Analytic dynamics is a branch of physics that describes the effects of forces on physical objects, using Sir Isaac Newton’s theories and calculus. Later scientists, such as Lagrange and Hamilton, generalized the behavior of physical systems with more advanced mathematics, leading to the study of field theories and quantum mechanics. Analytic dynamics uses scalar properties to analyze motion and Lagrangian mechanics forms the basis of most modern theories. Hamiltonian mechanics provides deeper insight into complex dynamical systems and can be applied to both classical and quantum mechanics.
Analytic dynamics is a modern formulation of classical mechanics; is the branch of physics that describes the effects of forces on the motion of physical objects. Sir Isaac Newton’s theories and the calculus he developed to formulate them underlie this field. Later scientists, such as Joseph-Louis Lagrange and William Rowan Hamilton, generalized the behavior of physical systems with the use of more advanced and descriptive mathematics. This work was important in the study of field theories, such as electromagnetism, and the subsequent development of quantum mechanics.
In Newton’s physics, forces act on the motion of bodies as if the objects were infinitely small. Rotating objects were treated as rigid or non-deformable due to their motion. These assumptions produce very accurate approximations of the real world and are particularly amenable to solution by Newton’s calculus. Mathematically, force has been treated as a vector, a quantity having both direction and magnitude. The aim was to calculate, given the initial position and velocity of an object, its position at an arbitrary time in the future.
The methodology of analytical dynamics extends the scope of Newtonian mechanics by becoming a more abstract description. His mathematics does not simply describe the position of objects, but can also apply to general physical systems. Among these are field theories such as those describing electromagnetism and general relativity. To each point of a field can be associated, among other things, a vector or a scalar, a quantity having only magnitude and no direction. In general, analytic dynamics uses two scalar properties, kinetic and potential energy, to analyze motion rather than vectors.
Lagrangian mechanics, introduced in the late 18th century, combined Newton’s second law, the conservation of momentum, with the first law of thermodynamics, the conservation of energy. This formulation of analytic dynamics is powerful and forms the basis of most modern theories. Lagrangian equations reveal all relevant information about a system and can be used to describe everything from Newtonian mechanics to general relativity.
In 1833, a further refinement of analytic dynamics was presented in the form of Hamiltonian mechanics, which differs from the Lagrangian method in the way it describes the properties of a system. The aim was not to offer a more convenient method of troubleshooting, but to provide deeper insight into the nature of complex dynamical systems. In a further generalization, the Hamiltonian equations were later made applicable to the description of quantum as well as classical mechanics. The abstraction needed to deepen insight into analytic dynamics also broadened the scope of his inquiry to other areas of science.
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