We begin in this section with a quick introduction to some variational methods.

These ideas will later serve as motivation for the Pontryagin Maximum Principle.

Assume we are given a smooth function L : Rⁿ × Rⁿ → R, L = L(x, v); L is called the Lagrangian. Let T > 0, x⁰, x¹ ∈ Rⁿ be given.

BASIC PROBLEM OF THE CALCULUS OF VARIATIONS. Find a curve x^∗(.) : [0, T] → Rⁿ that minimizes the functional

(1.1)

among all functions x(.) satisfying x(0) = x0 and x(T) = x¹.

Now assume x^∗(.) solves our variational problem. The fundamental question is this: how can we characterize x^∗(.)?

1.1 DERIVATION OF EULER–LAGRANGE EQUATIONS.

NOTATION.We write L = L(x, v), and regard the variable x as denoting position,  the variable v as denoting velocity. The partial derivatives of L are

THEOREM 1.1 (EULER–LAGRANGE EQUATIONS). Let x^∗(.) solve the calculus of variations problem. Then x^∗(.) solves the Euler–Lagrange differential equations:

The significance of preceding theorem is that if we can solve the Euler–Lagrange equations (E-L), then the solution of our original calculus of variations problem  (assuming it exists) will be among the solutions.

Note that (E-L) is a quasilinear system of n second–order ODE. The i^th component of the system reads

Proof. 1. Select any smooth curve y[0, T] → Rn, satisfying y(0) = y(T) = 0.

Define

i(τ ) := I[x(.) + τy(.)]

for τ ∈ R and x(.) = x^∗(.). (To simplify we omit the superscript ∗.) Notice that x(.) + τy(.) takes on the proper values at the endpoints. Hence, since x(.) is

minimizer, we have

                            i(τ ) ≥ I[x(.)] = i(0).

Consequently i(.) has a minimum at τ = 0, and so i′(0) = 0.

2. We must compute i′ (τ ). Note first that

This equality holds for all choices of y : [0, T] → Rⁿ, with y(0) = y(T) = 0.

3. Fix any 1 ≤ j ≤ n. Choose y(.) so that

where ψ is an arbitary function. Use this choice of y(.) above:

Integrate by parts, recalling that ψ(0) = ψ(T) = 0:

This holds for all ψ : [0, T] → R, ψ(0) = ψ(T) = 0 and therefore

for all times 0 ≤ t ≤ T. To see this, observe that otherwise L_xj – d/dt (L_vj ) would be,  say, positive on some subinterval on I ⊆ [0, T]. Choose ψ ≡ 0 off I, ψ > 0 on I.

Then

a contradiction.

1.2 CONVERSION TO HAMILTON’S EQUATIONS.

DEFINITION. For the given curve x(.), define

                                 p(t) := ∇_vL(x(t), x˙ (t))                            (0 ≤ t ≤ T).

We call p(.) the generalized momentum.

Our intention now is to rewrite the Euler–Lagrange equations as a system of first–order ODE for x(.), p(.).

IMPORTANT HYPOTHESIS: Assume that for all x, p ∈ Rⁿ, we can solve the equation

(1.2)                               p = ∇_vL(x, v)

for v in terms of x and p. That is, we suppose we can solve the identity (1.2) for

                                            v = v(x, p).

DEFINITION. Define the dynamical systems Hamiltonian H : Rⁿ × Rⁿ → R by the formula

                         H(x, p) = p . v(x, p) − L(x, v(x, p)),

where v is defined above.

NOTATION. The partial derivatives of H are

and we write

∇_xH := (H_x1 , . . . ,H_xn), ∇_pH := (H_p1 , . . . ,H_pn).

THEOREM 1.2 (HAMILTONIAN DYNAMICS). Let x(.) solve the EulerLagrange equations (E-L) and define p(.)as above. Then the pair (x(.), p(.)) solves Hamilton’s equations:

Furthermore, the mapping t → H(x(t), p(t)) is constant.

Proof. Recall that H(x, p) = p . v(x, p) − L(x, v(x, p)), where v = v(x, p) or,  equivalently, p = ∇_vL(x, v). Then

because p = ∇_vL. Now p(t) = ∇_vL(x(t), x˙ (t)) if and only if x˙ (t) = v(x(t), p(t)).

Therefore (E-L) implies

A PHYSICAL EXAMPLE. We define the Lagrangian

which we interpret as the kinetic energy minus the potential energy V . Then

∇_xL = −∇V (x), ∇_vL = mv.

Therefore the Euler-Lagrange equation is

which is Newton’s law. Furthermore

                       p = ∇_vL(x, v) = mv

is the momentum, and the Hamiltonian is

the sum of the kinetic and potential energies. For this example, Hamilton’s equations read

References

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