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A variational method is shown to derive the equations describing the propagation of surfaces of weak discontinuity, i.e. of those surfaces across which at least one of the derivatives of the field variables suffer a jump. From the point of view of the calculus of variations this is the case of the broken extremals. The equations are derived from that variational principle which accounts for the non-broken extremals. The variational principle must be formulated at least in the frame of the special relativity.

The fundamental equations of an ideal—entropy current conserving—fluid are obtained from a Hamilton Principle. The Lagrangian is formulated in the usual way. The physical situation, however, requires the introduction of some constraints in the variational principle to restrict the variations of the original field quantities. The Lagrange multipliers entering in such a way are shown to lead to the possibility of vortices in the velocity field.

The classical relativistic equations of motion of a pole particle subjected to scalar fields are investigated. It can be shown that in the case of a free particle the equations of motion admit a rigorous solution which corresponds to the uniform rectilinear motion. The probable existence of the runaway solutions is mentioned. It is shown that in a special case the particle can move uniformly along a straight line even under the action of a constant external field. The variation of the rest mass is studied on hand of this example and it is pointed out that the proper field of the particle gives no contribution to the rest mass. An analogue ofEliezer’s integral theorem is derived for a particle moving in a central scalar field. Finally it is shown in a certain approximation that the particle cannot perform a uniform circular motion under the action of the force derived from the Yukawa potential.