"The human body is a machine whose movements are directed by the soul," wrote René Descartes in the early seventeenth century. The intrinsic mechanisms of this machine gradually became clear through the hard work of Renaissance scientists. Leonardo da Vinci is one such scientist from this period of enlightenment. In pursuit of knowledge, Leonardo dissected the bodies of more than 30 men and women. He sawed the bones lengthwise, to see their internal structure; he sawed the skull, cut through the vertebrae, and showed the spinal cord. In the process, he took extensive notes and made carefully detailed sketches. His drawings differentiated muscles that run across several joints from those muscles that act on a single joint. "Nature has made all the muscles appertaining to the motion of the toes attached to the bone of the leg and not to that of the thigh," wrote Leonardo in 1504 next to one of his sketches of the lower extremity, "because when the knee joint is flexed, if attached to the bone of the thigh, these muscles would be bound under the knee joint and would not be able to serve the toes. The same occurs in the hand owing to the flexion of the elbow."

Another Renaissance scholar who made fundamental contributions to the physiology of movement is Giovanni Alfonso Borelli. Born in 1604 in Naples, Borelli was a well-respected mathematician. While teaching at the University of Pisa, he collaborated with the faculty of theoretical medicine in the study of movement. Borelli showed that muscles and bones formed a system of levers. He showed that during some physical activity the hip and the knee transmit forces that are several times greater than the body weight. He spent many years trying to secure funding for the publication of his masterpiece On the Movement of Animals. Borelli died in 1679, a few weeks after Queen Catherine of Sweden agreed to pay for the publication costs of the book. The first volume of On the Movement of Animals was published the following year.

The advances in the understanding of human body structure and its relation to movement were soon followed by the formulation of nature's laws of motion. In his groundbreaking book Philosophie Naturalis Principia

Mathematica, published in 1687, Sir Isaac Newton presented these laws in mathematical language. The laws of motion can be summarized as follows: A body in our universe is subjected to a multitude of forces exerted by other bodies. The forces exchanged between any two bodies are equal in magnitude but opposite in direction. When the forces acting on a body balance each other, the body either remains at rest or, if it were in motion, moves with constant velocity. Otherwise, the body accelerates in the direction of the net unbalanced force.

Newton's contributions to mechanics were built on the wealth of knowledge accumulated by others. In this regard, perhaps the most critical advances were made by Galileo Galilei. Born in Italy on February 15, 1564, Galileo became fascinated with mathematics while studying medicine at the University of Pisa. At the university, he was perceived as an arrogant young man. He made many enemies with his defiant attitude toward the Aristotelian dogma and had to leave the university for financial reasons without receiving a degree. Galileo recognized early on the importance of experiments for advancing science. He observed that, for small oscillations of a pendulum, the period of oscillation was independent of the amplitude of oscillation. This discovery paved the way for making mechanical clocks. One of his stellar contributions to mechanics is the law of free fall. Published first in his 1638 book Discorsi, the law states that in a free fall distances from rest are proportional to the square of elapsed times from rest. Although Galileo found recognition and respect in his lifetime, he was nonetheless sentenced to prison at the age of 70 by the Catholic Church for having held and taught the Copernican doctrine that the Earth revolves around the Sun. He died while under house arrest.

Newton's laws were written for so-called particles, however large they may be. A particle is an idealized body for which the velocity is uniform within the body. In the eighteenth century, Leonhard Euler, Joseph-Louis Lagrange, and others generalized these laws to the study of solid bodies and systems of particles. Euler was the first to assign the same gravitational force to a body whether at rest or in motion. In 1760, his work Tho-ria Motus Corporum Solidurum seu Rigidorum described a solid object's resistance to changes in the rate of rotation. A few years later, in 1781, Charles-Augustin de Coulomb formulated the law of friction between two bodies: "In order to draw a weight along a horizontal plane it is necessary to deploy a force proportional to the weight ... ." Coulomb went on to discover one of the most important formulas in physics, that the force between two electrical charges is inversely proportional to the square of the distance between them. Analytical developments on solid mechanics continued with the publication in 1788 of Lagrange's elegant work Mechanic Analytique.

The foundation of classical mechanics set the stage for further studies of human and animal motion. "It seems that, as far as its physique is concerned, an animal may be considered as an assembly of particles sepa rated by more or less compressed springs," wrote Lazare Carnot in 1803. In the 1880s, Eadweard Muybridge in America and Ettiene-Jules Marey in France established the foundation of motion analysis. They took sequential photographs of athletes and horses during physical activity to gain insights into movement mechanics. Today, motion analysis finds particular use in physical education, professional sports, and medical diagnostics. Recent research suggests that the video recording of crawling infants may be used to diagnose autism at an early stage.

The sequential photography allows for the evaluation of velocities and accelerations of body segments. The analysis of forces involved in movement is much more challenging, however, because of the difficult mathematics of classical mechanics. To illustrate the point, scientists were intrigued in the nineteenth century about the righting movements of a freely falling cat. How does a falling cat turn over and fall on its feet? M. Marey and M. Guyou addressed the issue in separate papers published in Paris in 1894. About 40 years later, in 1935, G.G.J. Rademaker and J.W.G. Ter Braak came up with a mathematical model that captured the full turnover of the cat during a fall. The model was refined in 1969 by T.R. Kane and M.P. Schmer so that as observed in the motion of the falling cat the predicted backward bending would be much smaller than forward bending. The mechanism presented by Kane and Schmer is simple; it consists of two identical axisymmetric bodies that are linked together at one end. These bodies can bend relative to each other but do not twist. Space scientists found the model useful in teaching astronauts how to move with catlike ease in low gravity.

Although the mechanical model of a falling cat is simple conceptually, its mathematical formulation and subsequent solution are quite challenging. Since the development of the falling cat model, computational advances have made it easier to solve the differential equations of classical mechanics. Currently, there are a number of powerful software packages for solving multibody problems. Video recording is used to quantify complex modes of movement. Present technology also allows for the measurement of contact forces and the evaluation of the degree of activation of muscle groups associated with motion. Nowadays, the data obtained on biomechanics of movement can be overwhelming. A valid interpretation of the data requires an in-depth understanding of the laws of motion and the complex interplay between mechanics and human body structure. The main goal of this book is to present the principles of classical mechanics using case studies involving human movement. Unlike nonliving objects, humans and animals have the capacity to initiate movement and to modify motion through changes of shape. This capability makes the mechanics of human and animal movement all the more exciting.

I believe that Human Body Dynamics will stimulate the interests of engineering students in biomechanics. Quantitative studies of human movement bring to light the healthcare-related issues facing classical mechan ics in the twenty-first century. There are already a number of outstanding statics and dynamics books written for engineering students. In recent years, with each revision, these books have incorporated more examples, more problems, and more colored photographs and figures, a few of which touch on the mechanics of human movement. Nevertheless, the focus of these books remains almost exclusively on the mechanics of man-made structures. It is my hope that Human Body Dynamics exposes the reader not only to the principles of classical mechanics but also to the fascinating interplay between mechanics and human body structure.

The book assumes a background in calculus and physics. Vector algebra and vector differentiation are introduced in the text and are used to describe the motion of objects. Advanced topics such as three-dimensional motion mechanics are treated in some depth. Whenever possible, the analysis is presented graphically using schematic diagrams and software-created sequences of human movement in an athletic event or a dance performance. Each chapter contains illustrative examples and problem sets. I have spent long days in the library reading scientific journals on biomechanics, sports biomechanics, orthopaedics, and physical therapy so that I could conceive realistic examples for this book. The references included provide a list of sources that I used in the preparation of the text. The book contains mechanical analysis of dancing steps in classical ballet, jumping, running, kicking, throwing, weight lifting, pole vaulting, and three-dimensional diving. Also included are examples on crash mechanics, orthopaedic techniques, limb-lengthening, and overuse injuries associated with running.

Although the emphasis is on rigid body mechanics and human motion, the book delves into other fundamental topics of mechanics such as de-formability, internal stresses, and constitutive equations. If Human Body Dynamics is used as a textbook for a graduate-level course, I would recommend that student projects on sports biomechanics and orthopaedic engineering become an integral part of the course. The references cited at the end of the text provide a useful guide to the wealth of information on the biomechanics of movement.

Human Body Dynamics should be of great interest to orthopaedic surgeons, physical therapists, and professionals and graduate students in sports medicine, movement science, and athletics. They will find in this book concise definitions of terms such as linear momentum and angular velocity and their use in the study of human movement.

I wish to acknowledge my gratitude to all authors on whose work I have drawn. My colleagues and students at The Catholic University of America helped me refine my teaching skills in biomechanics. Professor Van Mow provided me with generous resources during my sabbatical at Columbia University where I prepared most of the text. I am deeply indebted to Professor H. Bulent Atabek of The Catholic University of America for his careful reading of the manuscript. Professor Atabek corrected countless equations and figures and provided valuable input to the contents of the manuscript. My teachers, Professors Maciej P. Bieniek and Frank L. DiMaggio of Columbia University, also spent considerable time reviewing the manuscript. I am very grateful to them for their corrections and constructive suggestions. Dr. Rukmini Rao Mirotznik enriched the text with her beautiful sketches and sublime figures. Barbara A. Chernow and her associates contributed to the book with careful editing and outstanding production. Finally, my thanks goes to Dr. Robin Smith and his associates at Springer-Verlag for bringing this book to life.

Washington, D.C.

Aydin Tözeren

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Getting Started With Dumbbells

Getting Started With Dumbbells

The use of dumbbells gives you a much more comprehensive strengthening effect because the workout engages your stabilizer muscles, in addition to the muscle you may be pin-pointing. Without all of the belts and artificial stabilizers of a machine, you also engage your core muscles, which are your body's natural stabilizers.

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