We present an account of transversality conditions of variational problems and show how they give rise to essential results in the analysis of different physical phenomena. We find such conditions to be a powerful and elegant approach to a diverse number of variational problems with free endpoints. In this work, we illustrate such condition with the study of a heavy inextensible rope sagging both symmetrically and asymmetrically between two variously shaped, steering-guide wires without friction. In such a situation, the transversality conditions lead to the orthogonality of the rope to the wires at endpoints of the rope, which we confirm experimentally. Freeing the endpoints of the rope yields exact analytical equations that predict the tension of the rope. Heavy ropes whose endpoints are free to slip between arbitrarily shaped, steering wires are discussed.

1.
C.
Lanczos
,
The Variational Principles of Mechanics
(
Dover Publications
,
NY
,
1970
), pp.
5
6
.
2.
J.-L.
Basdevant
,
Variational Principles in Physics
(
Vuibert
,
Paris, France
,
2007
), pp.
9
12
.
3.
L. D.
Landau
and
E. M.
Lifshitz
,
Mechanics
, 3rd ed. (
Butterworth-Heinemann
,
Oxford
,
1976
), Vol.
1
, pp.
2
4
.
4.
L. D.
Landau
and
E. M.
Lifshitz
,
The Classical Theory of Fields
, 4th ed. (
Butterworth-Heinemann
,
Oxford
,
1975
), Vol.
2
, pp.
24
25
.
5.
R. P.
Feynman
and
A. R.
Hibbs
,
Quantum Mechanics and Path Integrals
(
McGraw-Hill
,
NY
,
1965
), pp.
26
31
.
6.
L. M.
Martyushev
and
V. D.
Seleznev
, “
Maximum entropy production principle in physics, chemistry and biology
,”
Phys. Rep.
426
,
1
45
(
2006
).
7.
I.
Gyarmati
,
Non-Equilibrium Thermodynamics: Field Theory and Variational Principle
(
Springer
,
NY
,
1970
), pp.
10
15
.
8.
G. B.
Arfken
and
H. J.
Weber
,
Mathematical Methods for Physicists
, 5th ed. (
Harcourt Academic Press
,
San Diego
,
2001
), pp.
1017
1052
.
9.
I. M.
Gelfand
and
S. V.
Fomin
,
Calculus of Variations
(
Dover Books on Mathematics
,
Dover Publications
,
2000
), pp.
60
65
.
10.
A. B.
Whiting
, “
The Least-Action Principle: Theory of cosmological solutions and the radial velocity action
,”
Astrophys. J.
533
,
50
61
(
2000
).
11.
E.
Bormashenko
, “
Young, Boruvka–Neumann,Wenzel and Cassie–Baxter equations as the transversality conditions for the variational problem of wetting
,”
Colloids Surf. A
345
,
163
165
(
2009
).
12.
D. G. B.
Edelen
, “
Aspects of variational arguments in the theory of elasticity: Fact and folklore
,”
Int. J. Solids Struct.
17
,
729
740
(
1981
).
13.
E.
Bormashenko
,
Wetting of Real Surfaces
(
De Gruyter
,
Berlin
,
2013
).
14.
E.
Conversano
,
M.
Frangaviglia
,
M. G.
Lorenzi
, and
L.
Tedeschini-Lalli
, “
Persistence of form in art in architecture: catenaries, helicoids and sinusoids
,”
APLIMAT Journal of Applied Mathematics
,
4
,
101
112
(
2011
).
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