Symmetries and invariances in classical physics

Symmetry, intended as invariance with respect to a transformation (more precisely, with respect to a transformation group), has acquired more and more importance in modern physics. This Chapter explores in 8 Sections the meaning, application and interpretation of symmetry in classical physics. This is done both in general, and with attention to specific topics. The general topics include illustration of the distinctions between symmetries of objects and of laws, and between symmetry principles and symmetry arguments (such as Curie's principle), and reviewing the meaning and various types of symmetry that may be found in classical physics, along with different interpretative strategies that may be adopted. Specific topics discussed include the historical path by which group theory entered classical physics, transformation theory in classical mechanics, the relativity principle in Einstein's Special Theory of Relativity, general covariance in his General Theory of Relativity, and Noether's theorems. In bringing these diverse materials together in a single Chapter, we display the pervasive and powerful influence of symmetry in classical physics, and offer a possible framework for the further philosophical investigation of this topic

Lagrangian Formalism Over Graded Algebras

This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary first step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green's formula, the relation between integral and differential forms, conservation laws, Euler operator, Noether's theorem is considered.Comment: 26 pages, AMS-TeX 2.1, to appear in J. Geom. Phys. (resubmitted because of a TeX-error

A Cause Among Causes? God Acting in the Natural World

Contemporary debates on divine action tend to focus on finding a space in nature where there would be no natural causes, where nature offers indeterminacy, openness, and potentiality, to place God’s action. These places are found through the natural sciences, in particular quantum mechanics. God’s action is then located in those ontological ”causal-gaps’ offered by certain interpretations of quantum mechanics. In this view, God would determine what is left underdetermined in nature without disrupting the laws of nature. These contemporary proposals evidence at least two unexamined assumptions, which frame the discussion in such a way that they portray God as acting as a secondary cause or a ”cause among causes’. God is somewhat required to act within these ”gaps’, binding God to the laws of nature, and placing God’s action at the level of secondary causes. I suggest that understanding God’s action, following Thomas Aquinas, in terms of primary and secondary causation could help dissolve this difficulty. Aquinas moves away from this objection by suggesting to speak of an analogical notion of cause, allowing for an analogical understanding of God’s causality in nature. With a radically different understanding of the interplay between secondary causes and God, Aquinas manages to avoid conceiving God as a cause among causes, keeping the distinctive transcendent character of God’s causality safe from objections

Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics

I extract some philosophical morals from some aspects of Lagrangian mechanics. (A companion paper will present similar morals from Hamiltonian mechanics and Hamilton-Jacobi theory.) One main moral concerns methodology: Lagrangian mechanics provides a level of description of phenomena which has been largely ignored by philosophers, since it falls between their accustomed levels--``laws of nature'' and ``models''. Another main moral concerns ontology: the ontology of Lagrangian mechanics is both more subtle and more problematic than philosophers often realize. The treatment of Lagrangian mechanics provides an introduction to the subject for philosophers, and is technically elementary. In particular, it is confined to systems with a finite number of degrees of freedom, and for the most part eschews modern geometry. But it includes a presentation of Routhian reduction and of Noether's ``first theorem''.Comment: 106 pages, no figure

Perspectives in Fundamental Physics in Space

We discuss the fundamental principles underlying the current physical theories and the prospects of further improving their knowledge through experiments in space.Comment: Gravitational waves, gravitomagnetism, Equivalence Principle, Antimatter, Pioneer Anomaly, Lorentz invariance. To appear in IAA - Acta Astronautica Journal (2006