tokens: The two transonic trajectories with are shown in pink. Green lines show two subsonic trajectories with and and two supersonic trajectories with and . The trajectories that are discontinuous across position are shown in yellow for the same set of constants, , but represent different branches. 3 HAMILTONIAN DESCRIPTION Hamilton’s principle for a conservative system is where the difference of the kinetic and potential energies is the Lagrangian, . This may be generalized for a fluid with internal energy, U (Herivel 1955 ), In non-dimensional units, the Lagrangian for the BP flow is the same as the steady-state equation ( 6 ), where the non-dimensional terms, (log | u | + 2 log | x |), which derive from the dimensional pressure term in equation ( 1 ), are the internal energy. If we define the functional gradient as the directional derivative, for any variation, η, and constant, ε, then the necessary condition of Euler–Lagrange for x to be a critical function of J [ x ] is that the first variation is zero, where the velocity . Since the right-hand side is equivalent to the equation of motion ( 5 ) as can be verified by substitution, then 〈∇ J [ x ]〉 = 0, and the steady-state solution, , is the solution of stationary action toward which the time-dependent solutions evolve. This description is similar to the simple physics problem in which the Lagrangian determines the parabolic trajectory of a particle in a gravitational field and the exact trajectory is determined from the end points as a boundary value problem (BVP) or alternatively from the initial velocities as an IVP. However, the Lagrangian for the BP flow includes the internal energy of the fluid as a function of the density–pressure gradient that is constrained by the conservation equation. Therefore, it is not always possible to find an energy conserving trajectory between two arbitrary velocities. Considered as a BVP, not all boundary conditions are consistent with a steady-state solution. As in Lagrangian mechanics, the energy serves as a Lyapunov function for the stability. The definiteness in sign of the second variation evaluated at the steady state implies at least linear stability (Arnol’d 1965 , 1989 ). Equivalently, the integrand of the second variation, must be definite in sign for every non-zero variation η. Since the middle term with mixed partial derivatives is zero, and This is easily verified for the range x > 0 allowed for BP flows. This implies that the steady-state solution, including all the subsonic, transonic, and supersonic trajectories in the family, is at least linearly stable. The existence of a steady-state solution, a critical point in Lagrangian mechanics, is required for stability. An arbitrary IVP can evolve to the steady state, but the boundary conditions in a BVP must be consistent with the conservation of energy to allow a stable solution. This requirement itself does not determine the steady-state solution or its stability. For example, the derivation of
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