Null Lagrangians in linear theories of micropolar type and few other generalizations of elasticity


In the context of linear theories of generalized elasticity including those for homogeneous micropolar media, quasicrystals, and piezoelectric and piezomagnetic media, we explore the concept of null Lagrangians. For obtaining the family of null Lagrangians, we employ the sufficient conditions of H. Rund. In some cases, a nonzero null Lagrangian is found and the stored energy admits a split into a null Lagrangian and a remainder. However, the null Lagrangian vanishes whenever the relevant elasticity tensor obeys certain symmetry conditions which can be construed as an analogue of the Cauchy relations.

This is a preview of subscription content, access via your institution.


  1. 1.

    Cosserat, E., Cosserat, F.: Théorie des corps déformables. Librairie Scientifique A. Hermann et Fils (english translation by D. Delphenich, 2007), reprint 2009, Paris (1909)

  2. 2.

    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn. Cambridge University Presss, Cambridge (1927)

    MATH  Google Scholar 

  3. 3.

    Carathéodory, C.: Uber die Variationsrechnung bei mehrfachen Integralen. Acta Szeged Sect. Scient. Mathem. 4, 193 (1929)

    MATH  Google Scholar 

  4. 4.

    Landers, Jr., A. W.: Invariant Multiple Integrals in the Calculus of Variations. Univ. Chicago Press, Chicago (1942). Also: in Contributions to the Calculus of Variations, 1938-1941, 175 (Univ. Chicago Press, 1942)

  5. 5.

    Ericksen, J., Truesdell, C.: Exact theory of stress and strain in rods and shells. Arch. Rational Mech. Anal. 1, 295–323 (1958)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Edelen, D.G.B.: The null set of the Euler-Lagrange operator. Arch. Rational Mech. Anal. 11, 117–121 (1962)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Ericksen, J.: Nilpotent energies in liquid crystal theory. Arch. Rational Mech. Anal. 10, 189–196 (1962)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Picone, M.: Criteri sufficienti peril minima assoluto di un integrale pluridimensionale del primo ordine nel vettore minimante. Mem. Accad. Naz. Lincei V I, 281–338 (1962)

    MATH  Google Scholar 

  9. 9.

    Picone, M.: Criteri sufficienti peril minima assoluto di un integrale pluridimensionale del primo ordine nel vettore minimante in parte o completamente libero alla frontiera del dominio di integrazione. Mem Accad. Naz. Lincei V I, 341–374 (1964)

    MATH  Google Scholar 

  10. 10.

    De Franchis, M.: La piu generale funzione d’invarianza per criteri sufficienti di minima con condizioni di Dirichlet per integrali pluridimensionali del primo ordine dipendenti da un vettore a piu componenti. Rend. Accad. Naz. Lincei XXXVI I, 130–140 (1964)

    MATH  Google Scholar 

  11. 11.

    Aero, E. L., Kuvshinski, E. V.: Continuum theory of asymmetric elasticity. Microrotation effect, Solid State Physics, 5(9), 2591–2598 (in Russian) (1963) (English translation: Soviet Physics–Solid State, 5 (1964), 1892–1899)

  12. 12.

    Berlincourt, D.A., Curran, D.R., Jaffe, H.: Piezoelectric and piezomagnetic materials and their function in transducers. Phys. Acoust. Princ. Methods Part A(1), 169–270 (1964)

    Google Scholar 

  13. 13.

    Toupin, R.: Theories of elasticity with couple-stresses. Arch. Rational Mech. Anal. 17, 85–112 (1964)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Mindlin, R.: Microstructure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78 (1964)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Aero, E.L., Kuvshinski, E.V.: Continuum theory of asymmetric elasticity. Equilibrium of an isotropic body. Sov. Phys. Solid State 6, 2141–2148 (1965)

    MathSciNet  Google Scholar 

  16. 16.

    Eringen, A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 15, 909–923 (1966)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Rund, H.: The Hamiltonian-Jacobi Theory in the Calculus of Variation, Its Role in Mathematics and Physics. D. Van Nostrand Company Ltd, London (1966)

    MATH  Google Scholar 

  18. 18.

    Rund, H.: Integral formulae associated with the Euler-Lagrange operators of multiple integral problems in the calculus of variations. Aequationes Math. 11, 212–229 (1974)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Barnett, D.M., Lothe, J.: Dislocations and line charges in anisotropic piezoelectric insulators. Phys. Stat. Sol. (b) 67, 105–111 (1975)

    Google Scholar 

  20. 20.

    Eringen, A.C., Kadafar, C.B.: Polar field theories. In: Eringen, A.C. (ed.) Continuum Physics, vol. 4, pp. 1–73. Academic Press, New York (1976)

    Google Scholar 

  21. 21.

    Nowacki, J.P.: Green’s function for a hemitropic micropolar continuum. Green’s Funct Hemitr Micropolar Continuum 8, 235 (1977)

    MATH  Google Scholar 

  22. 22.

    Edelen, D.G.B.: Isovector Methods for Equations of Balance with Programs for Computer Assistance in Operator Calculations and an Exposition of Practical Topics of the Exterior Calculus. Sijthoff and Noordhoff, Amstredam (1980)

    MATH  Google Scholar 

  23. 23.

    Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, New York (1981)

    MATH  Google Scholar 

  24. 24.

    Ball, J.M., Currie, J.C., Olver, P.J.: Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41, 135–174 (1981)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Lakes, R.S., Benedict, R.L.: Noncentrosymmetry in micropolar elasticity. Int. J. Eng. Sci. 20(10), 1161–1167 (1982)

    MATH  Google Scholar 

  26. 26.

    Nowacki, W.: Theory of Asymmetric Elasticity, PWN-Polish Scientific Publishers. ISBN: 0-08-027584-2. (1986)

  27. 27.

    Edelen, D.G.B., Lagoudas, D.C.: Null Lagrangians, admissible tractions, and finite element methods. Int. J. Solids Struct. 22(6), 659–672 (1986)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Maugin, G.A.: Continuum Mechanics of Electromagnetic Solids. Elsevier, Oxford (1988)

    MATH  Google Scholar 

  29. 29.

    Olver, P.J., Sivaloganathan, J.: Structure of null Lagrangians. Nonlinearity 1, 389–398 (1988)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Eringen, A .C., Maugin, G .A.: Electrodynamics of Continua. Springer, New York (1990)

    Google Scholar 

  31. 31.

    Alshits, V.I., Darinskii, A.N., Lothe, J.: On the existence of interface waves in half-infinite anisotropic elastic media with piezoelectronic and piezomagnetic properties. Wave Motion 16, 265–283 (1992)

    MATH  Google Scholar 

  32. 32.

    Lakes, R.: Materials with structural hierarchy. Nature 361, 511–515 (1993)

    Google Scholar 

  33. 33.

    Ding, D.H., Wang, W., Hu, C., Yang, R.: Generalized elasticity theory of quasicrystals. Phys. Rev. B 48, 7003–7010 (1993)

    Google Scholar 

  34. 34.

    Campanella, A., Tonon, M.L.: A note on the Cauchy relations. Meccanica 29, 105–108 (1994)

    MATH  Google Scholar 

  35. 35.

    Virga, E.: Variational Theories for Liquid Crystals, Applied Mathematics and Computation, 8. Chapman & Hall, Baco Raton (1994)

    Google Scholar 

  36. 36.

    Lancia, M. R., Caffarelli, G. V., Podio-Guidugli, P.: Null Lagrangians in Linear Elasticity, Mathematical Models and Methods in Applied Sciences, pp. 415–427 (1995)

  37. 37.

    Eringen, A.C.: Microcontinuum Field Theories-Volume 1: Foundations and Solids. Springer, New York (1999)

    MATH  Google Scholar 

  38. 38.

    Hu, C., Wang, R., Ding, D.H.: Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep. Prog. Phys. 63, 1–39 (2000)

    MathSciNet  Google Scholar 

  39. 39.

    Lakes, R.: Elastic and viscoelastic behavior of chiral materials. Int. J. Mech. Sci. 43, 1579–1589 (2001)

    MATH  Google Scholar 

  40. 40.

    Hehl, F.W., Itin, Y.: The Cauchy relations in linear elasticity theory. J. Elast. 66, 185–192 (2002)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Carillo, S.: Null Lagrangians and surface interaction potentials in nonlinear elasticity. Ser. Adv. Math. Appl. Sci. 62, 9–18 (2002)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Sharma, P.: Size-dependent elastic fields of embedded inclusions in isotropic chiral solids. Int. J. Solids Struct. 41(22–23), 6317–6333 (2004)

    MATH  Google Scholar 

  43. 43.

    Natroshvili, D., Stratis, I.G.: Mathematical problems of the theory of elasticity of chiral materials for Lipschitz domains. Math. Methods Appl. Sci. 29, 445–478 (2006)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Pietraszkiewicz, W., Eremeyev, V.A.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46(3–4), 774–787 (2009)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Joumaa, H., Ostoja-Starzewski, M.: Stress and couple-stress invariance in non-centrosymmetric micropolar planar elasticity. Proc. R. Soc. Lond. A 467(2134), 2896–2911 (2011)

    MATH  Google Scholar 

  46. 46.

    Fan, T.Y.: Mathematical Theory of Elasticity of Quasicrystals and Its Applications. Science Press, Beijing (2011)

    Google Scholar 

  47. 47.

    Itin, Y., Hehl, F.W.: The constitutive tensor of linear elastic Its decompositions, Cauchy relations, Lagrangians, and wave propagation. J. Math. Phys. 54, 042903 (2013)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Sharma, B. L., Basak, N.: Null lagrangians in Cosserat elasticity. J. Elasticity. arXiv:2009.03490 (2020) (under review)

Download references


BLS acknowledges the partial support of SERB MATRICS grant MTR/2017/000013. The authors thank the anonymous reviewers for their constructive comments and suggestions. This work has been available free of peer review on the arXiv:2009.03487 since 09/09/2020.

Author information



Corresponding author

Correspondence to Basant Lal Sharma.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Characterization of null Lagrangians

Appendix A: Characterization of null Lagrangians

If the right-hand side of (1.33) is to vanish for all values of \(\frac{{\partial }^2y^j}{{\partial } x^{{\beta }}{\partial } x^{{\gamma }}}\), the coefficient of these quantities which is symmetric in \({{\beta }}\) and \({{\gamma }}\) must be skew-symmetric in these indices, i.e.

$$\begin{aligned} \frac{{\partial }^2\Psi }{{\partial } y^j_{{\beta }}{\partial } y^k_{{\gamma }}}=-\frac{{\partial }^2\Psi }{{\partial } y^j_{{\gamma }}{\partial } y^k_{{\beta }}}. \end{aligned}$$

Following the analysis of [17], in general, \(\Psi \) is a polynomial in its dependence on the derivative of y. In the class of Lagrangians which are interesting in this document, it is sufficient to consider \(M=2.\) Thus, \(\Psi \) can be written as

$$\begin{aligned}&\displaystyle \Psi =\Psi ^{(2)}+\Psi ^{(1)}+\Phi , \end{aligned}$$
$$\begin{aligned}&\displaystyle \text {where } \Psi ^{(1)}:=A^{\alpha _1}_{i_1}y^{i_1}_{\alpha _1},\qquad \Psi ^{(2)}:=\frac{1}{2}A^{\alpha _1\alpha _2}_{i_1i_2}y^{i_1}_{\alpha _1}y^{i_2}_{\alpha _2}. \end{aligned}$$

Here, \(\Phi ,\mathbf {A},{{\mathbb {A}}}\) are functions of \(\varvec{x}\) and \(\varvec{y}\) of class \(\mathcal {C}^2\). Clearly \(A^{\alpha _1\alpha _2}_{i_1i_2}=A^{\alpha _2\alpha _1}_{i_2i_1}.\) Then (1.34) becomes

$$\begin{aligned} \sum _{n=1}^2{\mathscr {E}}_k(\Psi ^{(n)})+{\mathscr {E}}_k(\Phi )\equiv 0. \end{aligned}$$

On substituting from (1.33), we see that this represents a set of PDEs for \(\mathbf {A}\) and \({{\mathbb {A}}}\), whose precise form we have to determine.

From (A.2), we have \(\frac{{\partial }\Psi ^{(2)}}{{\partial } y^k_{{\gamma }}}=A^{{{\gamma }}\alpha _2}_{ki_2}y^{i_2}_{\alpha _2}\qquad \frac{{\partial }\Psi ^{(1)}}{{\partial } y^k_{{\gamma }}}=A^{{\gamma }}_k.\) Therefore,

$$\begin{aligned} {\mathscr {E}}_k(\Psi ^{(2)})&=\frac{\partial }{{\partial } x^{{\gamma }}}\left( A^{{{\gamma }}\alpha _2}_{ki_2}\right) y^{i_2}_{\alpha _2}+\left[ \frac{\partial }{{\partial } y^{i_1}}\left( A^{\alpha _1\alpha _2}_{ki_2}\right) -\frac{1}{2}\frac{\partial }{{\partial } y^k}\left( A^{\alpha _1\alpha _2}_{i_1i_2}\right) \right] y^{i_1}_{\alpha _1}y^{i_2}_{\alpha _2}+A^{{{\gamma }}{{\beta }}}_{kj}\frac{{\partial }^2y^j}{{\partial } x^{{\beta }}{\partial } x^{{\gamma }}},\\ {\mathscr {E}}_k(\Psi ^{(1)})&=\frac{\partial }{{\partial } x^{{\gamma }}}\left( A^{{\gamma }}_k\right) +\left[ \frac{\partial }{{\partial } y^{i_1}}\left( A^{\alpha _1}_k\right) -\frac{\partial }{{\partial } y^k}\left( A^{\alpha _1}_{i_1}\right) \right] y^{i_1}_{\alpha _1}. \end{aligned}$$

From (A.3), we get

$$\begin{aligned}&\left[ \frac{\partial }{{\partial } x^{{\gamma }}}\left( A^{{{\gamma }}\alpha _2}_{ki_2}\right) +\frac{\partial }{{\partial } y^{i_2}}\left( A^{\alpha _2}_k\right) -\frac{\partial }{{\partial } y^k}\left( A^{\alpha _2}_{i_2}\right) \right] y^{i_2}_{\alpha _2}+\left[ \frac{\partial }{{\partial } y^{i_1}}\left( A^{\alpha _1\alpha _2}_{ki_2}\right) -\frac{1}{2}\frac{\partial }{{\partial } y^k}\left( A^{\alpha _1\alpha _2}_{i_1i_2}\right) \right] y^{i_1}_{\alpha _1}y^{i_2}_{\alpha _2}\nonumber \\&\quad +A^{{{\gamma }}{{\beta }}}_{kj}\frac{{\partial }^2y^j}{{\partial } x^{{\beta }}{\partial } x^{{\gamma }}}+\left[ \frac{\partial }{{\partial } x^{{\gamma }}}\left( A^{{\gamma }}_k\right) -\frac{{\partial }\Phi }{{\partial } y^k}\right] =0. \end{aligned}$$

This holds for all values of \(y^i_\alpha \) and \(\frac{{\partial }^2y^j}{{\partial } x^{{\beta }}{\partial } x^{{\gamma }}}\). So every term is zero separately. In particular,

$$\begin{aligned} \left[ \frac{\partial }{{\partial } x^{{\gamma }}}\left( A^{{{\gamma }}\alpha _2}_{ki_2}\right) +\frac{\partial }{{\partial } y^{i_2}}\left( A^{\alpha _2}_k\right) -\frac{\partial }{{\partial } y^k}\left( A^{\alpha _2}_{i_2}\right) \right] =0,\qquad \left[ \frac{\partial }{{\partial } x^{{\gamma }}}\left( A^{{\gamma }}_k\right) -\frac{{\partial }\Phi }{{\partial } y^k}\right] =0. \end{aligned}$$

for \(k=1,\dots ,N.\)

Let us consider 3 functions (of \(\mathcal {C}^2\) smoothness) \(\{S^\alpha (\varvec{x},\varvec{y})\}_{\alpha =1,2,3}\) and define

$$\begin{aligned} S^\alpha _i:=\frac{{\partial } S^\alpha }{{\partial } y^i}; \qquad S^\alpha _{|\beta }:=\frac{{\partial } S^\alpha }{{\partial } x^\beta }. \end{aligned}$$

Recall \(\mathfrak {D}\)s as defined by (1.36). Therefore,

$$\begin{aligned} \frac{{\partial } \mathfrak {D}(\alpha _1;i_1)}{{\partial } x^{\alpha _1}}&= \begin{vmatrix} S^{\alpha _1}_{i_1|\alpha _1}&S^{\alpha _2}_{i_1|\alpha _1}\\ S^{\alpha _1}_{|\alpha _2}&S^{\alpha _2}_{|\alpha _2} \end{vmatrix}+ \begin{vmatrix} S^{\alpha _1}_{i_1}&S^{\alpha _2}_{i_1}\\ S^{\alpha _1}_{|\alpha _1\alpha _2}&S^{\alpha _2}_{|\alpha _1\alpha _2} \end{vmatrix}\nonumber \\&\quad (\alpha _1 and \alpha _2\text { both being dummy indices, the latter determinant is }0)\end{aligned}$$
$$\begin{aligned}&=\frac{\partial }{{\partial } y^{i_1}} \begin{vmatrix} S^{\alpha _1}_{|\alpha _1}&S^{\alpha _2}_{|\alpha _1}\\ S^{\alpha _1}_{|\alpha _2}&S^{\alpha _2}_{|\alpha _2} \end{vmatrix} -\begin{vmatrix} S^{\alpha _1}_{|\alpha _1}&S^{\alpha _2}_{|\alpha _1}\\ S^{\alpha _1}_{i_1|\alpha _2}&S^{\alpha _2}_{i_1|\alpha _2} \end{vmatrix}=\frac{{\partial } \mathfrak {D}(0;0)}{{\partial } y^{i_1}}-\frac{{\partial } \mathfrak {D}(\alpha _1;i_1)}{{\partial } x^{\alpha _1}}\nonumber \\ \implies 2\frac{{\partial } \mathfrak {D}(\alpha _1;i_1)}{{\partial } x^{\alpha _1}}&=\frac{{\partial } \mathfrak {D}(0;0)}{{\partial } y^{i_1}}. \end{aligned}$$


$$\begin{aligned} \frac{{\partial } \mathfrak {D}(\alpha _1,\alpha _2;i_1,i_2)}{{\partial } x^{\alpha _2}}&= \begin{vmatrix} S^{\alpha _1}_{i_1|\alpha _2}&S^{\alpha _2}_{i_1|\alpha _2}\\ S^{\alpha _1}_{i_2}&S^{\alpha _2}_{i_2} \end{vmatrix} + \begin{vmatrix} S^{\alpha _1}_{i_1}&S^{\alpha _2}_{i_1}\\ S^{\alpha _1}_{i_2|\alpha _2}&S^{\alpha _2}_{i_2|\alpha _2} \end{vmatrix}\\&=\frac{\partial }{{\partial } y^{i_1}} \begin{vmatrix} S^{\alpha _1}_{|\alpha _2}&S^{\alpha _2}_{|\alpha _2}\\ S^{\alpha _1}_{i_2}&S^{\alpha _2}_{i_2} \end{vmatrix} +\frac{\partial }{{\partial } y^{i_2}} \begin{vmatrix} S^{\alpha _1}_{i_1}&S^{\alpha _2}_{i_1}\\ S^{\alpha _1}_{|\alpha _2}&S^{\alpha _2}_{|\alpha _2} \end{vmatrix} - \begin{vmatrix} S^{\alpha _1}_{|\alpha _2}&S^{\alpha _2}_{|\alpha _2}\\ S^{\alpha _1}_{i_1i_2}&S^{\alpha _2}_{i_1i_2} \end{vmatrix} - \begin{vmatrix} S^{\alpha _1}_{i_1i_2}&S^{\alpha _2}_{i_1i_2}\\ S^{\alpha _1}_{|\alpha _2}&S^{\alpha _2}_{|\alpha _2} \end{vmatrix}\\&=-\frac{{\partial } \mathfrak {D}(\alpha _1;i_2)}{{\partial } y^{i_1}}+\frac{{\partial } \mathfrak {D}(\alpha _1;i_1)}{{\partial } y^{i_2}}\\&\quad \text {(the remaining two determinants cancel each other)}. \end{aligned}$$

Comparing this with (A.5)\({}_1\), we get

$$\begin{aligned} A^{\alpha _1\alpha _2}_{i_1i_2}=\mathfrak {D}(\alpha _1,\alpha _2;i_1,i_2),\qquad A^{\alpha _1}_{i_1}=\mathfrak {D}(\alpha _1;i_1). \end{aligned}$$

From (A.7), \(\frac{1}{2}\frac{{\partial } \mathfrak {D}(0;0)}{{\partial } y^{i_1}}=\frac{{\partial } A^{\alpha _1}_{i_1}}{{\partial } x^{\alpha _1}}=\frac{{\partial }\Phi }{{\partial } y^{i_1}}\) by (A.5)\({}_2\), which upon integrating yields

$$\begin{aligned} \Phi =\frac{1}{2}\mathfrak {D}(0;0). \end{aligned}$$

Therefore, reverting back to (A.1) we get (1.35).

Using (A.1), (A.2), (A.4) and (A.8), we can summarize the result as:

Theorem 1

([17], pg 257-258) Any function of the form (1.35) satisfies Euler–Lagrange equation (1.34) identically, where \(\mathcal {C}^2\) functions \(S^\alpha (\varvec{x},\varvec{y})\) are entirely arbitrary.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Basak, N., Sharma, B.L. Null Lagrangians in linear theories of micropolar type and few other generalizations of elasticity. Z. Angew. Math. Phys. 72, 9 (2021).

Download citation


  • Calculus of variations
  • Cosserat continuum
  • Electroelasticity
  • Magnetoelastic
  • Quasicrystals
  • Anisotropic elasticity
  • Phonon–phason coupling field
  • Isotropic material

Mathematics Subject Classification

  • Primary 74A35
  • Secondary 35E20
  • 49S05