Hassani S. Mathematical Methods: For Students of Physics and Related Fields

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30.2 Lagrangian Dynamics 741

By equating the ﬁrst term of (30.15) to the ﬁrst term of the last equation in

(30.37), we get

∂L

∂x

∂

∂x

(−Φ).

Antidiﬀerentiation yields L = −Φ(x, y, z)+f(y, z, ˙x, ˙y, ˙z), where f is the

“constant” of integration. If the partials of L with respect to y and z are to

be equal to the corresponding partials of −Φ, then f cannot depend on y and

z.So,f is a function of velocity components. If the second term of (30.37) is

to equal the second term of (30.15), then

m ˙x =

∂L

∂ ˙x

∂f

∂ ˙x

or f (˙x, ˙y, ˙z)=

m ˙x

+ g(˙y, ˙z),

where g(˙y, ˙z) is the “constant” of this new integration. Applying the same

argument to y and z, we conclude that f is just the kinetic energy. Therefore,

we arrive at the important conclusion that for a single particle with position

vector r, the extremization of

L(r,

r,t)=−Φ(r)+

m |

= −Φ(x, y, z)+

˙x

+˙y

+˙z

(30.38)

gives the equation of motion of the particle. L(r,

r,t) is called the Lagrangian

of a single particle moving in potential Φ.

For N non-interacting particles in an external potential, the Lagrangian

is the sum of the single-particle Lagrangians:

L =



i=1



i=1



−Φ





i=1

−Φ

˙x

+˙y

+˙z

where Φ

=Φ(x

). Note that this can be written as

L = KE − Φ, where KE =



i=1

˙x

+˙y

+˙z

, and Φ =



i=1

(30.39)

If the particles are interacting, then Φ is no longer the sum of individual

potentials, but a general function of all coordinates. It is therefore common to

collect all the N triple coordinates into one big 3N-component vector q and

call it the generalized coordinates vector. Then the Lagrangian is written

L (q,

q,t)=KE − Φ=



i=1

˙q

− Φ(q

,...,q

). (30.40)

We changed the mass to μ

to avoid confusion with the m

of the previous

equation. For example, for three particles interacting gravitationally,

Φ(q

,...,q

)=Φ(r

, r

)=−

− r

−

− r

−

− r

which can be written in terms of the q’s, once the latter are deﬁned in terms

of the position vectors. Note that many of the μ

’s in (30.40) are equal. For

instance, if q

= x

, q

= y

,andq

= z

,thenμ

= μ

= m

,etc.

742 Calculus of Variations

Figure 30.4: The inclined plane moves as m moves on it.

Example 30.2.1. Ablockofmassm slides on a frictionless inclined plane, which

has mass M and moves on a frictionless horizontal surface as shown in Figure 30.4.

The position of the incline is denoted by X and that of the block by r,or(x, y)with

x = X + r cos θ, y =(l −r)sinθ,

where l is the length of the inclined plane. The kinetic energy of the system is

KE =

˙x

+˙y





X +˙r cos θ



+˙r

sin





+˙r

X ˙r cos θ



and the potential energy

Φ=mgy = mg(l − r)sinθ,

giving rise to the Lagrangian

L =



+˙r

X ˙r cos θ



− mg(l − r)sinθ.

The equations of motion

∂L

∂X

−



∂L

∂



=0,

∂L

∂r

−



∂L

∂ ˙r



=0,

can now be calculated:

−M

X −m



X +¨r cos θ



=0,mgsin θ − m



¨r +

X cos θ



=0.

Solving for the two accelerations, we get

X =

−mg sin θ cos θ

M + m sin

, ¨r =

(m + M )g sin θ

M + m sin

Note that for an inﬁnitely heavy inclined plane,

X =0and¨r = g sin θ,as

expected.



Historical Notes

was born Giuseppe Luigi Lagrangia but adopted the French version of his name. He

was the eldest of eleven children, most of whom did not reach adulthood. His father

destined him for the law—a profession that one of his brothers later pursued—

and Lagrange oﬀered no objections. But having begun the study of physics and

geometry, he quickly became aware of his talents and henceforth devoted himself to

30.2 Lagrangian Dynamics 743

the exact sciences. Attracted ﬁrst by geometry, at the age of seventeen he turned

to analysis, then a rapidly developing ﬁeld.

In 1755, in a letter to the geometer Giulio da Fagnano, Lagrange speaks of

one of Euler’s papers published at Lausanne and Geneva in 1744. The same letter

shows that as early as the end of 1754 Lagrange had found interesting results in

this area, which was to become the calculus of variations (a term coined by Euler

Joseph Louis

Lagrange

1736–1813

in 1766). In the same year, Lagrange sent Euler a summary, written in Latin, of

the purely analytical method that he used for this type of problem. Euler replied

to Lagrange that he was very interested in the technique. Lagrange’s merit was

likewise recognized in Turin; and he was named, by a royal decree, professor at the

Royal Artillery School with an annual salary of 250 crowns—a sum never increased

in all the years he remained in his native country. Many years later, in a letter

to d´Alembert, Lagrange conﬁrmed that this method of maxima and minima was

the ﬁrst fruit of his studies—he was only nineteen when he devised it—and that he

regarded it as his best work in mathematics.

In 1756, in a letter to Euler that has been lost, Lagrange, applying the calculus of

variations to mechanics, generalized Euler’s earlier work on the trajectory described

by a material point subject to the inﬂuence of central forces to an arbitrary system

of bodies, and derived from it a procedure for solving all the problems of dynamics.

In 1757 some young Turin scientists, among them Lagrange, founded a scientiﬁc

society that was the origin of the Royal Academy of Sciences of Turin. One of the

main goals of this society was the publication of a miscellany in French and Latin,

Miscellanea Taurinensia ou M´elanges de Turin, to which Lagrange contributed fun-

damentally. These contributions included works on the calculus of variations, prob-

ability, vibrating strings, and the principle of least action.

To enter a competition for a prize, in 1763 Lagrange sent to the Paris Academy

of Sciences a memoir in which he provided a satisfactory explanation of the trans-

lational motion of the moon. In the meantime, the Marquis Caraccioli, ambassador

from the kingdom of Naples to the court of Turin, was transferred by his government

to London. He took along the young Lagrange, who until then seems never to have

left the immediate vicinity of Turin. Lagrange was warmly received in Paris, where

he had been preceded by his memoir on lunar libration. He may perhaps have been

treated too well in the Paris scientiﬁc community, where austerity was not a leading

virtue. Being of a delicate constitution, Lagrange fell ill and had to interrupt his

trip. In the spring of 1765 Lagrange returned to Turin by way of Geneva.

In the autumn of 1765 d´Alembert, who was on excellent terms with Frederick II

of Prussia, and familiar with Lagrange’s work through M´elanges de Turin, suggested

to Lagrange that he accept the vacant position in Berlin created by Euler’s departure

for St. Petersburg. It seems quite likely that Lagrange would gladly have remained

in Turin had the court of Turin been willing to improve his material and scientiﬁc

situation. On 26 April, d´Alembert transmitted to Lagrange the very precise and

advantageous propositions of the king of Prussia. Lagrange accepted the proposals

of the Prussian king and, not without diﬃculties, obtained his leave through the

intercession of Frederick II with the king of Sardinia. Eleven months after his arrival

in Berlin, Lagrange married his cousin Vittoria Conti who died in 1783 after a long

illness. With the death of Frederick II in August 1786 he also lost his strongest

support in Berlin. Advised of the situation, the princes of Italy zealously competed

in attracting him to their courts. In the meantime the French government decided

to bring Lagrange to Paris through an advantageous oﬀer. Of all the candidates,

Paris was victorious.

744 Calculus of Variations

Lagrange left Berlin on 18 May 1787 to become pensionnaire v´et´eran of the Paris

Academy of Sciences, of which he had been a foreign associate member since 1772.

Warmly welcomed in Paris, he experiencedacertainlassitudeanddidnotimme-

diately resume his research. Yet he astonished those around him by his extensive

knowledge of metaphysics, history, religion, linguistics, medicine, and botany.

In 1792 Lagrange married the daughter of his colleague at the Academy, the

astronomer Pierre Charles Le Monnier. This was a troubled period, about a year

after the ﬂight of the king and his arrest at Varennes. Nevertheless, on 3 June the

royal family signed the marriage contract “as a sign of its agreement to the union.”

Lagrange had no children from this second marriage, which, like the ﬁrst, was a

happy one.

When the academy was suppressed in 1793, many noted scientists, including

Lavoisier, Laplace,andCoulomb were purged from its membership; but Lagrange

remained as its chairman. For the next ten years, Lagrange survived the turmoil of

the aftermath of the French Revolution, but by March of 1813, he became seriously

ill. He died on the morning of 11 April 1813, and three days later his body was

carried to the Panth´eon. The funeral oration was given by Laplace in the name of

the Senate.

30.2.2 Lagrangian Densities

Particles are localized objects (indeed mathematical points), whose trajecto-

ries, determined by ordinary diﬀerential equations, describe curves in space.

A Lagrangian of the form (30.40), with one independent variable (time), is

therefore appropriate for particles.

Most of physical quantities, however, are not particles, but ﬁelds, which

are not localized. In order to apply the variational techniques to ﬁelds, one has

to consider a Lagrangian density L, whose integral over some volume gives

the Lagrangian, which can now be integrated over time as in (30.2). Thus,

in ﬁeld theories, the integration is over the 4-dimensional spacetime, a nat-

ural setting for relativity—which is very relevant because most ﬁeld theories

are relativistic—to operate. A physical ﬁeld usually has several components,

making Equation (30.23) relevant to the situation.

Electrodynamics Lagrangian

Section 17.3.2 derived the electromagnetic ﬁeld tensor F

αβ

and wrote the four

Maxwell’s equations in terms of it. Since F

αβ

seems to be so fundamental,

and the variational techniques seem to yield the (partial) diﬀerential equations

of physics, there may be a chance that electrodynamics can be described by

a Lagrangian density. In the language of tensors, a Lagrangian density is a

scalar. Thus, we have to construct a scalar out of F

αβ

. The simplest such

scalar is F

αβ

. Equation (17.47) showed that the ﬁeld tensor can be written

as derivatives of the 4-potential A

, which is therefore more “fundamental”

than F

αβ

. There is another 4-vector appearing in Maxwell’s equations, namely

the 4-current J

. Thus, by taking the dot product J

,weformanother

30.2 Lagrangian Dynamics 745

scalar. We therefore write

L = aF

αβ

+ bJ

= aη

αμ

βν

μν

αβ

+ bJ

where a and b are to be determined later. Writing the ﬁeld tensor in terms of

the 4-potential, we get

L = aη

αμ

βν

(∂

− ∂

)(∂

− ∂

)+bJ

≡ aη

αμ

βν

ν,μ

− A

μ,ν

)(A

β,α

− A

α,β

)+bJ

. (30.41)

The Euler-Lagrange equation for A

can be written as

∂L

∂A

−

∂

∂x

∂L

∂A

σ,ρ

=0. (30.42)

The ﬁrst term is easy to calculate:

∂L

∂A

= bJ

∂A

= bJ

The second term is only slightly more complicated once we realize that

∂A

β,α

∂A

σ,ρ

= δ

With this in mind, the second term of (30.42) can be shown to be

∂

∂x

∂L

∂A

σ,ρ

=4a∂

(∂

− ∂

)=4a∂

ρσ

and (30.42) becomes

4a∂

ρσ

= bJ

or 4a∂

ρσ

= bJ

This becomes Maxwell’s ﬁrst and fourth equations combined [see Equation

(17.45)] if a =

and b = μ

. Thus the Lagrangian density for electrodynamics

L =

αμ

βν

ν,μ

− A

μ,ν

)(A

β,α

− A

α,β

)+μ

. (30.43)

This, like any other Lagrangian, can be multiplied by a constant without

aﬀecting the Euler-Lagrange equations.

Example 30.2.2.

ChargedParticleinEMField Problem 30.18 shows that

the Lagrangian density (30.43) can be written as

L =

|B|

−|E|

+ μ

(ρΦ − J · A) ,

with the variational problem

L =



L d





dt.

746 Calculus of Variations

Now consider a single particle of charge q interacting with an electromagnetic ﬁeld.

For such a particle,

ρ = qδ(r − r



)andJ = ρv = qvδ(r − r



and L becomes

L =

|B|

−|E|



+ μ

qΦδ(r − r



) − qv · Aδ(r − r



)



|B|

−|E|



+ μ

dt {Φ(r,t) − v ·A(r,t)}.

The particle also has kinetic energy, which needs to be added to this Lagrangian.

When adding Lagrangians, one has to incorporate the freedom in multiplying La-

grangians by constants. In the case at hand, the kinetic energy of the particle should

be added to the negative of the scalar potential energy (recall that L = KE − Φ).

To assure this, we have to divide the entire EM Lagrangian by −1/μ

and add it to

the kinetic energy of the particle. Hence the total Lagrangian becomes

L = −

2μ

|B|

−|E|



m|v|

− qΦ(r,t)+qv · A(r,t)

Notice how the ﬁrst integral is four-dimensional while the second integral is over a

single variable.

We are interested in the motion of the particle. Therefore, the ﬁrst integral

is just a constant (independent of the coordinates and velocity components of the

particle) and can be dropped. Thus, substituting

r for v,wehave

part

− qΦ(r,t)+q

r · A(r,t)

with the LagrangianLagrangian of a

charged particle in

EM ﬁeld

L(r,

r,t)=

− qΦ(r,t)+q

r · A(r,t). (30.44)

Let’s look at the x-component of the motion:

∂L

∂x

−

∂L

∂ ˙x

=0 or − q

∂Φ

∂x

+ q

r ·

∂A

∂x

−

(m ˙x + qA

)=0,

m¨x + q

∂Φ

∂x

+ q



−

r ·

∂A

∂x



=0. (30.45)

Now note that

∂A

∂t

∂A

∂x

˙x +

∂A

∂y

˙y +

∂A

∂z

˙z,

and

r ·

∂A

∂x

=˙x

∂A

∂x

+˙y

∂A

∂x

+˙z

∂A

∂x

Putting these two equations in (30.45) and rearranging, we obtain

m¨x + q



∂Φ

∂x

∂A

∂t





 

=−E

by (15.31)

+q ˙y



∂A

∂y

−

∂A

∂x





 

=−B

by (15.31)

+q ˙z



∂A

∂z

−

∂A

∂x





 

by (15.31)

=0,

30.3 Hamiltonian Dynamics 747

m¨x − qE

− q (˙yB

− ˙zB

)=0. (30.46)

The expression in parentheses is just the x-component of v×B. Thus, (30.46) is the

x-component of the Lorentz force law, governing the motion of a charged particle in

an electromagnetic ﬁeld.



Klein-Gordon Lagrangian

One of the ﬁrst attempts at combining the special theory of relativity with

quantum mechanics was made by Oskar Klein and Walter Gordon. In fact,

Schr

dinger himself started with the relativistic version of his equation, but

abandoned it because of some diﬃculty he encountered when applying it to

hydrogen atom. By the usual substitution

E → i

∂

∂t

, p →−i∇

in the relativistic equation E

− p · p = m

, Klein and Gordon derived

the equation that now bears their names:

∂

∂t

−∇

φ +



φ =0,

which, in units  =1=c, becomes

∂

∂t

−∇

φ + m

φ =0.

This equation can also be obtained from the Lagrangian density

L = η

αβ

(∂

φ)(∂

φ) −

, (30.47)

as the reader can verify.

30.3 Hamiltonian Dynamics

The Lagrangian formulation of mechanics treated in the previous section is a

powerful tool for studying many diﬀerent dynamical systems and ﬁelds. Fur-

thermore, considerations of symmetry, an indispensable technique in the inves-

tigation of fundamental forces, is most adequately handled in the Lagrangian

language. Once the Lagrangian is known, the Euler-Lagrange equations pro-

vide second-order diﬀerential equations to be solved under given boundary (or

initial) conditions.

There is another formulation of mechanics, which instead of second-order

diﬀerential equations, yields twice as many ﬁrst-order DEs. It is called the

Hamiltonian formulation. We describe only the case of several dependent

and one independent variables, the other cases being very similar. Let us

748 Calculus of Variations

assume that our dynamical system has n generalized coordinates {q

}

i=1

and

a Lagrangian L (q,

q,t). In the simplest case (30.40), L = KE−ΦwhereKE

is a quadratic term in velocities alone and Φ dependent on the coordinates

alone. In such a case,

∂L

∂ ˙q

∂KE

∂ ˙q

= μ

˙q

which is the momentum associated with the jth generalized coordinate. It is

therefore natural to generalize the concept of momentum as well, write

≡

∂L(q,

q,t)

∂ ˙q

, (30.48)

and call p

so deﬁned the generalized momentum of the dynamical system.

The transition from Lagrangian to Hamiltonian formulation, from a strictly

mathematical standpoint, is to go from the set of variables (q,

q,t)to(q, p,t).

The procedure for making this transition is the Legendre transformation dis-

cussed in Section 2.2.2. To ﬁnd the variables involved, consider the diﬀerential

of the Lagrangian:

dL =



i=1



∂L

∂q

∂L

∂ ˙q

d ˙q



∂L

∂t

dt,

and use (30.48) and the Euler-Laggrange equation to rewrite the above as

dL =



i=1

(˙p

+ p

d ˙q

∂L

∂t

dt.

If we want to switch the independent variable from ˙q

to p

,thenwehaveto

deﬁne the Hamiltonian as

H (q, p,t)=



i=1

˙q

− L (q,

q,t) . (30.49)

Toverifythis,wenotethat

dH =



i=1

(˙q

+ p

d ˙q

)−dL=



i=1

(˙q

+ p

d ˙q

)−



i=1

(˙p

+ p

d ˙q

)−

∂L

∂t

dt.

Note that p

d ˙q

terms cancel and we are left with

dH =



i=1

(˙q

− ˙p

) −

∂L

∂t

dt.

On the other hand,

dH =



i=1



∂H

∂q

∂H

∂p



∂H

∂t

dt.

30.3 Hamiltonian Dynamics 749

Comparison of the last two equations gives

˙q

∂H

∂p

, ˙p

= −

∂H

∂q

, −

∂L

∂t

∂H

∂t

,i=1, 2,...,n, (30.50)

which are called Hamilton or canonical equations. Note that instead of n

second-order DEs, we now have 2n ﬁrst order DEs.

To discover the physical signiﬁcance of the Hamiltonian, consider the fa-

miliar simple Lagrangian L = KE −Φ, where KE is the usual kinetic energy

term and Φ is the potential energy which is independent of velocities. Then,

(30.48) yields p

= μ

˙q

and Hamiltonian is the

total energy.

H =



i=1

˙q

− L =



i=1

˙q

  

=2KE

−KE +Φ=KE +Φ.

So H is the sum of the kinetic and potential energies, i.e., the total energy.

Example 30.3.1.

Hamiltonian of a Charged Particle in EM Field The

Lagrangian of a charged particle in an electromagnetic ﬁeld is given by (30.44).

Let’s ﬁnd the Hamiltonian of this system. First we need the generalized momentum

(30.48):

∂L

∂ ˙x

= m ˙x

+ qA

or p = m

r + qA. (30.51)

This is an important equation in its own right. It says that the momentum of the

system is not just that of the particle, but that it also includes a contribution from

the EM ﬁeld. In particular, that EM ﬁeld has momentum.

To ﬁnd the Hamiltonian, compute

r from (30.51):

r =

p − qA

and substitute in the deﬁnition of the Hamiltonian (30.49), where, in this case the

sum is just the dot product:

H(r, p,t)=p ·



p − qA



−



p − qA



+ qΦ − q



p − qA



· A

=(p − qA) ·



p − qA



−

|p − qA|

+ qΦ,

H(r, p,t)=

|p − qA(r,t)|

+ qΦ(r,t). (30.52)

Thus, in the presence of an electromagnetic ﬁeld, the Hamiltonian of a particle takes

the same form as the total energy of a particle in a potential qΦ, except that in the

expression for the KE part, p − qA replaces p. Such a replacement is called the

minimal coupling and plays a key role in the quantum mechanical treatment of

charged particles interacting with EM ﬁelds.



This momentum is the source of radiation pressure.

750 Calculus of Variations

30.4 Problems

30.1. Show that, in Example 30.1.6, C

=0,λ = λ

,andC



− a

where λ

is the solution of the equation

λ sin



2λ



= a.

30.2. Find the extremal of the functional

L[x, y]=

π/2

(˙x

+˙y

+2xy) dt

subject to the boundary conditions

x(0) = 0,x(π/2) = 1,y(0) = 0,y(π/2) = 1.

30.3. Find the extremals of the following functionals:

(a) L[x, y]=

(˙x

+˙y

+˙x ˙y) dt, (b) L[x, y]=

(2xy − 2x

+˙x

− ˙y

) dt.

30.4. Find the extremal of a functional of the form

L[x, y]=

L(˙x, ˙y) dt,

given that

∂

∂ ˙x

∂

∂ ˙y

−



∂

∂ ˙x∂ ˙y



=0fora ≤ x ≤ b.

30.5. Find the extremal of the functional

L[x]=

(˙x

+ t

) dt,

subject to the boundary conditions

x(0) = 0,x(1) = 0,

dt =2.

30.6. Show that the extremization of (30.33) leads to the Euler-Lagrange

equations (30.34).

30.7. Among all triangles with a given base line and a ﬁxed perimeter, show

that the isosceles triangle has the largest area.

30.8. An airplane with ﬁxed air speed v

ﬂies for a time T on a closed curve.

The wind velocity u is constant in magnitude and direction and |u| <v

What closed curve encloses the largest area?