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In (mathematical) physics many equations of motion can be interpreted as Euler-Lagrange (EL) equations. The Maxwell equation for electromagnetic (EM) field (say in vacuum and in absence of charges) seems to me quite unusual in comparison to examples known in classical mechanics. This is a system of first order PDE on 6 components of EM field. To get the Lagrangian density, one takes the first pair of the Maxwell equations and deduces from it existence of electromagnetic potential. Substituting the potential into the second pair of Maxwell equations, one gets second order equations for the potential. They can be presented as EL-equations for the potential.

I am wondering if there is a way to present the Maxwell equations as an EL-equation in terms of electromagnetic field only rather than potential.

I think I can prove that this is impossible if one requires in addition that the Lagrangian density is quadratic in fields and their first derivatives and invariant under the Poincare group.

ADDED 1: By the Maxwell equations I mean $$\operatorname{div}\vec B=\operatorname{div} \vec E=0,\, \operatorname{rot}\vec E=-\frac{1}{c}\dot{\vec B},\, \operatorname{rot}\vec B=\frac{1}{c}\dot{\vec E}.$$ Thus I am looking for a Lagrangian depending on $\vec E,\vec B$ as independent fields and their derivatives such that the EL-equation is equivalent to all these equations.

ADDED 2: Let me reformulate my question on the language of some of the answers and comments below. The standard approach to interpret Maxwell equations as EL-equation in electrodynamics is as follows. One selects out of 8 Maxwell equations four equations and declares them to be constrains, and the other four - equations of motions. One considers the variations of the action functional $\int L$ only in the class of electromagnetic fields $(\vec B,\vec E)$ satisfying the constrains. Its extrema recover the four equations of motion. This separation of the Maxwell equations into to halves seems to me to be artifical and different from all other examples I know. I am wondering if there is a way to consider all 8 Maxwell equations on equal footing for this purpose.

In (mathematical) physics many equations of motion can be interpreted as Euler-Lagrange (EL) equations. The Maxwell equation for electromagnetic (EM) field (say in vacuum and in absence of charges) seems to me quite unusual in comparison to examples known in classical mechanics. This is a system of first order PDE on 6 components of EM field. To get the Lagrangian density, one takes the first pair of the Maxwell equations and deduces from it existence of electromagnetic potential. Substituting the potential into the second pair of Maxwell equations, one gets second order equations for the potential. They can be presented as EL-equations for the potential.

I am wondering if there is a way to present the Maxwell equations as an EL-equation in terms of electromagnetic field only rather than potential.

I think I can prove that this is impossible if one requires in addition that the Lagrangian density is quadratic in fields and their first derivatives and invariant under the Poincare group.

ADDED 1: By the Maxwell equations I mean $$\operatorname{div}\vec B=\operatorname{div} \vec E=0,\, \operatorname{rot}\vec E=-\frac{1}{c}\dot{\vec B},\, \operatorname{rot}\vec B=\frac{1}{c}\dot{\vec E}.$$ Thus I am looking for a Lagrangian depending on $\vec E,\vec B$ as independent fields and their derivatives such that the EL-equation is equivalent to all these equations.

ADDED 2: Let me reformulate my question on the language of some of the answers and comments below. The standard approach to interpret Maxwell equations as EL-equation in electrodynamics is as follows. One selects out of 8 Maxwell equations four equations and declares them to be constrains, and the other four - equations of motions. One considers the variations of the action functional $\int L$ only in the class of electromagnetic fields $(\vec B,\vec E)$ satisfying the constrains. Its extrema recover the four equations of motion. This separation of the Maxwell equations into two halves seems to me to be artifical and different from all other examples I know. I am wondering if there is a way to consider all 8 Maxwell equations on equal footing for this purpose.

In (mathematical) physics many equations of motion can be interpreted as Euler-Lagrange (EL) equations. The Maxwell equation for electromagnetic (EM) field (say in vacuum and in absence of charges) seems to me quite unusual in comparison to examples known in classical mechanics. This is a system of first order PDE on 6 components of EM field. To get the Lagrangian density, one takes the first pair of the Maxwell equations and deduces from it existence of electromagnetic potential. Substituting the potential into the second pair of Maxwell equations, one gets second order equations for the potential. They can be presented as EL-equations for the potential.

I am wondering if there is a way to present the Maxwell equations as an EL-equation in terms of electromagnetic field only rather than potential.

I think I can prove that this is impossible if one requires in addition that the Lagrangian density is quadratic in fields and their first derivatives and invariant under the Poincare group.

ADDED: By the Maxwell equations I mean $$\operatorname{div}\vec B=\operatorname{div} \vec E=0,\, \operatorname{rot}\vec E=-\frac{1}{c}\dot{\vec B},\, \operatorname{rot}\vec B=\frac{1}{c}\dot{\vec E}.$$ Thus I am looking for a Lagrangian depending on $\vec E,\vec B$ as independent fields and their derivatives such that the EL-equation is equivalent to all these equations.

In (mathematical) physics many equations of motion can be interpreted as Euler-Lagrange (EL) equations. The Maxwell equation for electromagnetic (EM) field (say in vacuum and in absence of charges) seems to me quite unusual in comparison to examples known in classical mechanics. This is a system of first order PDE on 6 components of EM field. To get the Lagrangian density, one takes the first pair of the Maxwell equations and deduces from it existence of electromagnetic potential. Substituting the potential into the second pair of Maxwell equations, one gets second order equations for the potential. They can be presented as EL-equations for the potential.

I am wondering if there is a way to present the Maxwell equations as an EL-equation in terms of electromagnetic field only rather than potential.

I think I can prove that this is impossible if one requires in addition that the Lagrangian density is quadratic in fields and their first derivatives and invariant under the Poincare group.

ADDED 1: By the Maxwell equations I mean $$\operatorname{div}\vec B=\operatorname{div} \vec E=0,\, \operatorname{rot}\vec E=-\frac{1}{c}\dot{\vec B},\, \operatorname{rot}\vec B=\frac{1}{c}\dot{\vec E}.$$ Thus I am looking for a Lagrangian depending on $\vec E,\vec B$ as independent fields and their derivatives such that the EL-equation is equivalent to all these equations.

ADDED 2: Let me reformulate my question on the language of some of the answers and comments below. The standard approach to interpret Maxwell equations as EL-equation in electrodynamics is as follows. One selects out of 8 Maxwell equations four equations and declares them to be constrains, and the other four - equations of motions. One considers the variations of the action functional $\int L$ only in the class of electromagnetic fields $(\vec B,\vec E)$ satisfying the constrains. Its extrema recover the four equations of motion. This separation of the Maxwell equations into to halves seems to me to be artifical and different from all other examples I know. I am wondering if there is a way to consider all 8 Maxwell equations on equal footing for this purpose.

In (mathematical) physics many equations of motion can be interpreted as Euler-Lagrange (EL) equaitons. The Maxwell equation for electromagnetic (EM) field (say in vacuum and in absence of charges) seems to me quite unusual in comparison to examples known in classical mechanics. This is a system of first order PDE on 6 components of EM field. To get the Lagrangian density, one takes the first pair of the Maxwell equaitons and deduces from it existence of electromagnetic potential. Substituting the potential into the second pair of Maxwell equations, one gets second order equations for the potential. They can be presented as EL-equations for the potential.

I am wondering if there is a way to present the Maxwell equations as an EL-equation in terms of electromagnetic field only rather than potential.

I think I can prove that this is impossible if one requires in addition that the Lagrangian density is quadratic in fields and their first derivatives and invariant under the Poincare group.

ADDED: By the Maxwell equations I mean $$div\vec B=div \vec E=0,\, rot\vec E=-\frac{1}{c}\dot{\vec B},\, rot\vec B=\frac{1}{c}\dot{\vec E}.$$ Thus I am looking for a Lagrnagian depending on $\vec E,\vec B$ as independent fields and their derivatives such that the EL-equation is equivalent to all these equations.

In (mathematical) physics many equations of motion can be interpreted as Euler-Lagrange (EL) equations. The Maxwell equation for electromagnetic (EM) field (say in vacuum and in absence of charges) seems to me quite unusual in comparison to examples known in classical mechanics. This is a system of first order PDE on 6 components of EM field. To get the Lagrangian density, one takes the first pair of the Maxwell equations and deduces from it existence of electromagnetic potential. Substituting the potential into the second pair of Maxwell equations, one gets second order equations for the potential. They can be presented as EL-equations for the potential.

I am wondering if there is a way to present the Maxwell equations as an EL-equation in terms of electromagnetic field only rather than potential.

I think I can prove that this is impossible if one requires in addition that the Lagrangian density is quadratic in fields and their first derivatives and invariant under the Poincare group.

ADDED: By the Maxwell equations I mean $$\operatorname{div}\vec B=\operatorname{div} \vec E=0,\, \operatorname{rot}\vec E=-\frac{1}{c}\dot{\vec B},\, \operatorname{rot}\vec B=\frac{1}{c}\dot{\vec E}.$$ Thus I am looking for a Lagrangian depending on $\vec E,\vec B$ as independent fields and their derivatives such that the EL-equation is equivalent to all these equations.