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The many-worlds interpretation has usually been regarded as immune to nonlocality, and similar claims have been made for the consistent-histories interpretation (1-3). However, for a thought experiment of the Hardy type, the argument for nonlocality in the usual interpretation can be extended to both these other interpretations.

It is shown that nonlocality gives rise to an undecidable proposition, meaning it cannot be proved true nor proved false from the usual assumptions, but is independent of them. A variation on the usual thought experiment is considered in which the observers are timelike separated, but the nonlocality fails to become a “precognition” effect because of this independence result.

In the previous Paper I, the Variational Principle has been presented as a principle of natural transformation between conserved quantities and infinitesimal in variances, which preserves the geometrical character of the various standard objects under point transformations. So as to establish the theory as easily as possible, Paper I (2) confined itself to the simple case of autonomous dynamical systems, and their 2n dimensional space. The aim of the present paper is to extend the work to the 2n + 1 dimensional theory, in which time is included as an extra variable, and time dependent systems can be handled. It will be seen that within a suitable framework the extension is straightforward, and there are some rather more satisfactory features of the 2n + 1 dimensional theory - even for autonomous systems when included in the new formalism - than in the previous 2n dimensional theory. The exposition will now be rather concise; readers are requested to refer to Paper I for fuller discussion, particularly of the underlying ideas.

When a system of differential equations is derivable from a variational principle, an outstanding consequence is the connection which ensues between symmetries and conservation laws. The idea behind the present work is to use this consequence as a new characterization of the variational principle itself. For general systems of equations there is a rather weak form of the connection, as will be demonstrated; but it has highly arbitrary and non-invariant features, and, for instance, there is not a formula associating a particular infinitesimal invariance with a particular constant of the motion. In terms of the modern technical concept which will be employed, the transformation is not natural. However, if the system is derivable from a variational principle, there follows a preferred transformation; and it has certain invariance properties which accord with the concept of naturality.

An allegation of a flaw in the theory developed by Dirac for the Hamiltonian formation of singular dynamical systems is denied. It is shown that the Dirac theory is perfectly correct, and adequate as a Hamiltonian theory-but also there is a gain to be had by looking at the structural features involved in the original Lagrangian picture, including those 'canonical' features necessary for the purpose of quantisation; it is indicated how the controversy about first-class secondary constraints may be examined in this way.

A new approach is suggested to the teaching of analytical dynamics, in which the Lagrangian and Hamiltonian pictures are presented simultaneously. It is used to explain the connection between symmetries and conservation laws, and the passage to quantisation.

Bondi’s Newtonian Poynting Vector is extended to other ‘Bondi-type’ conserved quantities, which are also derived by variational methods. By means of the technique of Seliger & Whitham it is shown that they are the expressions which arise most naturally in contexts concerned with the equations of motion. The pseudo-tensor derived from the Lagrangian R √(— g) is the corresponding expression in General Relativity; direct calculation verifies that it tends to Bondi’s vector in the Newtonian limit.

Some general considerations are applied to the Lenz vector, and a recent work on the subject is discussed.