10

Variational equations of motion

We encountered variational forms of equations of motion in prior chapters, for

example, when solving the brachistocrone problem in Section 1.4.2. Several

dynamic equations of motion will be derived from variational principles in this

chapter.

10.1 Legendre’s dual transformation

This transformation invented by Legendre is of fundamental importance. Let

us consider the function of n variables

f = f (u

1

,u

2

, ..., u

n

).

Legendre proposed to introduce a new set of variables by the transformation of

v

i

=

∂f

∂u

i

,i=1, 2, ..., n.

The Hessian matrix of this transformation is

H(f)=

⎡

⎢

⎢

⎢

⎢

⎣

∂

2

f

∂u

2

1

∂

2

f

∂u

1

∂u

2

...

∂

2

f

∂u

1

∂u

n

∂

2

f

∂u

2

∂u

1

∂

2

f

∂u

2

2

...

∂

2

f

∂u

2

∂u

n

... ... ... ...

∂

2

f

∂u

n

∂u

1

∂

2

f

∂u

n

∂u

2

...

∂

2

f

∂u

2

n

⎤

⎥

⎥

⎥

⎥

⎦

.

If the determinant of this matrix, sometimes called the Hessian, is not zero,

then the variables of the new set are independent. This means that they could

also be obtained as functions of the original variables.

We deﬁne a new function in terms of the new variables

g = g(v

1

,v

2

, ..., v

n

).

The two functions are related as

g =Σ

n

i=1

u

i

v

i

− f.

143

144 Applied calculus of variations for engineers

The notable consequence is the spectacular duality between the two sets. The

original variables can now be expressed as

u

i

=

∂g

∂v

i

,i=1, 2, ..., n.

and the original function regained as

f =Σ

n

i=1

u

i

v

i

− g.

Legendre’s transformation is completely symmetric.

Let us now look at a function of two sets of variables:

f = f (u

1

,u

2

, ..., u

n

,w

1

,w

2

, ..., w

n

).

If the variables of the second set are independent of the ﬁrst, they are consid-

ered to be parameters and the transformation will retain them as such:

g = g(v

1

,v

2

, ..., v

n

,w

1

,w

2

, ..., w

n

).

The relationship between the two functions regarding the parameters is

∂f

∂w

i

= −

∂g

∂w

i

,i=1, 2, ..., n.

This transformation will be instrumental when applied to the functions intro-

duced in the next sections.

10.2 Hamilton’s principle for mechanical systems

Hamilton’s principle was brieﬂy mentioned earlier in Section 1.4.2 in connec-

tion with the problem of a particle moving under the inﬂuence of a gravity

ﬁeld. The principle, however, is much more general and it is applicable to

complex mechanical systems. For conservative (energy preserving) systems,

Hamilton’s principle states that the motion between two points is deﬁned by

the variational problem of

t

1

t

0

(E

k

− E

p

)dt = extremum,

where E

k

and E

p

are the kinetic and potential energy, respectively. Introduc-

ing the Lagrange function

L = E

k

− E

p

,

Variational equations of motion 145

the principle may also be stated as

t

1

t

0

Ldt = extremum,

where the extremum is not always zero. The advantageous feature of Hamil-

ton’s principle is that it is stated in terms of energies, which are independent

of the selection of coordinate systems. Hamilton’s principle is of fundamental

importance because many of the general physical laws may be derived from

it as we will see in the next sections.

10.2.1 Newton’s law of motion

We consider the simplest mechanical system of a mass particle, but since

any complex mechanical system may be considered a collection of many mass

particles, the following is valid for those as well. Let the mass of the particle

be m and its position deﬁned at a certain time t by the coordinates:

q

i

(t),i=1, 2, 3,

where q

1

(t)=x(t),q

2

(t)=y(t),q

3

(t)=z(t).

The kinetic energy of the particle is then

E

k

=Σ

3

i=1

1

2

m ˙q

2

i

.

The mass particle is moving from its position at time t

0

to a position at

time t

1

. We assume that there is a force F acting on the particle to result in

this motion and the mechanical system is conservative, hence there exists a

force potential such that

F

i

= −

∂E

p

∂q

i

,i=1, 2, 3.

Here F

i

are the components of the force in the coordinate directions. Hamil-

ton’s principle dictates that

t

1

t

0

Ldt =

t

1

t

0

(E

k

− E

p

)dt = extremum.

Substituting the kinetic energy results in

t

1

t

0

(Σ

3

i=1

1

2

m ˙q

2

i

− E

p

)dt = extremum.

Applying the Euler-Lagrange equation

∂L

∂q

i

−

d

dt

∂L

∂ ˙q

i

=0

146 Applied calculus of variations for engineers

for this variational problem yields

∂E

p

∂q

i

+

d

dt

(m ˙q

i

),i=1, 2, 3.

The ﬁrst term yields the acting force components, the diﬀerentiation produces

the acceleration, and the formula becomes

m¨q

i

= F

i

,i=1, 2, 3.

This is Newton’s second law of motion, better known in the form of

F = ma.

10.3 Lagrange’s equations of motion

We now consider a mechanical system of n mass points with distinct masses

m

j

,j =1, ..., n and generalize the coordinates as

q

1

= x

1

,q

2

= y

1

,q

3

= z

1

; q

4

= x

2

,q

5

= y

2

,q

6

= z

2

; ...

and

q

3n−2

= x

n

,q

3n−1

= y

n

,q

3n

= z

n

.

They are gathered into a vector q. The mass particles are

m

1

= m

2

= m

3

= m

1

; m

4

= m

5

= m

6

= m

2

; ...

and

m

3n−2

= m

n

,m

3n−1

= m

n

,m

3n

= m

n

.

Note the distinction between the subscripts and superscripts. To represent

the system with Newton’s law we simply extend the form derived in the last

section to this case as

m

i

¨q

i

= f

i

,i=1, 2, 3, ..., 3n.

Now our variational problem contains multiple functions

I(q

, ˙q, ¨q)=extremum.

The system of Euler-Lagrange equations of this problem is called Lagrange’s

equations of motion. Note the emphasis on the plural. With the Lagrangian

expression

L = E

k

− E

p

,

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