Calculus of Variations: Piecewise-Smooth Extremals and the Weierstrass-Erdmann Corner Conditions

Calculus of Variations: Piecewise-Smooth Extremals and the Weierstrass-Erdmann Corner Conditions

The classical Euler–Lagrange derivation assumes that the candidate extremal is smooth enough across the whole interval. Many physical and geometric problems naturally produce a different situation: the candidate is smooth on separate subintervals but may have an interior point at which its slope changes. Such a curve is called piecewise smooth, and the interior point is commonly called a corner.

The central question of this entry is simple to state:

If each smooth segment already satisfies the Euler–Lagrange equation, what additional conditions must hold where the segments meet?

The answer is given by the Weierstrass–Erdmann corner conditions. For a freely movable corner, two quantities must agree across the corner: the generalized momentum

(1)

and the Hamiltonian-like combination

(2)

Equivalently, F − y′F_y′ is continuous. These conditions are classical necessary conditions for broken extremals; see, for example, [1, 2, 3, 4, 5].

1 Piecewise-smooth admissible curves

Consider the first-order functional

(3)

Suppose a candidate curve is smooth on the two intervals

(4)

and is continuous at x = c, but the one-sided slopes may differ:

(5)

(6)

in general. We write the two pieces as y₋(x) and y₊(x).

The action naturally splits into

(7)

The notation F₋ and F₊ allows the two sides to have different constitutive data, as at an optical or material interface. When the same smooth integrand applies everywhere, simply set F₋ = F₊ = F.

2 Why Euler–Lagrange is not enough

If variations are supported strictly inside (a,c), the usual argument gives

(8)

on the left arc. Similarly,

(9)

on the right arc.

These equations describe what happens between corners. They do not determine how two extremal arcs may be joined. That information comes from the boundary terms generated at the internal splitting point c.

This is the same structural idea encountered in CV05: integration by parts separates interior Euler–Lagrange information from endpoint information. Here, however, the “endpoint” is shared by two neighboring arcs.

3 A fixed corner location

Begin with the simpler case in which the corner abscissa c is fixed but its ordinate y_c is free to vary. Since the total curve remains continuous, the two variation fields have the same value at the corner:

(10)

After integrating by parts separately on the two subintervals and imposing Euler–Lagrange on each interior, the corner contribution is

(11)

Therefore

(12)

Because the corner ordinate is free, η_c is arbitrary. Hence

(13)

Thus even a fixed interface requires continuity of generalized momentum if the crossing point can slide along that interface.

If both c and y_c are prescribed, then there is no free corner variation and no additional matching condition is produced. The two arcs simply meet at a prescribed pin point.

4 Movable corner geometry

Now allow the corner itself to move. Let

(14)

and

(15)

The subtle point is that the ordinary variation field η(x) measures change at a fixed value of x, whereas δy_c measures the actual change of the moving corner point. These are not the same quantity.

For the left arc,

(16)

Differentiate with respect to 𝜖 at 𝜖 = 0:

(17)

Therefore

(18)

The same reasoning on the right gives

(19)

These two relations are the bookkeeping step that produces the second corner condition.

5 Variation of an integral with a moving limit

For the left piece,

(20)

Differentiating gives the usual first-variation integral plus the moving-upper-limit contribution:

(21)

After integration by parts and use of Euler–Lagrange on the left arc, the corner term becomes

(22)

Substitute the moving-corner relation:

δJ−(c)	= (F −)y′ + F−δc	(23)
	= (F−)y′δyc + δc.	(24)

For the right piece, the corner is a moving lower limit, so the sign reverses:

(25)

Adding the two contributions gives

δJcorner =	δyc	(26)
	+ δc.	(27)

6 The Weierstrass–Erdmann corner conditions

theorem. Let a piecewise-C¹ stationary curve have a freely movable interior corner C = (c,y _c). Assume each smooth arc satisfies the Euler–Lagrange equation and the one-sided values needed below exist. If δc and δy_c may be varied independently, then the corner must satisfy

(28)

and

(29)

These are the Weierstrass–Erdmann corner conditions.

Proof. The preceding derivation gives

(30)

where

(31)

and

(32)

Because δy_c and δc are independent, first set δc = 0 and vary δy_c to obtain A = 0. Then set δy_c = 0 and vary δc to obtain B = 0. □

It is often convenient to define

(33)

and

(34)

Then the conditions become simply

(35)

and

(36)

If the jump notation is

(37)

then

(38)

7 What the two conditions mean

The first condition comes from moving the corner vertically while holding its abscissa fixed. It therefore matches the generalized momentum transmitted from one arc to the other.

The second comes from moving the corner horizontally. It matches the quantity conjugate to shifts in the independent variable. In mechanics, where the independent variable is time, this is the Hamiltonian. In autonomous mechanical systems it is the energy.

This is not a coincidence. CV05 already showed that the boundary variation of a first-order integral may be written schematically as

(39)

At an internal corner, the same endpoint expression appears once from the left and once from the right. Stationarity requires the two endpoint covectors to match.

8 A corner constrained to a prescribed curve

A useful intermediate case occurs when the corner may move, but only along a prescribed interface

(40)

Then the corner displacement satisfies

(41)

The two corner variations are no longer independent. Substituting into the general corner term gives the single transversality condition

0 =	ψ′(c)	(42)
	+ F−− y−′(F−)y′− F+ + y+′(F+)y′.	(43)

Thus the fully free Weierstrass–Erdmann pair and the CV05 transversality condition are two versions of the same boundary-variation geometry.

For a vertical interface x = c, one has δc = 0 while δy_c is free, so only generalized momentum continuity is required.

9 Example: optical refraction and Snell’s law

Fermat’s principle makes refraction a particularly transparent corner problem. Suppose Light travels from a medium of refractive index n₁ to a medium of refractive index n₂ across the fixed vertical interface x = c. Apart from a constant factor, the optical travel-time functional is

(44)

with

(45)

On each side,

(46)

The generalized momentum is

(47)

Let 𝜃_i be the angle the ray makes with the Normal to the vertical interface. Since

(48)

we have

(49)

The interface location x = c is fixed, but the crossing height is free. Therefore the fixed-interface corner condition gives

(50)

or

(51)

This is Snell’s law.

The example also explains why a true slope corner is possible: the integrand changes across the interface because n₁≠n₂.

10 Example: a quadratic interface problem

Consider

(52)

where c is a fixed interface and the crossing ordinate is free. Each side satisfies

(53)

so the stationary pieces are straight lines. The corner condition is

(54)

Unless k₁ = k₂, the slope generally changes at the interface. The transmitted quantity is not slope itself but the generalized momentum p = ky′.

This is mathematically analogous to continuity of flux across many interfaces in diffusion, heat conduction, elasticity, and wave problems.

11 Why a regular homogeneous integrand usually forbids a genuine corner

Suppose the same smooth integrand F(x,y,y′) applies on both sides of the corner. The first Weierstrass–Erdmann condition says

(55)

If, for fixed (x,y), the map

(56)

is one-to-one, then necessarily

(57)

A common sufficient local condition is strict convexity in the derivative:

(58)

Then F_y′ is strictly increasing as a function of y′, so momentum continuity forces slope continuity.

This is an important conceptual point: corner conditions do not predict that corners must occur. Often they prove that a stationary curve cannot have one.

12 Mechanics and a free switching time

Let a trajectory q(t) be governed by one Lagrangian before a switching time t_c and another afterward:

(59)

If both the switching state q(t_c) and the switching time t_c are free, the Weierstrass–Erdmann conditions become

(60)

and

(61)

For several coordinates, p becomes a vector and the momentum condition is componentwise.

For an ordinary natural mechanical Lagrangian

(62)

we have

(63)

and

(64)

If the same regular Lagrangian applies on both sides and there is no impulse, momentum continuity forces velocity continuity. Thus a spontaneous velocity kink is incompatible with stationarity. An actual impulsive force changes the variational model and can produce a momentum jump.

13 Vector-valued corner conditions

For a vector path q(x) ∈ ℝⁿ and

(65)

define the generalized momentum vector

(66)

At a freely movable corner the conditions become

(67)

and

(68)

These are precisely the multi-coordinate forms that recur in mechanics, ray tracing, hybrid systems, and later optimal-control switching conditions.

14 Several corners

If a broken extremal contains corners

(69)

then the first variation splits across all subintervals. Each freely movable interior corner independently contributes a matching condition. Therefore at every c_j,

(70)

The logic is local: moving one corner while holding the others fixed isolates its own jump terms.

15 What if only one corner coordinate is free?

The number of independent corner conditions equals the number of independent corner displacements.

• If c and y_c are both fixed: no corner condition is generated.
• If c is fixed and y_c is free: continuity of p.
• If y_c is fixed and c is free: continuity of F − y′p, equivalently H.
• If both are free: both Weierstrass–Erdmann conditions.
• If the corner is constrained to a one-dimensional interface: one transversality combination.

This counting principle is a useful way to avoid imposing too many conditions.

16 Necessary, not sufficient

The Weierstrass–Erdmann conditions are necessary stationarity conditions. They do not by themselves prove that a broken extremal minimizes the functional.

A candidate may satisfy Euler–Lagrange on every arc and satisfy both corner conditions while still being a maximum or saddle. Stronger minimum theory requires second-variation conditions, convexity, the Weierstrass excess function, and related results developed later in the series.

Likewise, corner conditions do not establish existence. A variational problem may fail to possess a minimizer even though its formal necessary conditions are well defined.

17 Common mistakes

• Applying Euler–Lagrange separately and stopping. Interior equations do not contain the corner matching information.
• Assuming η₋(c) = η₊(c) when the corner moves horizontally. For a movable corner, η_±(c) = δy_c − y_±′δc.
• Forgetting the moving-limit term F δc. Without it the second Weierstrass–Erdmann condition is lost.
• Using the same sign for the left and right corner contributions. The corner is an upper limit for the left integral and a lower limit for the right integral.
• Imposing two conditions when the corner is constrained to an interface. Only independent corner motions produce independent conditions.
• Assuming a slope kink is always allowed. For a regular strictly convex homogeneous integrand, momentum continuity typically forces the slope to be continuous.
• Confusing physical impulses with ordinary corners. An impulse introduces a momentum jump and therefore changes the stationarity conditions/model.
• Treating the corner conditions as a minimum proof. They are necessary conditions only.

18 A compact workflow for broken extremals

For a piecewise-smooth candidate, use the following sequence.

1. Identify the subintervals and the corner coordinates that are allowed to move.
2. Split the functional at every corner.
3. Apply Euler–Lagrange separately on each smooth arc.
4. Integrate by parts without discarding the internal boundary terms.
5. Include moving-limit contributions when a corner abscissa is free.
6. Relate fixed-x variations to actual corner displacement through

   (71)

7. Collect coefficients of the independent corner displacements.
8. Set those coefficients to zero.
9. Interpret the resulting conditions as continuity of generalized momentum and, when the corner location is free, continuity of the Hamiltonian-like quantity.
10. Separately test whether the stationary broken curve is actually minimizing.

19 What CV11 adds

CV10 has completed the main first-order necessary-condition structure for smooth and piecewise-smooth extremals. The next question is classification: when does a stationary curve actually behave like a minimum?

CV11 introduces the second variation. The second variation is a quadratic functional in the perturbation and provides the variational analogue of the Hessian in multivariable calculus. Its simplest local consequence is the Legendre necessary condition, which constrains the sign of F_y′y′ along a minimizing extremal.

Summary

For a piecewise-smooth extremal with a freely movable corner, Euler–Lagrange must hold separately on every smooth arc, and the internal boundary terms must vanish under independent corner motion. The resulting Weierstrass–Erdmann conditions are

(72)

and

(73)

Equivalently, with

(74)

we have

(75)

The first condition matches generalized momentum; the second matches the quantity conjugate to movement of the corner in the independent variable. Fixed interfaces, movable interfaces, optical refraction, mechanical switching times, and vector trajectories are all variations of this same internal-boundary principle.

References