \; = \; K \;- \; V \; = \; (1/2)m \dot{x}^2 \, - \; (1/2)k x^2 \ ,\) where m is the mass and k is the force constant. The partial derivatives of \( L \) are \(\partial L/ \partial \dot{x} \; = \; m \dot{x} \) and \( \partial L/ \partial x \, = \; -kx \) so that the Euler-Lagrange equation (5) gives \( m \ddot{x} \; + \; kx \; = \; 0 \ ,\) which is Newton's equation of motion for this system. The well known general solution is \( x(t) \; = \; C_1 sin \omega t \; + \; C_2 cos \omega t \ ,\) where \( \omega \; = \; (k/m)^{1/2} \) is the frequency. The constants \( C_1 \) and \( C_2 \) are chosen to satisfy the constraints \( x \; = \; x_A \) at \( t \; = \; 0 \) and \( x \; = \; x_B \) at \( t \; = \; T \)--see next paragraph.
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 (\( q \) and \( \dot{q} \) are specified at one point \( q_A \)), 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 with coordinate x (see preceding paragraph) and specifying x = 0 at t = 0 and at t = T (one period \(2\pi/\omega\)) gives an infinite number of solutions, i.e. \(x(t) = A sin \omega t\) with one solution for each value of the amplitude A, which is arbitrary. The same system with the constraints x = 0 at t = 0 and x = A at t = T/4 has the unique solution \(x(t) = A sin \omega t\ ,\) and for the constraints x = 0 at t = 0 and x = C at t = T/2 no solution exists for nonzero C. In practice, one usually has initial conditions in mind, where the solution is unique, 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. Thus for the harmonic oscillator example with specified initial conditions, \(x=0\) and \( \dot{x}=v_0\) at \(t=0\), we simply choose \(C_1 = v_0/\omega\) and \(C_2=0\) in the general solution given in the paragraph above.
Another system exhibiting multiple solutions under space-time boundary condition constraints is the quartic oscillator, discussed in Sections 5 and 8. In Fig.1 note that two true trajectories (labelled 1 and 0) connect the initial space-time event P at the origin and the final space-time event denoted by a square symbol where the two trajectories intersect.
As an example giving multiple solutions with Maupertuis principle constraints (specified initial and final positions, and specified energy), consider throwing a ball in a uniform gravitational field from a specified position P and with specified energy E (which corresponds to a specific initial speed). Ignore air friction. If we throw the ball twice, in the same vertical plane, with two different angles of elevation of the initial velocity, one with 45 degrees and the other with 75 degrees, but the same initial speed, the two parabolic spatial paths will recross at some point in the plane, call it R. Thus specifying P and R and E does not in general determine a unique true trajectory, as we have found two true trajectories here with the same values of P and R and E.
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 usual action principles to nonholonomic systems have been controversial and ultimately unsuccessful (Papastavridis 2002). Hamilton's principle in its standard form (2) is not valid, but a more general and correct Galerkin-d'Alembert form has been derived. For a holonomic system with n coordinates and c constraints, the number of independent coordinates (degrees of freedom) is f = n-c. Thus for the example of the particle confined to a spherical surface we have n = 3 coordinates, c = 1 constraint, and hence f = 2 independent coordinates. These can be chosen as any two of the particle's three Cartesian coordinates with respect to the center of the sphere, or as latitude and longitude coordinates on the sphere surface, etc. One can implement holonomic constraints as in Sections 1 and 3 by using a Lagrangian L with any set of \(~\) f \(~\) independent coordinates q, or one can treat the n coordinates symmetrically by expressing L as a function of all of them and using the method of Lagrange multipliers (Lanczos 1970, Morse and Feshbach 1953, Fox 1950) to take account of the constraints. In essence, the Lagrange multipliers relax the constraints, with one multiplier for each constraint relaxed.
In the literature (e.g., Dirac 1964) a second type of velocity-dependent constraint, nongeometic and called "kinematic" in Gray et al (2004), has been discussed. The usual action principles are valid for this type of velocity-dependent constraint. As simple examples, for conservative systems one could impose the additional constraint of fixed energy \( K(\dot{q}) \; + \; V(q) \) on the trial trajectories in the Hamilton principle, and the fundamental constraints in the Hamilton and Maupertuis principles involve the velocities. The Dirac-type constraints are implemented by the method of Lagrange multipliers. In Section 7 we use Lagrange multipliers to relax the fundamental constraints of the Hamilton and Maupertuis principles.
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 (see Gray et al 2004 for a brief review, and Chandrasekhar et al 2007 for more recent developments).
More generally, the question of whether a Lagrangian and corresponding action principle exist for a particular dynamical system, given the equations 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).
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 (at second order the action is larger for some nearby trial trajectories and smaller for others, compared to the true trajectory action). Action is never a local maximum. (In relativistic mechanics (see Section 9) two sign conventions for the action have been employed, and whether the action is never a maximum or never a minimum depends on which convention is used. In our convention it is never a minimum.) We discuss here 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. Establishing the existence of a kinetic focus using this criterion is discussed by Fox (1950). An equivalent and more intuitive definition of a kinetic focus can be given. As an example, consider a family of true trajectories \( x(t,v_0) \) for the quartic oscillator with \(V(x) = (1/4) Cx^4\ ,\) all starting at \(P (x = 0 \) at \( t = 0)\ ,\) and with various initial velocities \(v_0 > 0\ .\) Three trajectories of the family, denoted \(0\ ,\) \(1\ ,\) and \( 2\ ,\) are shown in Figure 1. These true trajectories intersect each other – note the open squares in Figure 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 derived (Gray and Taylor 2007), i.e., \(\partial x(t,v_0)/ \partial v_0 = 0\ ,\) and for a quartic oscillator trajectory starting at P(0,0) the kinetic focus Q occurs at time \( t_Q \) given approximately by \(t_Q = 0.646(T/2)\ ,\) where T is the period, as shown in Fig.1 for trajectory 0. This is the first kinetic focus, usually called simply the kinetic focus. Subsequent kinetic foci may exist but we will not be concerned with them.
The other trajectories shown in Figure 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 Figure 1.
Thus, for trajectory \(0\) in Figure 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.
By an argument due originally to Jacobi, it is easy to see intuitively that action S can never be a local maximum (Morin 2008, Gray and Taylor 2007). Note that for any true trajectory the action S in (1) can be increased by considering a varied trajectory with wiggles added somewhere in the middle. The wiggles are to be of very high frequency and very small amplitude so that there is increased kinetic energy K compared to the original trajectory but only a small change in potential energy V. The Lagrangian L = K - V in the region of the wiggles is then larger for the varied trajectory and so is the integral S. Thus S cannot be a maximum for the original true trajectory. A similar intuitive argument due originally to Routh shows that action W also cannot be a local maximum for true trajectories (Gray and Taylor 2007).
For the purpose of determining the true trajectories, the nature of the stationary action (minimum or saddle point) is usually not of interest. However, there are situations where this is of interest, such as investigating whether a trajectory is stable or unstable (Papastavridis 1986), and in semiclassical mechanics where the phase of the propagator depends on the true classical trajectory action and its stationary nature; the latter dependence is expressed in terms of the number of kinetic foci occurring between the end-points of the true trajectory (Schulman 1981). In general relativity kinetic foci play a key role in establishing the Hawking-Penrose singularity theorems for the gravitational field (Wald 1984). Kinetic foci are also of importance in electron and particle beam optics. Finally, in seeking stationary action trajectories numerically (Basile and Gray 1992, Beck et al 1989, Marsden and West 2001), it is useful to know whether one is seeking a minimum or a saddle point, since the choice of algorithm often depends on the nature of the stationary point. If a minimum is being sought, comparison of the action at successive stages of the calculation gives an indication of the error in the trajectory at a given stage since the action should approach the minimum value monotonically from above as the trajectory is refined. The error sensitivity is, unfortunately, not particularly good, as, due the the stationarity of the action, the error in the action is of second order in the error of the trajectory. Thus a relatively large error in the trajectory can produce a small error in the action.
Relation of Hamilton and Maupertuis Principles
For conservative (time-invariant) systems the Hamilton and Maupertuis principles are related by a Legendre transformation (Gray et al 1996a, 2004). Recall first that the Lagrangian \(L \left(q\; ,\; \dot{q}\right)\) and Hamiltonian \(H(q, p)\) are so-related, i.e. \[\tag{6} H \left(q\; ,\; p\right)\; =\; p \dot{q}\; -\; L \left(q\; ,\; \dot{q}\right)\quad , \]
where in general \( p \dot{q} \) stands for \( p_1 \dot{q_1} + p_2 \dot{q_2} + ... \ .\) 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 \[\tag{7} S\; =\; W\; -\; \bar{E}\; T\quad , \]
where \(\bar{E}\; \equiv \; \int _{0}^{T}d t\; H/T \) is the mean energy along the trial trajectory. (Along a true trajectory of a conservative system, 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, 1996a, 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 which determine the true trajectories, 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.
Generalizations
We again restrict the discussion to time-invariant (conservative) systems. If we vary the trial trajectory \(q(t)\) in (7), with no variation in end positions \(q_A\) and \(q_B\) but allowing a variation in end-time T, the corresponding variations \(\delta S\ ,\) \(\delta W\ ,\) \(\delta \bar{E}\) and \(\delta T\) are seen to be related by \[\tag{8} \delta S\; +\; \bar{E}\; \delta \; T\; =\; \delta \; W\; -\; T \; \delta \; \bar{E} \; \; . \]
Next one can show (Gray et al 1996a) 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 for conservative systems, which is called the unconstrained Hamiltonian principle. This can be written in the standard form for a variational relation with a relaxed constraint\[\delta S = \lambda \delta T\ ,\] where \(\lambda\) is a constant Lagrange multiplier, here determined as \(\lambda = -E\) (negative of energy of the true trajectory). If we constrain \(T\) 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 in the standard form of a variational principle with a relaxed constraint, i.e. \(\delta W = \lambda \delta \bar{E}\) where \(\lambda = T\) (duration of true trajectory) is a constant Lagrange multiplier. If we constrain \(\bar{E}\) to be fixed for the trial trajectories, we get (\(\delta W)_\bar{E} = 0\ ,\) 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 various 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-positions \(q_A\) and \(q_B\) fixed. It is possible to derive additional generalized principles (Gray et al 2004) which allow variations in the end-positions. A word on notation may be appropriate in this regard: the quantities \( \delta S \ ,\) \( \delta W \ ,\) \( \delta T \) and \( \delta \bar{E} \) denote unambiguously the differences in the values of S etc between the original and varied trajectories, and \( q(t) \) and \( q(t) + \delta q(t) \) denote the original and varied trajectory positions at time t. In considering a generalized principle involving a trajectory variation which includes an end-position variation of, say, \( q_B \ ,\) one needs a more elaborate notation (Whittaker 1937, Papastavridis 2002) in order to distinguish between the variation in position at the end-time \( t_B \) of the original trajectory, i.e. \( \delta q_B \equiv \delta q(t = t_B = T) \ ,\) and the total variation in end-position \( \Delta q_B \) which includes the contribution due to the end-time variation \( \delta t_B \equiv \delta T \) if it is nonzero, i.e. \( \Delta q_B = \delta q_B + \dot{q}_B \delta T \ .\) Since we consider only variational principles with fixed end-positions in this review (i.e.\( \Delta q_B = 0 \)), we do not need to pursue this issue here.
As we shall see in the next section and in Section 10, the alternative formulations of the action principles we have considered, 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.
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 example and refer the reader elsewhere for other pedagogical examples and more complicated examples dealing with research problems (Gray et al 1996a, 1996b, 2004). Consider a one-dimensional quartic oscillator, with Hamiltonian \[\tag{9} H\; =\; \frac{p^{2} }{2 m} \; +\; \frac{1}{4} \; C\; x^{4} \quad . \]
Unlike a harmonic oscillator, the frequency \(\omega\) will depend on the amplitude or energy of motion, as is evident in Fig.1. We wish to estimate this dependence. We consider a one-cycle trajectory and for simplicity we choose x = 0 at t = 0 and at t = T (the period \( 2 \pi / \omega \)). As a trial trajectory we take \[\tag{10} x(t) = A \sin \omega t\ ,\]
where the amplitude A is regarded as known and where we treat \(\omega\) as a variational parameter; we will vary \(\omega\) such that an action principle is satisfied. For illustration, 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
\[\tag{11} \bar{E}\; =\; \frac{\omega }{4 \pi} \; W\; +\; C\; \frac{3\; W^{2} }{32 \pi ^{2} m^2 \omega ^{2} } \quad , \]
\[\tag{12} W\; =\; \pi \; \omega \; m\; A^{2} \quad . \]
Treating \(\omega\) as a variational parameter in (11) and applying \(\left(\partial \bar{E}/\partial \omega \right)_{W} \; =\; 0\) gives \[\tag{13} \omega \; =\; \left(\frac{3\; C\; W}{4\; \pi \; m^{2} } \right)^{1/3} \quad . \]
Substituting (13) in (11) gives for \(\bar{E}\) \[\tag{14} \bar{E}\; =\; \frac{1}{2} \; \left(\frac{C}{m^{2} } \right)^{1/3} \left(\frac{3\; W}{4 \pi } \right)^{4/3} \quad . \]
Eq. (13) can be combined with (12) or (14) to give \[\tag{15} \omega = \; \left(\frac{3\; C\; }{4m } \right)^{1/2} A = \; \left(\frac{2\; C\; \bar{E}}{m^{2} } \right)^{1/4} \quad, \]
i.e. a variational estimate of the frequency as a function of the amplitude or energy. The frequency increases with amplitude, confirming what is seen in Fig.1.
This problem is simple enough that the exact solution can be found in terms of an elliptic integral (Gray et al 1996b), with the result \( \omega_{exact}/ \omega_{approx} = 2^{3/4} \pi \Gamma(3/4)/ \Gamma(1/2) \Gamma(1/4) = 1.0075\ .\) Thus 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 \(x(t)\ .\)
Direct variational methods have been used relatively infrequently in classical mechanics (Gray et al 2004) and in quantum field theory (Polley and Pottinger 1988). These methods are widely used in quantum mechanics (Epstein 1974, Adhikari 1998), classical continuum mechanics (Reddy 2002), and classical field theory (Milton and Schwinger 2006).
Relativistic Systems
The Hamilton and Maupertuis principles, and the generalizations discussed above in Section 7, can be made relativistic and put in either Lorentz covariant or noncovariant forms (Gray et al 2004). As an example of the relativistic Hamilton principle treated covariantly, consider a particle of mass m and charge e in an external electromagnetic field with a four-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 \(\phi(x)\) and \(A_i(x)\) (for \(i = 1, 2, 3\)) are the usual scalar and vector potentials respectively. Here \( x = (x^0, x^1, x^2, x^3) \) denotes a point in space-time. A Lorentz invariant form for the Hamilton action for this system is (Jackson 1999, Landau and Lifshitz 1962, Lanczos 1970) \[\tag{16} S\; =\; m\; \int d s\; +\; e\; \int A_{\alpha } \; d x^{\alpha } \quad . \]
The sign of the Lagrangian and corresponding action can be chosen arbitrarily since the action principle and equations of motion do not depend on this sign; here we choose the sign of Lanczos (1970) in (16), opposite to that of Jackson (1999). An advantage of the choice of sign of Lagrangian L implied by (16), as discussed briefly by Gray et al (2004) and in detail by Brizard (2009) who relates this advantage to the consistent choice of sign of the metric (given just below), is that the standard definitions of the canonical momentum and Hamiltonian can be employed - with the other choice unorthodox minus signs are required in these definitions (Jackson 1999). A disadvantage of our choice of sign is that our L approaches the negative of the standard nonrelativistic Lagrangian in the nonrelativistic limit (Brizard 2009). The four-dimensional path in (16) runs from the initial space-time point \(x_{A} \) to the final space-time point \( x_{B} \ ,\) 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 \(A_\alpha \rightarrow A_\alpha + \partial_{\alpha} f \) (for arbitrary \(f(x)\)), where \( \partial_{\alpha} = \partial / \partial x^{\alpha} \ ,\) 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 \[\tag{17} m\; \frac{d\, v_{\alpha } }{d\, s} \; =\; e\; F_{\alpha \beta } \; v^{\beta } \quad , \]
where \(F_{\alpha \beta} = \partial_\alpha A_\beta - \partial_\beta A_\alpha \) is the electromagnetic field tensor, 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 below, the equations for the field (Maxwell equations) can also be derived from an action principle.
Action principles are important also in general relativity. First 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 test particle in a gravitational field (Taylor and Wheeler 2000). For "long" true trajectories ("long" and "short" trajectories are defined in Section 5) the proper time is a saddle point (Misner et al 1973, Wald 1984, Gray and Poisson 2011). 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).
General relativity is perhaps the best example of a field where new physics was derived heuristically from action principles, since Einstein and Hilbert were both motivated by action principles, at least partly, in establishing the field equations, and the principle of stationary proper time was used to obtain the equation of motion of a test particle in a gravitational field (the Euler-Lagrange equation here is the relativistic geodesic equation). A second example is modern (Yang-Mills type) gauge field theory. Some of the pioneers (e.g., Weyl, Klein, Utiyama) explicitly used action principles to implement their ideas, and others, including Yang and Mills, used them implicitly by working with the Lagrangian (O'Raifeartaigh and Straumann, 2000). Some of the early gauge theories were unified field theories of gravity and electromagnetism interacting with matter, and other early unified field theories developed by Einstein, Hilbert and others were also based on action principles (Vizgin 1994). Modern quantum field theories under development, for gravity alone (Rovelli 2004) or unified theories (Freedman and Van Proeyen 2012, Zwiebach 2009, Weinberg 2000), are usually based on action principles. As for the role of action principles in the creation of quantum mechanics, there was a near miss in the case of wave mechanics, as described in the next section.
Relation to Quantum Variational Principles
We discuss here only the Schrödinger time-independent quantum variational principle; apart from a few remarks at the end of this section, for discussion and references to the various quantum time-dependent principles, we refer to Gray et al (2004), Feynman and Hibbs (1965),Yourgrau and Mandelstam (1968), and Toms (2007). As is well known (e.g. Merzbacher 1998), the time-independent Schrödinger equation \[\tag{18} \hat{H}\; \psi _{n} \; =\; E_{n} \; \psi _{n} \]
for the stationary states \(\psi_n\ ,\) with energies \(E_n\ ,\) is equivalent to the variational principle of stationary mean energy \[\tag{19} \left(\delta \; \frac{\left\langle \psi \; \left|\, \hat{H}\, \right|\; \psi \right\rangle }{\left\langle \psi \; |\; \psi \right\rangle } \right)_{n} \; =\; 0\quad , \]
where \(\hat{H}\) is the Hamiltonian operator corresponding to the classical Hamiltonian \(H(q, p)\), \(\left\langle \psi_1 \vert \psi_2 \right\rangle \) denotes the scalar product of two states, and trial state \(\psi\) in (19) has arbitrary normalization. (The word stationary is used in this section with two different meanings.) Equation (18) is the Euler-Lagrange equation for (19). 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, for which (19) is a minimum mean energy principle, 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 approximately classical dynamics problems (see Section 8 above).
The reader will notice the striking similarity of (19) to one of the classical variational principles discussed above in Section 7, i.e. the reciprocal Maupertuis principle applied to the case of stationary (steady-state) motions: \[\tag{20} \left(\delta \bar{E}\right)_{W} \; =\; 0\quad . \]
Here the time average \( \bar{E} \; \equiv \; \int _{0}^{T}dt \; H/T \) is over a period for periodic motions, and is over an infinite time interval for other stationary motions, i.e., quasiperiodic and chaotic. The classical mean energy \(\bar{E}\; \equiv \; \int _{0}^{T}d t\; H/T \) in (20) is clearly analogous to the quantum mean energy \(\left\langle \psi \; \left|\, \hat{H}\, \right|\; \psi \right\rangle /\left\langle \psi \; |\; \psi \right\rangle \) in (19). 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 1996a). 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, in his second paper (Schrödinger 1926b), abandoned this route to quantum mechanics. In this second paper 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. Thus, from \(W_n=nh\) and eq.(14), for a quartic oscillator we find the semiclassical estimate \(E_n=(1/2)(C/m^2)^{1/3} (3nh/4\pi)^{4/3}\), with \(n=1, 2, ...~\) .
In classical mechanics per se there is no particular physical reason for the existence of a principle of stationary action. However, as first discussed by Dirac and Feynman, Hamilton's principle can be derived in the classical limit of the path integral formulation of quantum mechanics (Feynman and Hibbs 1965, Schulman 1981). In the limit \( h \to 0 \) the phase factors \( exp(i 2 \pi S /h) \) contributed by all the virtual paths to the quantum propagator between two given space-time events A and B cancel by destructive interference, with the exception of the contribution of the stationary phase path satisfying \( \delta S = 0 \ ;\) the latter is the classical path. Thus, in the classical limit, the classical Hamilton principle of stationary action is a consequence of the quantum stationary phase condition for constructive interference.
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, Melia 2001, 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).
As our first example, we consider the classical nonrelativistic one-dimensional vibrating string with fixed ends, following Brizard (2008). Assuming small displacements from equilibrium, we find the equation of motion for the transverse displacement \(\phi(x, t)\) is \[\tag{21} \rho \; \frac{\partial ^{2} \; \phi }{\partial \; t^{2} } \; -\; \tau \; \frac{\partial ^{2} \; \phi }{\partial \; x^{2} } \; =\; 0\quad , \]
where \(\rho\) is the density and \(\tau\) the tension. Eq. (21) is the well known classical linear wave equation. It is assumed that \(\phi (x, t)\) is zero at all times at the two ends, \(x = 0 \) and \( x = X\ ,\) and that \(\phi (x, t)\) is given for all positions 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 \[\tag{22} S\; =\; \int _{0}^{T}d t\; \int _{0}^{X}d x\; \mathcal{L}\; \left(\phi \; ,\; \partial _{t} \phi \; ,\; \partial _{x} \phi \right)\quad , \]
with \[\tag{23} \mathcal{L}\; \left(\phi \; ,\; \partial _{t} \phi \; ,\; \partial _{x} \phi \right) = \frac{1}{2} \; \rho \left(\frac{\partial \phi }{\partial t} \right)^{2} \; -\; \frac{1}{2} \; \tau \left(\frac{\partial \phi }{\partial x} \right)^{2} \]
the Lagrangian density \(\left(\int_0^{X} dx\; \mathcal{L}\; =\; L \right) , ~ \partial _{t} \phi = \partial \phi / \partial t ~~\text{and}~~\partial _{x} \phi = \partial \phi / \partial x. \) 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 natural generalization of (5), \[\tag{24} \frac{\partial }{\partial t} \; \left(\frac{\partial \mathcal{L}} {\partial \left(\partial _{t} \phi \right)} \right)\; +\; \frac{\partial }{\partial x} \; \left(\frac{\partial \mathcal{L}}{\partial \left(\partial _{x} \phi \right)} \right)\; -\; \frac{\partial \mathcal{L}}{\partial \phi } \; =\; 0\quad , \]
which also gives (21) for the Lagrangian density (23).
As a second example we consider the classical relativistic description of a source-free electromagnetic field \( F_{\alpha \beta}(x) \) enclosed in a volume V, where \( x \) denotes a space-time point and we use covariant notation (see Section 9 above). Because of the structure of the two Maxwell equations which never have source terms (due to the absence of magnetic monopoles) the field \( F_{\alpha \beta}(x) \) can be represented in terms of the four-potential \( A_{\alpha}(x) \) by \( F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha} \) (Jackson 1999). As in (23) we assume that in general the Lagrangian density \( \mathcal{L}(A_{\alpha},\partial_{\beta} A_{\alpha}) \ ,\) here a Lorentz scalar, depends at most on the potential and its first derivatives so that the Lorentz invariant action S is given by
\[\tag{25}
S \; = \; \int d^4 x \; \mathcal{L}(A_{\alpha},\; \partial_{\beta} A_{\alpha}) \quad,
\]
where the space-time integration is over the spatial volume V and time interval T. Assuming \( A_{\alpha}(x) \) is fixed on the boundary of (V,T) and setting \( \delta S = 0 \) gives the Lorentz covariant Euler-Lagrange equations \[\tag{26} \partial_{\beta} \; \left( \frac{\partial{} \mathcal{L}} {\partial{} \left( \partial _{\beta} A_{\alpha} \right)} \right) \; - \; \frac{\partial \mathcal{L}}{\partial A_{\alpha}} \; = \; 0 \quad , \; \; \alpha \; = \; 0,1,2,3 \quad . \]
For a source-free field the Lagrangian density is given by (Jackson 1999, Melia 2001) \[\tag{27} \mathcal{L}(A_{\alpha}, \; \partial_{\beta} A_{\alpha}) \; = \; \frac{g_{\mu \mu'} g_{\nu \nu'}}{16 \pi} (\partial_{\mu} A_{\nu} \; - \; \partial_{\nu} A_{\mu}) (\partial_{\mu'} A_{\nu'} \; - \; \partial_{\nu'}A_{\mu'}) \quad , \]
where \( g_{\mu \mu'} \) is the Lorentz metric tensor defined earlier. Again we have a choice of sign in (27) and have chosen that of Melia (2001), opposite to that of Jackson (1999). \( \mathcal{L} \) defined by (27) is proportional to \( F_{\mu \nu}F^{\mu \nu} \) and is therefore gauge invariant. From (27) and (26) we find the field equations \[\tag{28} \partial_{\beta} \partial^{\beta} A_{\alpha} \; - \; \partial_{\alpha} \partial^{\beta} A_{\beta} \; = \; 0 \quad , \]
which represent the source-free version of the two Maxwell equations which in general contain source terms. As mentioned above, the other two Maxwell equations are satisfied identically by the representation of the field in terms of the four-potential, i.e. \( F_{\alpha \beta} \; = \; \partial_{\alpha} A_{\beta} \; - \; \partial_{\beta} A_{\alpha} \ .\) Eq.(28) is valid for any choice of gauge; in the Lorenz gauge (\( \partial^{\beta} A_{\beta} \; = \; 0 \)) (28) reduces to the simpler form \( \partial_{\beta} \partial^{\beta} A_{\alpha} \; = \; 0 \ ,\) which is the 3D homogeneous wave equation of type (21). (Note that \( \partial_{\beta} \partial^{\beta} = \partial^2/\partial t^2 - \nabla^2 \ ,\) where \( \nabla^2 \) is the Laplacian.)
So far we have assumed a source-free field and the Lagrangian density \( \mathcal{L}(A_{\alpha}, \; \partial_{\beta} A_{\alpha}) \) given by (27) is actually independent of \( A_{\alpha} \ .\) If a prescribed source four-current density \( J_{\alpha}(x)\;=\; (\rho(x),\; -J_i(x)) \) is present, where \( \rho \) and \( J_i \) are the charge and three-current densities, respectively, one adds to (27) a term (assuming \( c \; = \; 1 \)) \( J^{\mu} A_{\mu} \) (Melia 2001). The Euler-Lagrange equation (26) now gives the inhomogeneous wave equation \( \partial^{\beta} \partial_{\beta} A_{\alpha} \; = \; 4 \pi J_{\alpha} \ ,\) where we have again assumed the Lorenz gauge.
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 (Brizard 2008, Goldstein et al 2002, Melia 2001, Lanczos 1970, Oliver 1994, Schwinger et al 1998). Since Noether's theorem is to be discussed elsewhere in Scholarpedia, we do not go into detail here.
References (historical)
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Internal references
- Ian Gladwell (2008) Boundary value problem. Scholarpedia, 3(1):2853.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
- Jean Zinn-Justin and Riccardo Guida (2008) Gauge invariance. Scholarpedia, 3(12):8287.
- James Meiss (2007) Hamiltonian systems. Scholarpedia, 2(8):1943.
- Lawrence F. Shampine and Skip Thompson (2007) Initial value problems. Scholarpedia, 2(3):2861.
- Graham W Griffiths and William E. Schiesser (2009) Linear and nonlinear waves. Scholarpedia, 4(7):4308.
- Kendall E. Atkinson (2007) Numerical analysis. Scholarpedia, 2(8):3163.
- Jean Zinn-Justin (2009) Path integral. Scholarpedia, 4(2):8674.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
Further reading
- Doughty, N. A. (1990). Lagrangian Interaction, Addison-Wesley, Reading.
- Feynman, R.P., R.B. Leighton and M. Sands (1963). The Feynman Lectures on Physics, Vol.II, Ch.19, Addison-Wesley, Reading.
- Gerjouy, E., A. R. P. Rau and L. Spruch (1983). "A Unified Formulation of the Construction of Variational Principles", Rev. Mod. Phys. 55, 725-774.
- Greiner, W. and J. Reinhardt (1996). Field Quantization, Springer, Berlin.
- Henneaux, M. and C. T. Teitelboim (1992). Quantization of Gauge Systems, Princeton U.P., Princeton.
- Hildebrandt, S. and A. Tromba (1996). The Parsimonious Universe, Springer, New York.
- Moiseiwitsch, B.L. (1966). Variational Principles, Interscience, New York.
- Nesbet, R.K. (2003). Variational Principles and Methods in Theoretical Physics and Chemistry, Cambridge U.P., Cambridge.
- Tabarrok, B. and F. P. J. Rimrott (1994). Variational Methods and Complementary Formulations in Dynamics, Kluwer, Dordrecht.
See Also
Dynamical systems, Gauge invariance, Hamilton-Jacobi equation, Hamiltonian systems, Path integral, Quasiperiodicity, Chaos, Lagrangian mechanics, General relativity, Noether's Theorem.
