BBA IInd SEMESTER EXAMINATION 2008-09

... What are generalized coordinates? Distinguish a system on the basis of time dependence of generalized coordinates. Show that for a single particle with constant mass, the differential equation for kinetic energy is ...

... What are generalized coordinates? Distinguish a system on the basis of time dependence of generalized coordinates. Show that for a single particle with constant mass, the differential equation for kinetic energy is ...

Classical Mechanics 420

... Classical Mechanics 420 J. D. Gunton Lewis Lab 418 [email protected] ...

... Classical Mechanics 420 J. D. Gunton Lewis Lab 418 [email protected] ...

Fiz 235 Mechanics 2002

... due to an internal explosion, it breaks into three parts. One part, whose mass is 10 kg, moves away from the point of explosion with a speed of 100 m/s along the positive y axis. A second fragment with a mass 4 kg moves along negative x-axis with a speed of 500 m/s. a) What is the velocity of third ...

... due to an internal explosion, it breaks into three parts. One part, whose mass is 10 kg, moves away from the point of explosion with a speed of 100 m/s along the positive y axis. A second fragment with a mass 4 kg moves along negative x-axis with a speed of 500 m/s. a) What is the velocity of third ...

Transformations and conservation laws

... Galilean transformation We mainly use inertial frames in which a free body (no forces applied) moves with a constant velocity. A frame moving with a constant velocity with respect to an inertial frame is inertial, too. Thus there is an infinite number of inertial frames. The equations of motion are ...

... Galilean transformation We mainly use inertial frames in which a free body (no forces applied) moves with a constant velocity. A frame moving with a constant velocity with respect to an inertial frame is inertial, too. Thus there is an infinite number of inertial frames. The equations of motion are ...

Lagrange`s equations of motion in generalized coordinates

... Canonical equations of motion – Hamiltonian mechanics. if the potential energy of a system is velocity independent, then the linear momentum components in rectangular coordinates are given by ...

... Canonical equations of motion – Hamiltonian mechanics. if the potential energy of a system is velocity independent, then the linear momentum components in rectangular coordinates are given by ...

Lagrange Multiplier Frames - University of Colorado Boulder

... system of constrained rigid bodies in Newtonian Mechanics Formulation. 1- Treat the problem as if all bodies are entirely free and formulate the virtual work by summing up the contributions of each free body. 2- Identify constraint equations and multiply each by an indeterminate coefficient. Then ta ...

... system of constrained rigid bodies in Newtonian Mechanics Formulation. 1- Treat the problem as if all bodies are entirely free and formulate the virtual work by summing up the contributions of each free body. 2- Identify constraint equations and multiply each by an indeterminate coefficient. Then ta ...

On the Shoulders of Giants”

... Hero of Alexandria found that light, traveling from one point to another by a reflection from a plane mirror, always takes the shortest possible path. ...

... Hero of Alexandria found that light, traveling from one point to another by a reflection from a plane mirror, always takes the shortest possible path. ...

Energy Methods - MIT OpenCourseWare

... example would be polar coordinates where for a two-dimensional position of a mass particle, x1 and x2 could be given by r and θ. A two-degree of freedom system remains two-degree so that the number of coordinate variables required remains two. r and θ and their counterparts in other coordinate syst ...

... example would be polar coordinates where for a two-dimensional position of a mass particle, x1 and x2 could be given by r and θ. A two-degree of freedom system remains two-degree so that the number of coordinate variables required remains two. r and θ and their counterparts in other coordinate syst ...

Slide 1

... Joseph-Louise Lagrange: 1736-1813. Born in Italy and lived in Berlin and Paris. Studied to be a lawyer. Contemporary of Euler, Bernoulli, D’Alembert, Laplace, and Newton. He was interested in math. Contribution: – Calculus of variations. – Calculus of probabilities. – Integration of differential equ ...

... Joseph-Louise Lagrange: 1736-1813. Born in Italy and lived in Berlin and Paris. Studied to be a lawyer. Contemporary of Euler, Bernoulli, D’Alembert, Laplace, and Newton. He was interested in math. Contribution: – Calculus of variations. – Calculus of probabilities. – Integration of differential equ ...

L7+8-AST1420 - University of Toronto

... From the example of Sun-Earth-Moon system we find that: integrals of motion guarantee no-escape from the allowed regions of motion for an infinite period of time, which is better than either the general or the special perturbation theory can do but only if the assumptions of the theory are satisfied ...

... From the example of Sun-Earth-Moon system we find that: integrals of motion guarantee no-escape from the allowed regions of motion for an infinite period of time, which is better than either the general or the special perturbation theory can do but only if the assumptions of the theory are satisfied ...

Energy3

... HAMILTON’S EQUATIONS AND ENERGY If, as in Example 2, the kinetic energy is a quadratic function of the generalized velocities and the forces are conservative (arising from a potential energy U) then the total energy is given by the Hamiltonian ...

... HAMILTON’S EQUATIONS AND ENERGY If, as in Example 2, the kinetic energy is a quadratic function of the generalized velocities and the forces are conservative (arising from a potential energy U) then the total energy is given by the Hamiltonian ...

Lagrange Multiplier Form of the EOM - SBEL

... Implicit Function Theorem gives us the answer: the Jacobian must be nonsingular ...

... Implicit Function Theorem gives us the answer: the Jacobian must be nonsingular ...

Materialy/01/Applied Mechanics-Lectures/Applied Mechanics

... Let us consider that the particle follows during the time interval [t1, t2] a motion trajectory u i* distinct from the real one ui. This allows us to define the virtual displacement of the particle the relationship ...

... Let us consider that the particle follows during the time interval [t1, t2] a motion trajectory u i* distinct from the real one ui. This allows us to define the virtual displacement of the particle the relationship ...

Nonholonomic dynamics as limit of friction an introduction by example

... Lagrange showed that equations (2) appear as the variational equations for this functional. That is, a curve x(t) extremizes S precisely if it satisfies (2). Since this property is coordinate invariant, it follows that the Lagrange equations yield the correct equations of motion in any coordinate sy ...

... Lagrange showed that equations (2) appear as the variational equations for this functional. That is, a curve x(t) extremizes S precisely if it satisfies (2). Since this property is coordinate invariant, it follows that the Lagrange equations yield the correct equations of motion in any coordinate sy ...

Lagrange`s Equation

... Lagrange’s Method • Newton’s method of developing equations of motion requires taking elements apart • When forces at interconnections are not of primary interest, more advantageous to derive equations of motion by considering energies in the system • Lagrange’s equations: – Indirect approach that c ...

... Lagrange’s Method • Newton’s method of developing equations of motion requires taking elements apart • When forces at interconnections are not of primary interest, more advantageous to derive equations of motion by considering energies in the system • Lagrange’s equations: – Indirect approach that c ...

lecture 10

... A particle moving on a plane curve 1 Two particles moving on a space curve and having a constant distance between them ...

... A particle moving on a plane curve 1 Two particles moving on a space curve and having a constant distance between them ...

Lagrangian and Hamiltonian Dynamics

... • Seek generalization of coordinates. • Consider mechanical systems consisting of a collection of n discrete point particles. • Rigid bodies will be discussed later… • We need n position vectors, I.e. 3n quantities. • If there are m constraint equations that limit the motion of particle by for insta ...

... • Seek generalization of coordinates. • Consider mechanical systems consisting of a collection of n discrete point particles. • Rigid bodies will be discussed later… • We need n position vectors, I.e. 3n quantities. • If there are m constraint equations that limit the motion of particle by for insta ...

An introduction to Lagrangian and Hamiltonian mechanics

... Formulating the Euler–Lagrange equations in these coordinates and then solving them will tell us that the extremizing solution is a straight line (only it will be expressed in polar coordinates). 3. Let Y denote a function space. In the context above Y was the space of twice continuously differentia ...

... Formulating the Euler–Lagrange equations in these coordinates and then solving them will tell us that the extremizing solution is a straight line (only it will be expressed in polar coordinates). 3. Let Y denote a function space. In the context above Y was the space of twice continuously differentia ...

Calculus Before Newton and Leibniz: Part I

... "Along with Isaac Newton (1642-1727), Leibniz is generally credited as one of the inventors or discoverers of calculus." -- Thomas P. Dick and Charles M. Patton, Calculus. History has a way of focusing credit for any invention or discovery on one or two individuals in one time and place. The truth i ...

... "Along with Isaac Newton (1642-1727), Leibniz is generally credited as one of the inventors or discoverers of calculus." -- Thomas P. Dick and Charles M. Patton, Calculus. History has a way of focusing credit for any invention or discovery on one or two individuals in one time and place. The truth i ...

the Lagrangian formulation

... oceans and the atmosphere. For a particle travelling on the surface of the rotating earth, the direction of the coriolis force is drawn in figure 4. We see that a particle thrown in the northern hemisphere will be seen to rotate in a clockwise direction; a particle thrown in the southern hemisphere ...

... oceans and the atmosphere. For a particle travelling on the surface of the rotating earth, the direction of the coriolis force is drawn in figure 4. We see that a particle thrown in the northern hemisphere will be seen to rotate in a clockwise direction; a particle thrown in the southern hemisphere ...

Non-Linear Forces and Irreversibility Problem in Classical Mechanics

... reversible due to formal descriptions of classical mechanics and at the same time it is irreversible due to the determined mechanism described above. It is obvious that we cannot consider the explanation of irreversibility as a complete one until we resolve this contradiction. We can resolve the con ...

... reversible due to formal descriptions of classical mechanics and at the same time it is irreversible due to the determined mechanism described above. It is obvious that we cannot consider the explanation of irreversibility as a complete one until we resolve this contradiction. We can resolve the con ...

ppt - SBEL - University of Wisconsin–Madison

... 3rd Law To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and always opposite in direction. ...

... 3rd Law To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and always opposite in direction. ...

Generalized Coordinates, Lagrange`s Equations, and Constraints 1

... 2 Principle of Virtual Displacements Virtual displacements δri are any displacements consistent with the constraints of the system. The principle of virtual displacements1 says that the work of real external forces through virtual external displacements equals the work of the real internal forces ar ...

... 2 Principle of Virtual Displacements Virtual displacements δri are any displacements consistent with the constraints of the system. The principle of virtual displacements1 says that the work of real external forces through virtual external displacements equals the work of the real internal forces ar ...

Joseph-Louis Lagrange (/ləˈɡrɑːndʒ/ or /ləˈɡreɪndʒ/; French: [laˈgrɑ̃ʒ]),born Giuseppe Lodovico Lagrangia or Giuseppe Ludovico De la Grange Tournier (also reported as Giuseppe Luigi Lagrange or Lagrangia) (25 January 1736 – 10 April 1813) was an Italian Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.In 1766, on the recommendation of Euler and d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, Prussia, where he stayed for over twenty years, producing volumes of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics (Mécanique Analytique, 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1888–89), written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century.In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy. He remained in France until the end of his life. He was significantly involved in the decimalisation in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, founding member of the Bureau des Longitudes and Senator in 1799.