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In Noether's original presentation of her celebrated theorem of 1918, allowances were made for the dependence of the coefficient functions of the differential operator which generated the infinitesimal transformation of the Action Integral upon the derivatives of the dependent variable(s), the so-called generalized, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to only point transformations. In recent decades, this diminution of the power of Noether's Theorem has been partly countered, in particular, in the review of Sarlet and Cantrijn. In this Special Issue, we emphasize the generality of Noether's Theorem in its original form and explore the applicability of even more general coefficient functions by allowing for nonlocal terms. We also look at the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence upon the independent variables.

Symmetry.

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This reprint is dedicated to Professor Delfim F. M. Torres for his 50th birthday. Professor Delfim F. M. Torres is a recognized researcher, particularly in the calculus of variations, optimal control, and mathematical biology. In this reprint, several studies in his research fields are presented.

Symmetry (Physics)

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The Max Born Symposia started in 1991 and have been held once or twice a year in different places in Lower Silesia. The scientific topics of the Symposia are closely related with front-line research subjects in theoretical physics. This volume deals with new concepts of symmetries in the theory of fundamental intersections as well as integrable dynamical systems.

Symmetry (Physics) --- Mathematical physics

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Seeking Symmetry: Finding patterns in human health offers a guide through the overwhelming mass of data generated by contemporary science. Starved for the knowledge that would best help us stay healthy, we are simultaneously glutted with an overload of information about the human body. Amidst ubiquitous talk that patient-centered care and lifestyle changes are the keys to personal health, self-neglect and medical overtreatment nevertheless prevail.The body is rich with symmetries, many of them unknown to us who live in these bodies. Symmetry-seeking reveals certain patterns for understanding the information we have about the body, patterns whose roots lie in embryonic development and in evolution.The book's exploration will guide readers through the parts of their own bodies and introduce tangible, visible examples of symmetry, not only right and left but up and down, male and female, inside and out, as well as symmetries between humans and other species.It presents the symmetries of the body's internal structures that, despite their complexity, are nevertheless simple to understand when viewed with an eye for pattern.Through both words and images, this book will illustrate the most foundational of the principles, structures, and processes that decide how bodies function.A core purpose of the book is to present this knowledge through a lens that makes the information meaningful, by modeling the habit of symmetry-seeking.

Health. --- Symmetry (Biology).

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Whenever systems are governed by continuous chains of causes and effects, their behavior exhibits the consequences of dynamical symmetries, many of them far from obvious. ""Dynamical Symmetry"" introduces the reader to Sophus Lie's discoveries of the connections between differential equations and continuous groups that underlie this observation. It develops and applies the mathematical relations between dynamics and geometry that result. Systematic methods for uncovering dynamical symmetries are described, and put to use. Much material in the book is new and some has only recently appeared in

Symmetry (Physics) --- Hamiltonian systems.