Analytical Mechanics
1.10 Complements
1.10.1 Higher-Order Variational Calculus
The variational calculus described in Sect.1.2can be extended to cases were the functiongin (1.1) depends on derivatives of a higher order than the first. Consider for instance the functional
G[w]= b
a
g(w,w,˙ w,¨ ξ) dξ. (1.60) Following the procedure of Sect.1.2and assuming that the derivativeη˙ vanishes ataandb, yields the following differential equation for the extremum functions of (1.60):
− d2 dξ2
∂g
∂w¨ + d dξ
∂g
∂w˙ = ∂g
∂w. (1.61)
4The letter “µ” stands for “molecule”, whereas the letter “γ” in the term “γ-space” stands for “gas”.
20 1 Analytical Mechanics
1.10.2 Lagrangian Invariance and Gauge Invariance
It is shown in Sect.1.2that the extremum functionswi(ξ) are invariant under addition togof the total derivative of an arbitrary functionhthat depends onwandξ only (refer to Eq. (1.8)). Then, it is mentioned in Sect.1.3.2that theEandBfields are invariant under the gauge transformation (1.19), whereh(r,t) is an arbitrary function.
These two properties have in fact the same origin, namely, the description based upon a Lagrangian function. In fact, as shown in Sect. 4.2, a Lagrangian description is possible also in the case of a system having a continuous distribution of the degrees of freedom like, for instance, the electromagnetic field.
1.10.3 Variational Calculus with Constraints
In several problems it is required that the functionw, introduced in Sect.1.2as the extremum function of functional (1.1), be able to fulfill one or more constraints. By way of example consider the constraint
G0 = b
a
g0(w,w,˙ ξ) dξ, (1.62)
where the function g0 and the number G0 are prescribed. A typical case where (1.62) occurs is that of finding the maximum area bounded by a perimeter of given length (Dido’s problem). For this reason, extremum problems having a constraint like (1.62) are calledisoperimetriceven when they have no relation with geometry [115, Par. 4-1].
To tackle the problem one extends the definition of the variation ofwby letting δw=α1η1+α2η2, whereη1(ξ),η2(ξ) are arbitrary functions that are differentiable in the interior of [a,b] and fulfill the conditionsη1(a)=η1(b)=0,η2(a)=η2(b)=0.
Ifwis an extremum function ofGthat fulfills (1.62), replacingwwithw+δw transforms (1.1,1.62) to a pair of functions of theα1,α2parameters, namely,
G=G(α1,α2), G0(α1,α2)=G0(0, 0)=const. (1.63) The first of (1.63) has an extremum atα1 =α2=0, while the second one establishes a relation betweenα1andα2. The problem is thus reduced to that of calculating a constrained extremum, and is solved by the method of theLagrange multipliers.
For this, one considers the functionGλ =G(α1,α2)+λ G0(α1,α2), withλan indeterminate parameter, and calculates the free extremum ofGλby letting
∂Gλ
∂α1
0
=0,
∂Gλ
∂α2
0
=0, (1.64)
where index 0 stands forα1 =α2 = 0. The rest of the calculation is the same as in Sect.1.2; the two relations (1.64) turn out to be equivalent to each other and
provide the same Euler equation. More specifically, from the definition ofGandG0 as integrals ofgandg0one finds that the Euler equation of this case is obtained from that of Sect.1.2by replacinggwithgλ=g+λg0:
d dξ
∂gλ
∂w˙ = ∂gλ
∂w. (1.65)
As (1.65) is a second-order equation, its solutionwcontains two integration con- stants. Theλmultiplier is an additional indeterminate constant. The three constants are found from the constraint (1.62) and from the two relations provided by the boundary or initial conditions ofw.
1.10.4 An Interesting Example of Extremum Equation
Consider the Hamilton–Jacobi Eq. (1.51) for a single particle of massm. Using the Cartesian coordinates and a Hamiltonian function of the form
H= p2
2m+V(x1,x2,x3,t), p2=p21+p22+p23, (1.66) the Hamilton–Jacobi equation reads
∂S
∂t +|gradS|2
2m +V(x1,x2,x3,t)=0, pi = ∂S
∂qi. (1.67) IfV is independent of time, thenH = E and the separation S = W −E t (Sect. 2.4) yields∂S/∂t= −E, gradS=gradW=p. It follows
|gradW|2
2m +V(x1,x2,x3)=E. (1.68) Both Hamilton’s principal (S) and characteristic (W) functions have the dimensions of an action and are defined apart from an additive constant. Also, the form of|gradW| is uniquely defined by that ofV−E. In turn,Eis prescribed by the initial conditions of the particle’s motion.
Consider now the case whereE≥V within a closed domainwhose boundary is∂. As gradW is real, the motion of the particle is confined within, and gradW vanishes at the boundary∂. The Hamilton–Jacobi equation forW(1.68) is recast in a different form by introducing an auxiliary functionwsuch that
w=w0 exp (W/μ), (1.69)
withμa constant having the dimensions of an action. The other constantw0is used for prescribing the dimensions ofw. Apart from this, the choice ofw0is arbitrary due to the arbitrariness of the additive constant ofW. Taking the gradient of (1.69) yieldsμgradw =wgradW, withw =0 due to the definition. As gradW vanishes
22 1 Analytical Mechanics at the boundary, gradwvanishes there as well. As a consequence,wis constant over the boundary. Inserting (1.69) into (1.68) yields
μ2 2m
|gradw|2
w2 +V(x1,x2,x3)=E, (1.70) which determines|gradw/w| as a function ofV −E. Rearranging the above and observing that div(wgradw)=w∇2w+ |gradw|2(Sect. A.1) provides
μ2 2m
div(wgradw)−w∇2w
+(V −E)w2=0. (1.71) Integrating (1.71) overand remembering that gradwvanishes at the boundary,
w
−μ2
2m∇2w+(V −E)w
d=0. (1.72)
The term in brackets of (1.72) does not necessarily vanish. In fact, the form ofwis such that only the integral as a whole vanishes. On the other hand, by imposing that the term in brackets vanishes, and replacingμwith the reduced Planck constanth,¯ yields
− ¯h2
2m∇2w+(V −E)w=0, (1.73)
namely, the Schrödinger equation independent of time (7.45). This result shows that the Schrödinger equation derives from a stronger constraint than that prescribed by the Hamilton–Jacobi equation.
An immediate consequence of replacing the integral relation (1.72) with the dif- ferential Eq. (1.73) is that the domain ofwis not limited any more by the condition E≥V, but may extend to infinity.
Another consequence is that, if the boundary conditions are such thatwvanishes over the boundary (which, as said above, may also be placed at infinity), then (1.73) is solvable only for specific values of E, that form its spectrum of eigenvalues.
Moreover it can be demonstrated, basing on the form of the Schrodinger equation, that the conditionE≥Vminmust be fulfilled (Sect. 8.2.3).
It is interesting to note another relation between the Schrödinger and the Hamilton–Jacobi equations. For the sake of simplicity one takes the one-dimensional case of the Hamilton–Jacobi equation expressed in terms ofw(1.70):
μ2
2m(w′)2+V(x)w2 =Ew2, (1.74) where the prime indicates the derivative with respect tox. The left hand side of the equation may be considered the generating functiong=g(w,w′,x) of a functional G, defined over an interval of thexaxis that may extend to infinity:
G[w]= b
a
μ2
2m(w′)2+V w2
dx. (1.75)
One then seeks the extremum functionwofGthat fulfills the constraint G0[w]=
b a
w2dx =1. (1.76)
The problem is solved by the method of Sect.1.10.3, namely, by lettingg0 =w2, gE=g−E g0, and applying the Euler equation togE:
d dx
∂gE
∂w′ = d dx
μ2
m w′=μ2
m w′′, ∂gE
∂w =2 (V −E)w, (1.77)
showing that the Schrödinger equation is actually the Euler equation of the functional Gsubjected to the constraintG0, with the eigenvalueEprovided by the Lagrange multiplier. This result holds also in the higher-dimensional cases, and is in fact the method originally used by Schrödinger to determine the time-independent equation [94, Eqs. (23, 24)].
1.10.5 Constant-Energy Surfaces
Consider theγ-space for a system havingndegrees of freedom (Sect. 1.9). If the system is conservative, the relationH(q1,. . .,qn, p1,. . .,pn) = E introduces a constraint among the canonical coordinates. Due to this, at each instant of time the latter must belong to the (2n−1)-dimensional surfaceH =Eof the phase space, that is calledconstant-energy surface. AsEis prescribed by the initial conditions, the phase point of a conservative system always belongs to the same constant-energy surface.
For a system having one degree of freedom the relation describing the constant- energy surface reduces toH(q,p)=E, that describes a curve in theq pplane. The corresponding state trajectory is a curve of the three-dimensionalq p tspace.