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Reports of the Academy of Sciences of the USSR
- Volume 154, No. 2
PHYSICS
K. P. STANYUKOVICH
THE LAGRANGIAN OF A CONTINUOUS MEDIUM IN RIEMANNIAN SPACE
(Presented by Academician N. N. Bogolyubov on 12 IX 1963)
The Hamilton–Jacobi equation for a material point in Riemannian space (in the general theory of relativity), as is well known, has the form \((^{1a}, \S 85)\)
\[ g^{ik}\frac{\partial s^*}{\partial x^i}\frac{\partial s^*}{\partial x^k} = - m^2 c^2 = -\frac{\bar E^2}{c^2}, \tag{1} \]
where \(\bar E\) is the energy of the particle; \(s^*\) is the action;
\[ \frac{\partial s^*}{\partial x^i}=p_i=mcu_i \tag{2} \]
is the 4-momentum of the particle;
\[ \frac{\partial s^*}{\partial x^\alpha} = p_\alpha = \frac{m a_\alpha}{\sqrt{1-a^2/c^2}} = \frac{m a_\alpha}{\theta} \tag{3} \]
is the 3-momentum;
\[ c\partial s^*/\partial x^0=E^* \tag{4} \]
is the total energy of the particle.
In the case of a continuous medium, the energy attributable to one particle \((^2)\),
\[ \bar E=(\partial e/\partial n)_\sigma=m w, \tag{5} \]
where \(w\) is the heat content of a unit of rest mass (including internal energy), \(\sigma\) is entropy, and \(n\) is the density of the number of particles.
In this case equation (1) can be written in the form
\[ g^{ik}\frac{\partial s^*}{\partial x^i}\frac{\partial s^*}{\partial x^k} = - m^2 \frac{w^2}{c^2}. \tag{6} \]
Introducing actions calculated per unit mass, \(s=s^*/m\), we finally write equation (6), which is the Hamilton–Jacobi equation for a continuous medium, in the form
\[ g^{ik}\frac{\partial s}{\partial x^i}\frac{\partial s}{\partial x^k} + \frac{w^2}{c^2} = 0, \tag{7} \]
with
\[ c\,\partial s/\partial x^i=w u_i. \tag{8} \]
The Lagrangian in the mechanics of continuous media is the pressure \((^{3,4})\)
\[ p=L=(w-E)/v, \tag{9} \]
where \(v=1/mn\) is the specific volume.
From (7) and (9) we find that
\[ L=p=\frac{1}{v}\left[-E+ic\sqrt{g^{ik}s_i s_k}\right], \tag{10} \]
where, to shorten the calculations, \(s_i=\partial s/\partial x^i\) has been introduced.
First of all, let us find the energy–momentum tensor for the Lagrangian (10). As is known (([16]), § 94),
\[ \frac{\sqrt{-g}}{2}\,T_{ik} = -\frac{\partial(\sqrt{-g}L)}{\partial g^{ik}} + \frac{\partial}{\partial x^l} \frac{\partial(\sqrt{-g}L)}{\partial g^{ik}_{/\,\partial x^l}} +\cdots \tag{11} \]
Here the quantities \(s_i\) are regarded as constant.
If one takes into account that
\[ dE = T\,d\sigma - p\,dv \tag{12} \]
and that for adiabatic processes \(d\sigma=0\), then the calculations give
\[ T_{ik}=(p+\varepsilon)u_i u_k+g_{ik}p . \tag{13} \]
The same expression can be obtained if one uses another variational equation of Lagrange:
\[ \sqrt{-g}\,T_i^{\,k} = \sqrt{-g}\,L\delta_i^k - \frac{\partial s}{\partial x^i} \frac{\partial(\sqrt{-g}L)}{\partial\,\partial s/\partial x^k}. \tag{14} \]
Here the quantities \(g^{ik}\) are regarded as constant.
The field equations have the form
\[ \frac{\partial(\sqrt{-g}L)}{\partial s} - \frac{\partial}{\partial x^k} \frac{\partial(\sqrt{-g}L)}{\partial s_k} =0 . \tag{15} \]
The calculations give
\[ \frac{\partial}{\partial x^k} \left[ \sqrt{-g}\,\frac{c g^{ik}s_i}{v w} \right] = \frac{\partial}{\partial x^k} \left[ \sqrt{-g}\,\frac{u_k}{v} \right] =0, \tag{16} \]
which is the equation of continuity. The quantities \(\dot{s}\) and \(1/vw=\delta\) are canonically conjugate.
For a medium obeying the equation of state
\[ p v^k = A=\mathrm{const}, \tag{17} \]
\[ w=\alpha c^2+\frac{k}{k-1}\,A^{1/k}p^{(k-1)/k}, \tag{18} \]
where \(\alpha=1,0\), respectively, for the relativistic and ultrarelativistic cases. In this case
\[ L=\mathrm{const}\,\left[i\sqrt{g^{ik}s_i s_k}-\alpha c\right]^{k/(k-1)} . \tag{19} \]
(Equation (11) again gives the correct field tensor, which is quite natural.)
The field equations lead to an important equation, convenient for solution:
\[ g^{kr}s_r s_l \left[ g^{il}s_{ik} + \frac{1}{2}s_i\frac{\partial g^{il}}{\partial x^k} \right] \left[ \frac{2-k}{k-1}\sqrt{-g^{mn}s_m s_n}+\alpha c \right] + \]
\[ + \left[ g^{kl}s_{kl} + s_l\frac{\partial g^{kl}}{\partial x^k} \right] g^{ir}s_i s_r \left[ \sqrt{-g^{mn}s_m s_n}-\alpha c \right] + \]
\[ + \frac{\partial \ln\sqrt{-g}}{\partial x^k} s_r s_i s_m g^{kr}g^{im} \left[ \sqrt{-g^{mn}s_m s_n}-\alpha c \right] =0. \tag{20} \]
The particular case of motion of an ultrarelativistic gas \((\alpha>0)\) is easily investigated:
\[ (s_i s^i)^{\frac{2-k}{2(k-1)}}\,s^k \frac{\partial \ln\sqrt{-g}}{\partial x^k} + \frac{\partial}{\partial x^k} \left[ s^k(s_i s^i)^{\frac{2-k}{2(k-1)}} \right] =0. \tag{21} \]
In the particular, but most important and natural case \(k=4/3\), then
\[ s_j s^j s^k \frac{\partial \ln\sqrt{-g}}{\partial x^k} + \frac{\partial}{\partial x^k} (s_i s^i s^k) =0. \tag{22} \]
For \(k=2\) we have \(s^k \dfrac{\partial \ln \sqrt{-g}}{\partial x^k}+\dfrac{\partial s^k}{\partial x^k}=0\), or
\[ \frac{\partial\left(\sqrt{-g}\,s^k\right)}{\partial x^k}=0. \tag{23} \]
The results obtained are easily generalized to the case of motion of charged particles of a continuous medium in an electromagnetic field. In this case, as is well known, for a particle \(\left((1^a), \S 87\right)\)
\[ \frac{\partial s^*}{\partial x^i}=mcu_i+\frac{e}{c}A_i,\quad \left(\frac{\partial s^*}{\partial x^i}-\frac{e}{c}A_i\right) \left(\frac{\partial s^*}{\partial x^k}-\frac{e}{c}A_k\right)g^{ik}+m^2c^2=0, \tag{24} \]
where \(A_i\) are the components of the electromagnetic-field vector.
For a continuous medium, analogously, we shall have
\[ cs_i=c\,\frac{\partial s}{\partial x^i}=\omega u_i+\frac{e}{m}A_i, \tag{25} \]
where \(m\) is the total mass of the particle. Further we have
\[ \frac{\omega^2}{c^2}=-\left(s_i-\frac{e}{mc}A_i\right) \left(s^i-\frac{e}{mc}A^i\right). \tag{26} \]
It is now easy to verify that all the equations derived above are generalized to the case of an electromagnetic field by the simple replacement in them
\[ s_i\to \hat{s}_i=s_i-\frac{e}{mc}A_i. \tag{27} \]
Let us verify the relations obtained.
The conservation laws for matter together with the electromagnetic field are given by the equations
\[ \hat{T}^{k}_{i;k}=\left(T^k_i+\overline{T}^{\,k}_i\right)_{;k}=0, \tag{28} \]
where
\[ \overline{T}^{\,k}_i=\frac{1}{4\pi} \left[F_{il}F^{kl}-\frac{1}{4}\delta^k_i F_{lm}F^{lm}\right] \tag{29} \]
is the energy-momentum tensor of the electromagnetic field.
Write (28) in the form:
\[ T^k_{i;k}=-\overline{T}^{\,k}_{i;k}=f_i, \tag{30} \]
where \(f_i\) is the 4-force of interaction of matter with the electromagnetic field. We compute
\[ T^k_{i;k}= \frac{1}{\sqrt{-g}}\, \frac{\partial\left(\sqrt{-g}\,T^k_i\right)}{\partial x^k} -\frac{T^{kl}}{2}\frac{\partial g_{kl}}{\partial x^i} =f_i. \tag{31} \]
Multiplying scalarly by \(u^i\) and using the thermodynamic condition
\[ d\omega=v\,dp+T\,d\sigma, \tag{32} \]
we find
\[ u^k\left[ \frac{\partial(\omega u_i)}{\partial x^k} -\frac{\partial(\omega u_k)}{\partial x^i} \right] =v f_i+T\,\frac{\partial\sigma}{\partial x^i}. \tag{33} \]
Let us now compute \(\overline{T}^{\,k}_{i;k}=-f_i\). Simple calculations show that
\[ \overline{T}^{\,k}_{i;k}=-f_i=-\frac{1}{c}F_{ik}j^k, \]
where
\[ F_{ik}=\frac{\partial A_k}{\partial x^i}-\frac{\partial A_i}{\partial x^k}. \]
The components of the electromagnetic field and of the current vector are related by Maxwell’s equations
\[ \frac{\partial F_{ik}}{\partial x^l} + \frac{\partial F_{li}}{\partial x^k} + \frac{\partial F_{kl}}{\partial x^i} =0, \qquad \frac{4\pi}{c}j^i = \frac{1}{\sqrt{-g}}\, \frac{\partial\left(\sqrt{-g}F^{ik}\right)}{\partial x^k}, \]
where the obvious continuity equation for the current holds,
\(\partial(\sqrt{-g}j^i)/\partial x^i=0\). In this case (33) takes the form
\[ \frac{d(wu_i)}{ds} + \frac{\partial w}{\partial x^i} - \frac{w}{2}u^k u^l \frac{\partial g_{kl}}{\partial x^i} = -\frac{v}{c}F_{ik}j^k + T\frac{\partial \sigma}{\partial x^i}. \tag{34} \]
If we consider processes without energy dissipation, then
\[ f_i u^i=0;\qquad d\sigma/ds=0;\qquad \sigma=\mathrm{const}. \]
Then
\[ j^k= \frac{\delta c}{\sqrt{-g}}\frac{dx^k}{dx^0} = \delta c u^k\frac{ds}{\sqrt{-g}\,dx^0} = \delta c\,\frac{u^k}{\sqrt{-g}\,u^0}, \]
where \(u^k=dx^k/ds\);
\[ -\frac{v}{c}F_{ik}j^k = -\frac{e}{m}F_{ik}u^k, \]
with \(\delta v/\sqrt{-g}\,u^0=e/m\),
\[ v f_i = -v\overline{T}^{\,k}_{i;k} = -\frac{v}{c}F_{ik}j^k = -\frac{e}{m}F_{ik}u^k = \frac{e}{m}u^k \left[ \frac{\partial A_k}{\partial x^i} - \frac{\partial A_i}{\partial x^k} \right]. \]
In this case (33) takes the form:
\[ u^k \left[ \frac{\partial\left[wu_i+\frac{e}{m}A_i\right]}{\partial x^k} - \frac{\partial\left[wu_k+\frac{e}{m}A_k\right]}{\partial x^i} \right] =0, \tag{35} \]
whence
\[ c\frac{\partial s}{\partial x^i}=wu_i+\frac{e}{m}A_i, \]
and we arrive at the equation (25) found above. Let us now calculate the quantities
\[ \frac{\partial(\sqrt{-g}L)}{\partial x^0} = \frac{\partial(\sqrt{-g}L)}{\partial g^{lm}} \frac{\partial g^{lm}}{\partial x^i} + \sqrt{-g}\, \frac{\partial L}{\partial s_k} \frac{\partial s_k}{\partial x^i}, \]
transforming and using the field equations (15), we find that
\[ \frac{\partial(\sqrt{-g}T_i^{\,k})}{\partial x^k} - \frac{T^{lm}}{2}\frac{\partial g_{lm}}{\partial x^i} = T^{\,k}_{i;k} = 0, \]
i.e. we find in a natural way the equation of conservation of energy-momentum.
In the case of a medium obeying the equation of state
\(p=(k-1)\varepsilon\) (an ultrarelativistic gas), equations (35) and (25), with the replacement
\(w\to c^2(p/p_0)^{\frac{k-1}{k}}\), where \(p_0\) is some initial pressure, will hold also for \(\sigma\ne\mathrm{const}\), but under the condition \(d\sigma/ds=0\), i.e. not only for isentropic, but also for adiabatic motions \((^5)\).
Received
31 VIII 1963
CITED LITERATURE
- L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, a) 2nd ed., Moscow, 1948; b) 3rd ed., Moscow, 1960.
- I. M. Khalatnikov, ZhETF, 27, 529 (1954).
- D. L. Landau, ZhETF, 5, 71 (1941).
- K. P. Stanyukovich, DAN, 145, No. 1 (1962).
- K. P. Stanyukovich, ZhETF, 43, issue 7, 199 (1962).