The complete system of divergent equations of the electromagnetic dynamics of an ideal gas

A non-stationary analogue of Chaplygin's equations in one-dimensional gas dynamics

“…Some of these retain their meaning for the (x, t) plane also. It should be emphasised that on page 19 in [12], which was prepared for press and published later than [11], it is stated that the addition to the system of equations of gas dynamics investigated in [11,12], of new relations, which do not contradict it, may extend the system of conservation laws constructed in [11,12]. These relations include the potentiality condition, relations which occur when the number of variables is reduced, etc.…”

“…Equation (2.6) gives an unbounded set of conservation laws in the (x, t) plane. The finite number of conservation laws of gas dynamics in (x, y, z, t) space was constructed earlier in [10][11][12]. Some of these retain their meaning for the (x, t) plane also.…”

“…Thus, in the papers mentioned above [15,16], the construction of an additional conservation law is used to symmetrize the equations of gas dynamics. The problem of additional conservation laws was not discussed in [11,12], but a simple analysis of the divergent equations derived in [11,12] enables us to indicate several linear combinations which vanish.…”

“…Solution (3.9) and system (3.10) corresponding to it as well as the second-order equation (3.11) are fairly new, whereas the conservation law (3.12) in the (x, t) plane in the (% t) plane (the second of Eqs (3.10)) is the well-known law of conservation of energy [2][3][4][5][10][11][12]. Using this case, we note that this law of conservation enables us to reduce the fairly complex problem of constructing the optimum motion of a piston (optimum from the point of view of achieving maximum work) to a one-dimensional variational problem for the closing characteristic and enables us to obtain the conditions of optimality in the form of finite algebraic relations [14].…”