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Journal articles on the topic 'Calculus of variations. Multiple integrals'

Author: Grafiati

Published: 4 June 2021

Last updated: 8 February 2022

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1

Bousquet, Pierre. "The Euler Equation in the Multiple Integrals Calculus of Variations." SIAM Journal on Control and Optimization 51, no. 2 (January 2013): 1047–62. http://dx.doi.org/10.1137/120882561.

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2

Arcoya, David, and Lucio Boccardo. "Critical points for multiple integrals of the calculus of variations." Archive for Rational Mechanics and Analysis 134, no. 3 (1996): 249–74. http://dx.doi.org/10.1007/bf00379536.

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3

Ball, J. M., and K. W. Zhang. "Lower semicontinuity of multiple integrals and the Biting Lemma." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 114, no. 3-4 (1990): 367–79. http://dx.doi.org/10.1017/s0308210500024483.

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SynopsisWeak lower semicontinuity theorems in the sense of Chacon's Biting Lemma are proved for multiple integrals of the calculus of variations. A general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions. The “biting”weak limit of the corresponding integrands is characterised via the Young measure, and related to the weak* limit in the sense of measures. Finally, an example is given which shows that the Young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case.

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4

Rund, Hanno. "LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS." Quaestiones Mathematicae 12, no. 2 (January 1989): 205–29. http://dx.doi.org/10.1080/16073606.1989.9632177.

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5

Rund, Hanno. "LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS." Quaestiones Mathematicae 12, no. 3 (January 1989): 315–39. http://dx.doi.org/10.1080/16073606.1989.9632186.

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6

Almeida, Ricardo, Agnieszka B. Malinowska, and Delfim F. M. Torres. "A fractional calculus of variations for multiple integrals with application to vibrating string." Journal of Mathematical Physics 51, no. 3 (2010): 033503. http://dx.doi.org/10.1063/1.3319559.

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7

Cesari, L., P. Brandi, and A. Salvadori. "Existence theorems for multiple integrals of the calculus of variations for discontinuous solutions." Annali di Matematica Pura ed Applicata 152, no. 1 (December 1988): 95–121. http://dx.doi.org/10.1007/bf01766143.

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8

Fonseca, Irene, and Giovanni Leoni. "On lower semicontinuity and relaxation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 3 (June 2001): 519–65. http://dx.doi.org/10.1017/s0308210500000998.

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Lower semicontinuity and relaxation results in BV are obtained for multiple integrals where the energy density f satisfies lower semicontinuity conditions with respect to (x, u) and is not subjected to coercivity hypotheses. These results call for methods recently developed in the calculus of variations.

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9

Fonseca, Irene, and Giovanni Leoni. "On lower semicontinuity and relaxation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 3 (June 2001): 519–65. http://dx.doi.org/10.1017/s0308210501000245.

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Lower semicontinuity and relaxation results in BV are obtained for multiple integrals where the energy density f satisfies lower semicontinuity conditions with respect to (x, u) and is not subjected to coercivity hypotheses. These results call for methods recently developed in the calculus of variations.

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10

Taheri, Ali. "Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 1 (February 2001): 155–84. http://dx.doi.org/10.1017/s0308210500000822.

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Let Ω ⊂ Rn be a bounded domain and let f : Ω × RN × RN×n → R. Consider the functional over the class of Sobolev functions W1,q(Ω;RN) (1 ≤ q ≤ ∞) for which the integral on the right is well defined. In this paper we establish sufficient conditions on a given function u0 and f to ensure that u0 provides an Lr local minimizer for I where 1 ≤ r ≤ ∞. The case r = ∞ is somewhat known and there is a considerable literature on the subject treating the case min(n, N) = 1, mostly based on the field theory of the calculus of variations. The main contribution here is to present a set of sufficient conditions for the case 1 ≤ r < ∞. Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of ‘directional convergence’.

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11

Cesari, Lamberto. "Existence ofBV absolute discontinuous minima for modified multiple integrals of the calculus of variations." Rendiconti del Circolo Matematico di Palermo 39, no. 2 (May 1990): 169–209. http://dx.doi.org/10.1007/bf02844756.

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12

Edelen, Dominic G. B., and Izak M. Snyman. "Cartan forms for multiple integral problems in the calculus of variations." Journal of Mathematical Analysis and Applications 120, no. 1 (November 1986): 218–39. http://dx.doi.org/10.1016/0022-247x(86)90212-x.

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13

Crampin, M., and D. J. Saunders. "Some concepts of regularity for parametric multiple-integral problems in the calculus of variations." Czechoslovak Mathematical Journal 59, no. 3 (September 2009): 741–58. http://dx.doi.org/10.1007/s10587-009-0044-0.

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14

WINTER, MATTHIAS. "An example of microstructure with multiple scales." European Journal of Applied Mathematics 8, no. 2 (April 1997): 185–207. http://dx.doi.org/10.1017/s0956792597003021.

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This paper studies a vectorial problem in the calculus of variations arising in the theory of martensitic microstructure. The functional has an integral representation where the integrand is a non-convex function of the gradient with exactly four minima. We prove that the Young measure corresponding to a minimizing sequence is homogeneous and unique for certain linear boundary conditions. We also consider the singular perturbation of the problem by higher-order gradients. We study an example of microstructure involving infinite sequential lamination and calculate its energy and length scales in the zero limit of the perturbation.

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15

Cicalese, Marco, and Nicola Fusco. "A note on relaxation with constraints on the determinant." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 41. http://dx.doi.org/10.1051/cocv/2018030.

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We consider multiple integrals of the Calculus of Variations of the form E(u) = ∫ W(x, u(x), Du(x)) dx where W is a Carathéodory function finite on matrices satisfying an orientation preserving or an incompressibility constraint of the type, det Du > 0 or det Du = 1, respectively. Under suitable growth and lower semicontinuity assumptions in the u variable we prove that the functional ∫ Wqc(x, u(x), Du(x)) dx is an upper bound for the relaxation of E and coincides with the relaxation if the quasiconvex envelope Wqc of W is polyconvex and satisfies p growth from below for p bigger then the ambient dimension. Our result generalises a previous one by Conti and Dolzmann [Arch. Rational Mech. Anal. 217 (2015) 413–437] relative to the case where W depends only on the gradient variable.

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16

Sudha, T. "Multi-Objective Optimization Based Multi-Objective Controller Tuning Method with Robust Stabilization of Fractional Calculus CSTR." WSEAS TRANSACTIONS ON SYSTEMS AND CONTROL 16 (July 8, 2021): 375–82. http://dx.doi.org/10.37394/23203.2021.16.32.

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In Continuous Stirred Tank Reactor (CSTR) have Fractional order PID with the nominal order PID controller has been used to Multi-Criteria Decision Making (MCDM) and EMO (Evolutionary Multi-objective Optimization) by adjustment of control parameters like Hybrid methods in Multi objective optimization. But, this Fractional order PID with the nominal PID controller has maximum performance estimation. Proposed research work focused the Flower Pollination Algorithm based on Multi objective optimization with Genetic evaluation and Fractional order PID with the nominal PID controller is provides CSTR results. When a flower is displayed to maximum variations in this practical state, the Genetic evaluation has been used to identify the variations. The FPID (Flower Pollination Integral Derivative) is used for tuning the parameters of a Fractional order PID with the nominal PID controller for each region to improve the multi-criteria decision making. FPID also denoted as Flower Optimization Integral Derivative (FOID). The Genetic evaluation scheduler has been combined with multiple local linear Fractional order PID with the nominal PID controller to check the stability of loop for entire regions with various levels of temperatures. MATLAB results demonstrate that the feasibility of using the proposed Fractional order PID with the nominal PID controller compared than the existing PID controller, and it shows the FOID attained better results.

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17

Tringali, Alessandro, and Silvio Cocuzza. "Globally Optimal Inverse Kinematics Method for a Redundant Robot Manipulator with Linear and Nonlinear Constraints." Robotics 9, no. 3 (July 31, 2020): 61. http://dx.doi.org/10.3390/robotics9030061.

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This paper presents a novel inverse kinematics global method for a redundant robot manipulator performing a tracking maneuver. The proposed method, based on the choice of appropriate initial joint trajectories that satisfy the kinematic constraints to be used as inputs for a multi-start optimization algorithm, allows for the optimization of different integral cost functions, such as kinetic energy and joint torques norm, and can provide solutions with a variety of constraints, both linear and nonlinear. Furthermore, it is suitable for multi-objective optimization, and it is able to find multiple optima with minimal input from the user, and to solve cyclic trajectories. Problems with a high number of parameters have been addressed providing a sequential version of the method based on successive stages of interpolation. The results of simulations with a three-Degrees-of-Freedom (DOF) redundant manipulator have been compared with a solution available in the literature based on the calculus of variations, thus leading to the validation of the method. Moreover, the effectiveness of the presented method has been shown when used to solve problems with constraints on joint displacement, velocity, torque, and power.

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18

Iglesias-Zemmour, Patrick. "Variations of Integrals in Diffeology." Canadian Journal of Mathematics 65, no. 6 (December 1, 2013): 1255–86. http://dx.doi.org/10.4153/cjm-2012-044-5.

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AbstractWe establish a formula for the variation of integrals of differential forms on cubic chains in the context of diffeological spaces. Then we establish the diffeological version of Stokes’ theorem, and we apply that to get the diffeological variant of the Cartan–Lie formula. Still in the context of Cartan–De Rham calculus in diffeology, we construct a chain-homotopy operator K, and we apply it here to get the homotopic invariance of De Rham cohomology for diffeological spaces. This is the chain-homotopy operator that is used in symplectic diffeology to construct the moment map.

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19

Dacorogna, B. "Convexity of certain integrals of the calculus of variations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 107, no. 1-2 (1987): 15–26. http://dx.doi.org/10.1017/s0308210500029322.

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SynopsisIn this paper we study the convexity of the integral over the space . We isolate a necessary condition on f and we find necessary and sufficient conditions in the case where f(x, u, u′) = a(u)u′2n or g(u) + h(u′).

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20

Almeida, Ricardo, and Delfim F. M. Torres. "Calculus of variations with fractional derivatives and fractional integrals." Applied Mathematics Letters 22, no. 12 (December 2009): 1816–20. http://dx.doi.org/10.1016/j.aml.2009.07.002.

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21

Sil, Swarnendu. "Calculus of variations: A differential form approach." Advances in Calculus of Variations 12, no. 1 (January 1, 2019): 57–84. http://dx.doi.org/10.1515/acv-2016-0058.

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AbstractWe study integrals of the form {\int_{\Omega}f(d\omega_{1},\dots,d\omega_{m})}, where {m\geq 1} is a given integer, {1\leq k_{i}\leq n} are integers, {\omega_{i}} is a {(k_{i}-1)}-form for all {1\leq i\leq m} and {f:\prod_{i=1}^{m}\Lambda^{k_{i}}(\mathbb{R}^{n})\rightarrow\mathbb{R}} is a continuous function. We introduce the appropriate notions of convexity, namely vectorial ext. one convexity, vectorial ext. quasiconvexity and vectorial ext. polyconvexity. We prove weak lower semicontinuity theorems and weak continuity theorems and conclude with applications to minimization problems. These results generalize the corresponding results in both classical vectorial calculus of variations and the calculus of variations for a single differential form.

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22

Oziewicz, Zbigniew. "Calculus of variations for multiple-valued functionals." Reports on Mathematical Physics 31, no. 1 (February 1992): 85–90. http://dx.doi.org/10.1016/0034-4877(92)90006-m.

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23

Cellina, A., A. Ferriero, and E. M. Marchini. "Reparametrizations and approximate values of integrals of the calculus of variations." Journal of Differential Equations 193, no. 2 (September 2003): 374–84. http://dx.doi.org/10.1016/s0022-0396(02)00176-6.

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24

Mascolo, Elvira, and Gloria Papi. "Local boundedness of minimizers of integrals of the calculus of variations." Annali di Matematica Pura ed Applicata 167, no. 1 (December 1994): 323–39. http://dx.doi.org/10.1007/bf01760338.

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25

Anza Hafsa, Omar, and Jean-Philippe Mandallena. "Γ-convergence of nonconvex integrals in Cheeger--Sobolev spaces and homogenization." Advances in Calculus of Variations 10, no. 4 (October 1, 2017): 381–405. http://dx.doi.org/10.1515/acv-2015-0053.

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26

Nualart, D., and S. Ortiz-Latorre. "Central limit theorems for multiple stochastic integrals and Malliavin calculus." Stochastic Processes and their Applications 118, no. 4 (April 2008): 614–28. http://dx.doi.org/10.1016/j.spa.2007.05.004.

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27

Ai, Zhenghai, and Zeshui Xu. "Multiple Definite Integrals of Intuitionistic Fuzzy Calculus and Isomorphic Mappings." IEEE Transactions on Fuzzy Systems 26, no. 2 (April 2018): 670–80. http://dx.doi.org/10.1109/tfuzz.2017.2687885.

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28

Li, Guanfeng, Yong Wang, and Gejun Bao. "Variational Integrals of a Class of Nonhomogeneous𝒜-Harmonic Equations." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/697974.

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We introduce a class of variational integrals whose Euler equations are nonhomogeneous𝒜-harmonic equations. We investigate the relationship between the minimization problem and the Euler equation and give a simple proof of the existence of some nonhomogeneous𝒜-harmonic equations by applying direct methods of the calculus of variations. Besides, we establish some interesting results on variational integrals.

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29

Rchid Sidi Ammi, Moulay, and Delfim F. M. Torres. "Regularity of solutions to higher-order integrals of the calculus of variations." International Journal of Systems Science 39, no. 9 (September 2008): 889–95. http://dx.doi.org/10.1080/00207720802184733.

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30

Qi, Tang. "Regularity of minimizers of non-isotropic integrals of the calculus of variations." Annali di Matematica Pura ed Applicata 164, no. 1 (December 1993): 77–87. http://dx.doi.org/10.1007/bf01759315.

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31

Brandi, Primo, and Anna Salvadori. "On the lower semicontinuity of certain integrals of the calculus of variations." Journal of Mathematical Analysis and Applications 144, no. 1 (November 1989): 183–205. http://dx.doi.org/10.1016/0022-247x(89)90368-5.

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32

Cesari, L., P. Brandi, and A. Salvadori. "Existence theorems concerning simple integrals of the calculus of variations for discontinuous solutions." Archive for Rational Mechanics and Analysis 98, no. 4 (1987): 307–28. http://dx.doi.org/10.1007/bf00276912.

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33

Eleuteri, Michela, Paolo Marcellini, and Elvira Mascolo. "Regularity for scalar integrals without structure conditions." Advances in Calculus of Variations 13, no. 3 (July 1, 2020): 279–300. http://dx.doi.org/10.1515/acv-2017-0037.

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AbstractIntegrals of the Calculus of Variations with {p,q}-growth may have not smooth minimizers, not even bounded, for general {p,q} exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand {f(x,\xi)} with dependence on the modulus of the gradient, i.e. {f(x,\xi)=g(x,|\xi|)}. Without imposing structure conditions, we prove that if {\frac{q}{p}} is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous.

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34

Ito, Yoshifusa, and Izumi Kubo. "Calculus on Gaussian and Poisson white noises." Nagoya Mathematical Journal 111 (September 1988): 41–84. http://dx.doi.org/10.1017/s0027763000000994.

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Recently one of the authors has introduced the concept of generalized Poisson functionals and discussed the differentiation, renormalization, stochastic integrals etc. ([8], [9]), analogously to the works of T. Hida ([3], [4], [5]). Here we introduce a transformation for Poisson fnnctionals with the idea as in the case of Gaussian white noise (cf. [10], [11], [12], [13]). Then we can discuss the differentiation, renormalization, multiple Wiener integrals etc. in a way completely parallel with the Gaussian case. The only one exceptional point, which is most significant, is that the multiplications are described by for the Gaussian case, for the Poisson case,as will be stated in Section 5. Conversely, those formulae characterize the types of white noises.

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35

Cesari, L., and Wei H. Yang. "Serrin integrals and second order problems of plasticity." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 117, no. 3-4 (1991): 193–207. http://dx.doi.org/10.1017/s0308210500024677.

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SynopsisWe use the modern tools of the duality principles and the calculus of variations to formulate, analyse and solve a class of plasticity problems involving second order partial derivatives. The Serrin-type integrals can most appropriately facilitate the existence statements for the extrema from either side of the duality relation in a larger class of BV functions, and interpret the solutions with possible discontinuities on sets of measure zero. The exact solutions of a beam and numerical solutions of a circular plate are presented to demonstrate the theoretical conclusions.

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36

De Philippis, G., S. Di Marino, and M. Focardi. "Lower semi-continuity for non-coercive polyconvex integrals in the limit case." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 146, no. 2 (December 10, 2015): 243–64. http://dx.doi.org/10.1017/s0308210515000438.

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Lower semi-continuity results for polyconvex functionals of the calculus of variations along sequences of maps u: Ω ⊂ ℝn → ℝm in W1,m, 2 ⩽ m⩽ n, weakly converging in W1,m-1, are established. In addition, for m = n + 1, we also consider the autonomous case for weakly converging maps in W1,n-1.

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37

ES-SEBAIY, KHALIFA, and CIPRIAN A. TUDOR. "MULTIDIMENSIONAL BIFRACTIONAL BROWNIAN MOTION: ITÔ AND TANAKA FORMULAS." Stochastics and Dynamics 07, no. 03 (September 2007): 365–88. http://dx.doi.org/10.1142/s0219493707002050.

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38

Odzijewicz, Tatiana, Agnieszka B. Malinowska, and Delfim F. M. Torres. "Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics." Abstract and Applied Analysis 2012 (2012): 1–24. http://dx.doi.org/10.1155/2012/871912.

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We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to physics discussed.

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39

Schemm, Sabine, and Thomas Schmidt. "Partial regularity of strong local minimizers of quasiconvex integrals with (p, q)-growth." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 139, no. 3 (May 26, 2009): 595–621. http://dx.doi.org/10.1017/s0308210507001278.

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We consider strictly quasiconvex integralsin the multi-dimensional calculus of variations. For the C2-integrand f : ℝNn → ℝ we impose (p, q)-growth conditionswith γ, Γ > 0 and 1 < p ≤ q < min {p + 1/n, p(2n − 1)/(2n − 2)}. Under these assumptions we prove partial C1, αloc-regularity for strong local minimizers of F and the associated relaxed functional F.

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40

Galperin, E. A. "Translational symmetry of certain integrals with application to the calculus of variations and optimal control." Mathematical and Computer Modelling 36, no. 6 (October 2002): 717–28. http://dx.doi.org/10.1016/s0895-7177(02)00169-3.

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41

Ferriero, Alessandro. "The Approximation of Higher-Order Integrals of the Calculus of Variations and the Lavrentiev Phenomenon." SIAM Journal on Control and Optimization 44, no. 1 (January 2005): 99–110. http://dx.doi.org/10.1137/s0363012903437721.

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42

Granucci, Tiziano. "Hölder continuity for scalar minimizers of integrals of the calculus of variations with general growths." Afrika Matematika 25, no. 1 (September 25, 2012): 197–212. http://dx.doi.org/10.1007/s13370-012-0109-3.

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43

Marcellini, Paolo. "Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions." Archive for Rational Mechanics and Analysis 105, no. 3 (September 1989): 267–84. http://dx.doi.org/10.1007/bf00251503.

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44

Ekeland, I., N. Ghoussoub, and H. Tehrani. "Multiple Solutions for a Classical Problem in the Calculus of Variations." Journal of Differential Equations 131, no. 2 (November 1996): 229–43. http://dx.doi.org/10.1006/jdeq.1996.0162.

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45

Almeida, Ricardo, and Delfim F. M. Torres. "An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order." Scientific World Journal 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/915437.

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We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type.

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46

Prabseang, Julalak, Kamsing Nonlaopon, Jessada Tariboon, and Sotiris K. Ntouyas. "Refinements of Hermite–Hadamard Inequalities for Continuous Convex Functions via (p,q)-Calculus." Mathematics 9, no. 4 (February 23, 2021): 446. http://dx.doi.org/10.3390/math9040446.

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In this paper, we present some new refinements of Hermite–Hadamard inequalities for continuous convex functions by using (p,q)-calculus. Moreover, we study some new (p,q)-Hermite–Hadamard inequalities for multiple integrals. Many results given in this paper provide extensions of others given in previous research.

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47

Almeida, Alexandre Marques de, Marcelo Kaminski Lenzi, and Ervin Kaminski Lenzi. "A Survey of Fractional Order Calculus Applications of Multiple-Input, Multiple-Output (MIMO) Process Control." Fractal and Fractional 4, no. 2 (May 19, 2020): 22. http://dx.doi.org/10.3390/fractalfract4020022.

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Multiple-input multiple-output (MIMO) systems are usually present in process systems engineering. Due to the interaction among the variables and loops in the MIMO system, designing efficient control systems for both servo and regulatory scenarios remains a challenging task. The literature reports the use of several techniques mainly based on classical approaches, such as the proportional-integral-derivative (PID) controller, for single-input single-output (SISO) systems control. Furthermore, control system design approaches based on derivatives and integrals of non-integer order, also known as fractional control or fractional order (FO) control, are frequently used for SISO systems control. A natural consequence, already reported in the literature, is the application of these techniques to MIMO systems to address some inherent issues. Therefore, this work discusses the state-of-the-art of fractional control applied to MIMO systems. It outlines different types of applications, fractional controllers, controller tuning rules, experimental validation, software, and appropriate loop decoupling techniques, leading to literature gaps and research opportunities. The span of publications explored in this survey ranged from the years 1997 to 2019.

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48

Bezuglov, Maxim A. "Calculation of master integrals in terms of elliptic multiple polylogarithms." International Journal of Modern Physics A 35, no. 13 (May 10, 2020): 2050063. http://dx.doi.org/10.1142/s0217751x20500633.

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In modern quantum field theory, one of the most important tasks is the calculation of loop integrals. Loop integrals appear when evaluating the Feynman diagrams with one or more loops by integrating over the internal momenta. Even though this problem has already been in place since the mid-twentieth century, we not only do not understand how to calculate all classes of these integrals beyond one loop, we do not even know in what class of functions the answer is expressed. To partially solve this problem, different variations of new functions usually called elliptic multiple polylogarithms have been introduced in the last decade. In this paper, we explore the possibilities and limitations of this class of functions. As a practical example, we chose the processes associated with the physics of heavy quarkonium at the two-loop level.

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49

Arthurs, A. M., and G. R. Walsh. "On Hammersley's minimum problem for a rolling sphere." Mathematical Proceedings of the Cambridge Philosophical Society 99, no. 3 (May 1986): 529–34. http://dx.doi.org/10.1017/s0305004100064471.

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AbstractThe problem posed by Hammersley (1983) of finding the shortest path along which a sphere can roll from one prescribed state to another is formulated by using quaternion calculus of variations and optimal control theory. This leads to a system of coupled nonlinear differential equations with prescribed end conditions. From the resulting expression for the curvature, it is shown that the differential equation of the required path in intrinsic coordinates is the same as the equation of motion of a simple pendulum, giving a solution in terms of elliptic integrals.

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Botteron, Bernard, and Paolo Marcellini. "A general approach to the existence of minimizers of one-dimensional non-coercive integrals of the calculus of variations." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 8, no. 2 (March 1991): 197–223. http://dx.doi.org/10.1016/s0294-1449(16)30272-4.

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