The Second Edition includes a larger selection of examples and problems with hints in each chapter and continues the strong emphasis of the First Edition on the development and application of mathematical methods mostly calculus to the solution of problems in Classical Mechanics. New material has been added to most chapters. For example, a new derivation of the Noether theorem for discrete Lagrangian systems is given and a modified Rutherford scattering problem is solved exactly to show that the total scattering cross section associated with a confined potential i.
The Frenet-Serret formulas for the Coriolis-corrected projectile motion are presented, where the Frenet-Serret torsion is shown to be directly related to the Coriolis deflection, and a new treatment of the sleeping-top problem is given. Read more Read less. From the Inside Flap An Introduction to Lagrangian Mechanics begins with a proper historical perspective on the Lagrangian method by presenting Fermat's Principle of Least Time as an introduction to the Calculus of Variations as well as the principles of Maupertuis, Jacobi, and d'Alembert that preceded Hamilton's formulation of the Principle of Least Action, from which the Euler Lagrange equations of motion are derived.
No customer reviews. In general, Hamiltonian systems are chaotic ; concepts of measure, completeness, integrability and stability are poorly defined.
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An important special case consists of those Hamiltonians that are quadratic forms , that is, Hamiltonians that can be written as. This Hamiltonian consists entirely of the kinetic term. If one considers a Riemannian manifold or a pseudo-Riemannian manifold , the Riemannian metric induces a linear isomorphism between the tangent and cotangent bundles.
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See Musical isomorphism. Using this isomorphism, one can define a cometric. In coordinates, the matrix defining the cometric is the inverse of the matrix defining the metric. The solutions to the Hamilton—Jacobi equations for this Hamiltonian are then the same as the geodesics on the manifold. In particular, the Hamiltonian flow in this case is the same thing as the geodesic flow. The existence of such solutions, and the completeness of the set of solutions, are discussed in detail in the article on geodesics. See also Geodesics as Hamiltonian flows. When the cometric is degenerate, then it is not invertible.
In this case, one does not have a Riemannian manifold, as one does not have a metric. However, the Hamiltonian still exists. In the case where the cometric is degenerate at every point q of the configuration space manifold Q , so that the rank of the cometric is less than the dimension of the manifold Q , one has a sub-Riemannian manifold. The Hamiltonian in this case is known as a sub-Riemannian Hamiltonian.
Every such Hamiltonian uniquely determines the cometric, and vice versa. This implies that every sub-Riemannian manifold is uniquely determined by its sub-Riemannian Hamiltonian, and that the converse is true: every sub-Riemannian manifold has a unique sub-Riemannian Hamiltonian. The existence of sub-Riemannian geodesics is given by the Chow—Rashevskii theorem. The continuous, real-valued Heisenberg group provides a simple example of a sub-Riemannian manifold.
For the Heisenberg group, the Hamiltonian is given by. Hamiltonian systems can be generalized in various ways.
Instead of simply looking at the algebra of smooth functions over a symplectic manifold , Hamiltonian mechanics can be formulated on general commutative unital real Poisson algebras. A state is a continuous linear functional on the Poisson algebra equipped with some suitable topology such that for any element A of the algebra, A 2 maps to a nonnegative real number. A further generalization is given by Nambu dynamics. A good illustration of Hamiltonian mechanics is given by the Hamiltonian of a charged particle in an electromagnetic field.
In Cartesian coordinates i. This is called minimal coupling. If we substitute the definition of the momenta, and the definitions of the velocities in terms of the momenta, into the definition of the Hamiltonian given above, and then simplify and rearrange, we get:.
This equation is used frequently in quantum mechanics. The Lagrangian for a relativistic charged particle is given by:. From this we get the force equation equivalent to the Euler—Lagrange equation. This has the advantage that p can be measured experimentally whereas P cannot.
From Wikipedia, the free encyclopedia. Branch of analytical mechanics and formulation of physics. Second law of motion. History Timeline. Newton's laws of motion. Analytical mechanics Lagrangian mechanics Hamiltonian mechanics Routhian mechanics Hamilton—Jacobi equation Appell's equation of motion Udwadia—Kalaba equation Koopman—von Neumann mechanics. Core topics.
Introduction to Lagrangian & Hamiltonian Mechanics ( edition) | Open Library
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Introduction to Lagrangian & Hamiltonian Mechanics
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