In Section 7 we discuss several of the most recent generalizations.
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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. For particle systems these equations reduce to the standard Newton equations of motion if one chooses Cartesian coordinates in an inertial frame.
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. Another system exhibiting multiple solutions under space-time boundary condition constraints is the quartic oscillator, discussed in Sections 5 and 8.
In Fig. Additional true trajectories with higher energies also satisfy the boundary conditions. Ignore air friction. If we throw the ball twice, in the same vertical plane, with two different angles of elevation of the initial velocity, say 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. We see in these examples of multiple solutions the roles of the differing constraints in the Hamilton and Maupertuis principles.
The action principles 2 and 4 are restricted to holonomic systems, i. 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 Hamilton's principle in its standard form 2 is not valid, but a more general and correct Galerkin-d'Alembert form has been derived.
These can be chosen as any two of the particle's three Cartesian coordinates with respect to axes with origin at the center of the sphere, or as latitude and longitude coordinates on the sphere surface, etc. In essence, the Lagrange multipliers relax the constraints, with one multiplier for each constraint relaxed.
In the literature e. The usual action principles are valid for this type of velocity-dependent constraint. 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.
Rigid Body Dynamics and Dynamical Systems
However, for some dissipative systems, including all one-dimensional ones, Lagrangians have been shown to exist, and Hamilton's principle then applies see Gray et al. 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 Additional freedom of choice will also often exist. By putting additional conditions on the Lagrangian we can narrow down the choice.
Thus for the free particle in one dimension, using an inertial frame of reference and by requiring the Lagrangian function to be invariant under Galilean transformations, i. In this review we restrict ourselves to the most common case where the Lagrangian depends on the coordinates and their first derivatives, but when higher derivatives occur in the Lagrangian the Euler-Lagrange equation generalizes in a natural way Fox, As an example, in considering the vibrational motion of elastic continuum systems Section 11 such as beams and plates, the standard Lagrangian contains spatial second derivatives, and the corresponding Euler-Lagrange equation of motion contains spatial fourth derivatives Reddy, As defined in Section 1, action is never a local maximum, as we shall discuss.
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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. Establishing the existence of a kinetic focus using this criterion is discussed by Fox An equivalent and more intuitive definition of a kinetic focus can be given. Based on this definition a simple prescription for finding the kinetic focus can be derived Gray and Taylor , i.
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. 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 , and in semiclassical mechanics where the phase of the propagator Section 10 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 In general relativity kinetic foci play a key role in establishing the Hawking-Penrose singularity theorems for the gravitational field Wald Kinetic foci are also of importance in electron and particle beam optics.
Finally, in seeking stationary action trajectories numerically Basile and Gray , Beck et al. 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 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. For conservative time-invariant systems the Hamilton and Maupertuis principles are related by a Legendre transformation Gray et al. 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 constraint parameters. 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.
There, as established by Gibbs, one introduces two free energies related by a Legendre transformation, i. We again restrict the discussion to time-invariant conservative systems. Next one can show Gray et al. 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.
It is possible to derive additional generalized principles Gray et al. 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. 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 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.
We wish to estimate this dependence. The frequency increases with amplitude, confirming what is seen in Fig. This problem is simple enough that the exact solution can be found in terms of an elliptic integral Gray et al.
Direct variational methods have been used relatively infrequently in classical mechanics Gray et al. These methods are widely used in quantum mechanics Epstein , Adhikari , classical continuum mechanics Reddy , and classical field theory Milton and Schwinger They are also used in mathematics to prove the existence of solutions of differential Euler-Lagrange equations Dacorogna 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.
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 in 16 , opposite to that of Jackson The Hamilton principle is thus gauge invariant. Specific examples, such as an electron in a uniform magnetic field, are discussed in the references Gray et al. 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.
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 stationary proper time, or maximal aging, is also valid in general relativity for the motion of a test particle in a gravitational field Taylor and Wheeler ; for "short" true trajectories the proper time is a maximum, and for "long" true trajectories "long" and "short" trajectories are defined in Section 5 the proper time is a saddle point Misner et al.
The corresponding Euler-Lagrange equation of motion is the relativistic geodesic equation.
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 , Misner et al. General relativity is perhaps the first, and still the best, example of a field where new laws of physics were 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.
A second example is modern Yang-Mills type gauge field theory. Some of the pioneers e. Some of the early gauge theories were unified field theories of gravitational and electromagnetic fields interacting with matter, and other early unified field theories developed by Einstein, Hilbert and others were also based on action principles Vizgin Modern quantum field theories under development, for gravity alone Rovelli or unified theories Freedman and Van Proeyen , Zwiebach , Weinberg , are usually based on action principles. The earliest general quantum field theory Heisenberg and Pauli , essentially the theory used in the s for quantum electrodynamics , strong, and weak interactions Wentzel , and the basis of one of the modern methods Weinberg , derives from action principles; commutation relations or anticommutation relations for fermion fields are applied to the field components and their conjugate momenta, with the latter being determined from the Hamilton principle and Lagrangian density for the classical fields Section As for the role of action principles in the creation of quantum mechanics in , in the case of wave mechanics, following hints given in de Broglie's Ph.
Heisenberg did not use action principles in creating matrix mechanics, but his close collaborators Born and Jordan immediately showed that the equations of motion in matrix mechanics can be derived from a matrix mechanics version of Hamilton's principle. Later, following a hint from Dirac in , in his Ph.
A very general quantum operator version of Hamilton's principle was devised by Schwinger in Schwinger , Toms It should be mentioned that "treating the fictitious forces like real forces" means, in particular, that fictitious forces as seen in a particular non-inertial frame transform as vectors under coordinate transformations made within that frame, that is, like real forces. Next, it is observed that time varying coordinates are used in both inertial and non-inertial frames of reference, so the use of time varying coordinates should not be confounded with a change of observer, but is only a change of the observer's choice of description.
Elaboration of this point and some citations on the subject follow. The term frame of reference is used often in a very broad sense, but for the present discussion its meaning is restricted to refer to an observer's state of motion , that is, to either an inertial frame of reference or a non-inertial frame of reference. The term coordinate system is used to differentiate between different possible choices for a set of variables to describe motion, choices available to any observer, regardless of their state of motion. Examples are Cartesian coordinates , polar coordinates and more generally curvilinear coordinates.
Here are two quotes relating "state of motion" and "coordinate system":  . We first introduce the notion of reference frame , itself related to the idea of observer : the reference frame is, in some sense, the "Euclidean space carried by the observer". Let us give a more mathematical definition:… the reference frame is In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas.
The first is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods. The second, the frame of reference, refers to an idealized system used to assign such numbers … To avoid unnecessary restrictions, we can divorce this arrangement from metrical notions. This comfortable circumstance ceases immediately once we begin to consider frames of reference in nonuniform motion even within special relativity.
In a general coordinate system, the basis vectors for the coordinates may vary in time at fixed positions, or they may vary with position at fixed times, or both. It may be noted that coordinate systems attached to both inertial frames and non-inertial frames can have basis vectors that vary in time, space or both, for example the description of a trajectory in polar coordinates as seen from an inertial frame. In discussion of a particle moving in a circular orbit,  in an inertial frame of reference one can identify the centripetal and tangential forces.
It then seems to be no problem to switch hats, change perspective, and talk about the fictitious forces commonly called the centrifugal and Euler force. But what underlies this switch in vocabulary is a change of observational frame of reference from the inertial frame where we started, where centripetal and tangential forces make sense, to a rotating frame of reference where the particle appears motionless and fictitious centrifugal and Euler forces have to be brought into play.
That switch is unconscious, but real.
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Suppose we sit on a particle in general planar motion not just a circular orbit. What analysis underlies a switch of hats to introduce fictitious centrifugal and Euler forces? To explore that question, begin in an inertial frame of reference. By using a coordinate system commonly used in planar motion, the so-called local coordinate system ,  as shown in Figure 1 , it becomes easy to identify formulas for the centripetal inward force normal to the trajectory in direction opposite to u n in Figure 1 , and the tangential force parallel to the trajectory in direction u t , as shown next.
To introduce the unit vectors of the local coordinate system shown in Figure 1 , one approach is to begin in Cartesian coordinates in an inertial framework and describe the local coordinates in terms of these Cartesian coordinates. In Figure 1 , the arc length s is the distance the particle has traveled along its path in time t.
The path r t with components x t , y t in Cartesian coordinates is described using arc length s t as: . One way to look at the use of s is to think of the path of the particle as sitting in space, like the trail left by a skywriter , independent of time. Any position on this path is described by stating its distance s from some starting point on the path.