Please forward this error screen to sharedip-1601531662. Newton’s vectorial forces solved problems in lagrangian and hamiltonian mechanics pdf individual particles.
A scalar is a quantity, whereas a vector is represented by quantity and direction. The kinetic and potential energies of the system are expressed using these generalized coordinates or momenta, and the equations of motion can be readily set up, thus analytical mechanics allows numerous mechanical problems to be solved with greater efficiency than fully vectorial methods. Analytical mechanics does not introduce new physics and is not more general than Newtonian mechanics. Rather it is a collection of equivalent formalisms which have broad application. The methods of analytical mechanics apply to discrete particles, each with a finite number of degrees of freedom. They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom.
The definitions and equations have a close analogy with those of mechanics. In physical systems, however, some structure or other system usually constrains the body’s motion from taking certain directions and pathways. So a full set of Cartesian coordinates is often unneeded, as the constraints determine the evolving relations among the coordinates, which relations can be modeled by equations corresponding to the constraints. In the Lagrangian and Hamiltonian formalisms, the constraints are incorporated into the motion’s geometry, reducing the number of coordinates to the minimum needed to model the motion. Generalized coordinates incorporate constraints on the system. Generalized coordinates are not the same as curvilinear coordinates.
The idea of a constraint is useful – since this limits what the system can do, and can provide steps to solving for the motion of the system. Hamilton’s equations of motion additionally to the others. Following are overlapping properties between the Lagrangian and Hamiltonian functions. In other words, the Lagrangian of a system is not unique. It can be shown that the Hamiltonian is also cyclic in exactly the same generalized coordinates. The path for which action is least is the path taken by the system. The transformation of coordinates and momenta can allow simplification for solving Hamilton’s equations for a given problem.
If this does not hold then the transformation is not canonical. Lagrangian and Hamiltonian mechanics, not often used but especially useful for removing cyclic coordinates. Hamiltonian equations and those which enter the Lagrangian equations is arbitrary. It is simply convenient to let the Hamiltonian equations remove the cyclic coordinates, leaving the non cyclic coordinates to the Lagrangian equations of motion. Generalized coordinates apply to discrete particles. It is a question of determining the correct Lagrangian density to generate the correct field equation. New York: Dover Publications Inc.