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Hamiltonian Mechanics - Wikipedia, the Free Encyclopedia Page 1 of 12 Hamiltonian mechanics - Wikipedia, the free encyclopedia Page 1 of 12 Hamiltonian mechanics From Wikipedia, the free encyclopedia Hamiltonian mechanics is a reformulation of classical Classical mechanics mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. Newton's Second Law It arose from Lagrangian mechanics, a previous History of classical mechanics · reformulation of classical mechanics introduced by Joseph Timeline of classical mechanics Louis Lagrange in 1788, but can be formulated without Branches recourse to Lagrangian mechanics using symplectic spaces (see Mathematical formalism , below). The Hamiltonian Statics · Dynamics / Kinetics · Kinematics · method differs from the Lagrangian method in that instead Applied mechanics · Celestial mechanics · of expressing second-order differential constraints on an n- Continuum mechanics · dimensional coordinate space (where n is the number of Statistical mechanics degrees of freedom of the system), it expresses first-order constraints on a 2 n-dimensional phase space.[1] Formulations Newtonian mechanics (Vectorial As with Lagrangian mechanics, Hamilton's equations mechanics) provide a new and equivalent way of looking at classical Analytical mechanics: mechanics. Generally, these equations do not provide a more convenient way of solving a particular problem. Lagrangian mechanics Rather, they provide deeper insights into both the general Hamiltonian mechanics structure of classical mechanics and its connection to Fundamental concepts quantum mechanics as understood through Hamiltonian mechanics, as well as its connection to other areas of Space · Time · Velocity · Speed · Mass · science. Acceleration · Gravity · Force · Impulse · Torque / Moment / Couple · Momentum · Contents Angular momentum · Inertia · Moment of inertia · Reference frame · 1 Simplified overview of uses Energy · Kinetic energy · Potential energy · 1.1 Basic physical interpretation Mechanical work · Virtual work · 1.2 Using Hamilton's equations D'Alembert's principle 1.3 Notes Core topics 2 Deriving Hamilton's equations Rigid body · Rigid body dynamics · 3 As a reformulation of Lagrangian mechanics 4 Geometry of Hamiltonian systems Euler's equations (rigid body dynamics) · 5 Generalization to quantum mechanics through Motion · Newton's laws of motion · Poisson bracket Newton's law of universal gravitation · 6 Mathematical formalism Equations of motion · 7 Riemannian manifolds Inertial frame of reference · 8 Sub-Riemannian manifolds Non-inertial reference frame · 9 Poisson algebras 10 Charged particle in an electromagnetic field Rotating reference frame · Fictitious force · 11 Relativistic charged particle in an Linear motion · electromagnetic field Mechanics of planar particle motion · 12 See also Displacement (vector) · Relative velocity · 13 Notes and references Friction · Simple harmonic motion · 13.1 Notes http://en.wikipedia.org/wiki/Hamiltonian_mechanics 5/23/2011 Hamiltonian mechanics - Wikipedia, the free encyclopedia Page 2 of 12 13.2 References Harmonic oscillator · Vibration · Damping · 14 External links Damping ratio · Rotational motion · Circular motion · Uniform circular motion · Simplified overview of uses Non-uniform circular motion · Centripetal force · Centrifugal force · The value of the Hamiltonian is the total energy of the Centrifugal force (rotating reference frame) · system being described. For a closed system, it is the sum Reactive centrifugal force · Coriolis force · of the kinetic and potential energy in the system. There is a set of differential equations known as the Hamilton Pendulum · Rotational speed · equations which give the time evolution of the system. Angular acceleration · Angular velocity · Hamiltonians can be used to describe such simple systems Angular frequency · Angular displacement as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back Scientists again over time. Hamiltonians can also be employed to Galileo Galilei · Isaac Newton · model the energy of other more complex dynamic systems Jeremiah Horrocks · Leonhard Euler · such as planetary orbits in celestial mechanics and also in Jean le Rond d'Alembert · Alexis Clairaut · [2] quantum mechanics. Joseph Louis Lagrange · The Hamilton equations are generally written as follows: Pierre-Simon Laplace · William Rowan Hamilton · Siméon-Denis Poisson In the above equations, the dot denotes the ordinary derivative with respect to time of the functions p = p(t) (called generalized momenta) and q = q(t) (called generalized coordinates), taking values in some vector space, and = is the so-called Hamiltonian, or (scalar valued) Hamiltonian function. Thus, more explicitly, one can equivalently write and specify the domain of values in which the parameter t (time) varies. For a detailed derivation of these equations from Lagrangian mechanics, see below. Basic physical interpretation The simplest interpretation of the Hamilton equations is as follows, applying them to a one-dimensional system consisting of one particle of mass m under time-independent boundary conditions: The Hamiltonian represents the energy of the system, which is the sum of kinetic and potential energy, traditionally denoted T and V, respectively. Here q is the x coordinate and p is the momentum, mv. Then http://en.wikipedia.org/wiki/Hamiltonian_mechanics 5/23/2011 Hamiltonian mechanics - Wikipedia, the free encyclopedia Page 3 of 12 Note that T is a function of p alone, while V is a function of x (or q) alone. Now the time -derivative of the momentum p equals the Newtonian force , and so here the first Hamilton equation means that the force on the particle equals the rate at which it loses potential energy with respect to changes in x, its location. (Force equals the negative gradient of potential energy.) The time-derivative of q here means the velocity: the second Hamilton equation here means that the particle’s velocity equals the derivative of its kinetic energy with respect to its momentum. (Because the derivative with respect to p of p2/2 m equals p/m = mv /m = v.) Using Hamilton's equations In terms of the generalized coordinates and generalized velocities 1. First write out the Lagrangian Express T and V as though you were going to use Lagrange's equation. 2. Calculate the momenta by differentiating the Lagrangian with respect to velocity: . 3. Express the velocities in terms of the momenta by inverting the expressions in step (2). 4. Calculate the Hamiltonian using the usual definition of H as the Legendre transformation of L: . Substitute for the velocities using the results in step (3). 5. Apply Hamilton's equations. Notes Hamilton's equations are appealing in view of their beautiful simplicity and (slightly broken ) symmetry. They have been analyzed under almost every imaginable angle of view, from basic physics up to symplectic geometry. A lot is known about solutions of these equations, yet the exact general case solution of the equations of motion cannot be given explicitly for a system of more than two massive point particles. The finding of conserved quantities plays an important role in the search for solutions or information about their nature. In models with an infinite number of degrees of freedom, this is of course even more complicated. An interesting and promising area of research is the study of integrable systems, where an infinite number of independent conserved quantities can be constructed. Deriving Hamilton's equations We can derive Hamilton's equations by looking at how the total differential of the Lagrangian depends on time, generalized positions and generalized velocities [3] http://en.wikipedia.org/wiki/Hamiltonian_mechanics 5/23/2011 Hamiltonian mechanics - Wikipedia, the free encyclopedia Page 4 of 12 Now the generalized momenta were defined as and Lagrange's equations tell us that We can rearrange this to get and substitute the result into the total differential of the Lagrangian We can rewrite this as and rearrange again to get The term on the left-hand side is just the Hamiltonian that we have defined before, so we find that where the second equality holds because of the definition of the total differential of in terms of its partial derivatives. Associating terms from both sides of the equation above yields Hamilton's equations As a reformulation of Lagrangian mechanics http://en.wikipedia.org/wiki/Hamiltonian_mechanics 5/23/2011 Hamiltonian mechanics - Wikipedia, the free encyclopedia Page 5 of 12 Starting with Lagrangian mechanics, the equations of motion are based on generalized coordinates and matching generalized velocities We write the Lagrangian as with the subscripted variables understood to represent all N variables of that type. Hamiltonian mechanics aims to replace the generalized velocity variables with generalized momentum variables, also known as conjugate momenta . By doing so, it is possible to handle certain systems, such as aspects of quantum mechanics, that would otherwise be even more complicated. For each generalized velocity, there is one corresponding conjugate momentum, defined as: In Cartesian coordinates, the generalized momenta are precisely the physical linear momenta. In circular polar coordinates, the generalized momentum corresponding to the angular velocity is the physical angular momentum. For an arbitrary choice of generalized coordinates, it may not be possible to obtain an intuitive interpretation of the conjugate momenta. One thing which is not too obvious in this
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