Energy characteristics of oscillatory systems with internal (Hidden) degrees of freedom

The general formulas are derived for calculating the kinetic and potential energies and other energy characteristics of linear oscillatory systems, a portion of the degrees of freedom of which are internal, i.e., inaccessible for direct measurements and/or excluded from consideration. The energy characteristics are expressed by means of parameters pertaining only to the part of accessible (or selected for consideration) degrees of freedom. The formulas are derived based on the use of certain novel properties of the so-called Shur matrix complement. The obtained formulas are applicable for calculating the energy characteristics of the elements of acoustic metamaterials by means of their effective inertial-elastic parameters. The main focus is the case of negative values of these parameters. Proceeding from the nonnegativity of the kinetic and potential energies of mechanical structures, limitations on the character of the frequency dependences of the effective parameters are formulated.