A method is given for obtaining explicitly an infinite number of exact invariants for a physical system described by the coupled set of first‐order hyperbolic partial differential equations
. Temporal and spatial invariants are constructed as integrals of temporally and spatially invariant densities T and X, over appropriate spatial and temporal intervals, respectively. For physical systems the energy and momentum densities are temporal invariant densities. These invariant densities are solutions of the hodograph transformed equations corresponding to (A1). For the case n = 1 every invariant density T satisfies an equation in conservation form: (Tu0)t − (Tu1)x = 0. The methods are applied to the equation
,and a denumerable infinity of invariant densities, each expressible as a polynomial, are calculated in two equivalent cases: the first when (A2) (with α = 1) is expressed in zero diagonal form ut = vx, vt = (1 + εu)ux, where u = yx and v = yt; and the second when (A2) is expressed in diagonal form rt−Φrx=0,st+Φsx=0⁠, where Φ=(12 ε(2+α)(r+s))α/(2+α) and r and s, the Riemann invariants of (A2), are related to u and v. A theorem of Noether is used to construct from the invariant densities continuous transformation groups that leave the action functional invariant. Using the methods of Kruskal we derive the adiabatic invariant for the continuous system (A2) (α = 1) which has nearly periodic solutions. To order ε the adiabatic invariant is identical with one of the exact invariants and gives no new information about the system.
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1966
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