Physics
Generalized Momenta
Generalized momenta in physics refer to a concept used in classical mechanics and quantum mechanics to describe the momentum of a system. Unlike regular momentum, generalized momenta can account for systems with varying degrees of freedom and non-conservative forces. In classical mechanics, generalized momenta are used in Hamiltonian mechanics, while in quantum mechanics, they are essential for describing the behavior of particles.
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8 Key excerpts on "Generalized Momenta"
eBook - PDF- Patrick Hamill(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
In this case, there will be n Lagrange equations of motion. 5 4.6 Generalized Momentum Consider again the problem of a mass m connected to a spring of constant k as illustrated in Figure 4.3. If the mass is moving in the x direction with speed ˙ x it has momentum p x = m ˙ x . We have seen that the Lagrangian for this system is L = 1 2 m ˙ x 2 − 1 2 kx 2 . Taking the derivative of the Lagrangian with respect to ˙ x yields ∂L ∂ ˙ x = ∂ ∂ ˙ x 1 2 m ˙ x 2 − 1 2 kx 2 = m ˙ x . 4 An equation giving the value of one coordinate in terms of the other coordinates and possibly the time is called a “holonomic” constraint. The relationship between coordinates allows us to reduce by one the number of degrees of freedom for each holonomic constraint. If the equation contains other quantities, such as velocities, it is “nonholonomic.” We will be assuming all constraints are holonomic. 5 I must warn you that physicists are somewhat careless in their usage of the term “generalized coordinate.” You will often hear it applied in situations in which the q i are not all independent. 4.6 Generalized Momentum 91 But m ˙ x is just the linear momentum! Therefore, for this system, the linear momentum is related to the Lagrangian by p x = ∂L ∂ ˙ x . Next, consider the problem of a simple pendulum consisting of a mass m hanging from a string of length l . According to Equation (4.4), the Lagrangian for this system is L = 1 2 ml 2 ˙ θ 2 + mgl cos θ . Taking the derivative of the Lagrangian with respect to ˙ θ yields ∂L ∂ ˙ θ = ml 2 ˙ θ . But ml 2 ˙ θ is the angular momentum of the pendulum! In this case the derivative of the Lagrangian with respect to the angular velocity ˙ θ is the angular momentum. In the first case, the Lagrangian was expressed in terms of x and ˙ x . That is, L = L(x, ˙ x). The generalized coordinate was x and the generalized velocity was ˙ x . In the second case, the Lagrangian was a function of θ and ˙ θ, that is, L = L(θ, ˙ θ).
Available until 25 Jan |Learn more- Tai L. Chow(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
The generalized momentum conjugate to q j is defined as p L q j j = ∂ ∂ dotnosp . We are tempted, at this point, to search for a new way of describing the complete mechanical state of a system by giving q j and p j as functions of time, rather than q i and dotnosp q i . Hamilton first devel-oped the way in which this can be done. He replaced the Lagrangian function with a quantity H , called the Hamiltonian or Hamilton’s function of the system, by the defining relationship H p q L j j j = -∑ dotnosp (5.1) 5 126 Classical Mechanics © 2010 Taylor & Francis Group, LLC which we already saw in Chapter 4. In mathematics, Equation 5.1 is called a Legendre transformation, which is a procedure for the passage from one set of independent variables to another. Although dotnosp q i explicitly appears in the defining expression (Equation 5.1), H is a function of the generalized coordinates q j , the Generalized Momenta p j , and the time t . This is because we can solve the defining expressions p L q j j = ∂ ∂ / dotnosp explicitly for the dotnosp q s ’ in terms of p j , q j , and t. There are excep-tion cases where the transformation from the Lagrangian to the Hamiltonian is hindered by the fact that the equation p L q j j = ∂ ∂ / dotnosp cannot be solved for dotnosp q i as functions of p , q , and t. Such cases are called degenerate cases , and they were usually handled by special methods for each Lagrangian. A discus-sion of the general theory of this special topic goes beyond our syllabus. 5.1.1 P HASE S PACE The q ’ s and p ’ s are now treated on equal footing: H = H ( q j, p j , t ) . There are two quantities for each degree of freedom of the mechanical system: the generalized coordinate itself and the conjugate generalized momentum. Just as with the configuration spaces that are spanned by the n independent q ’ s, we can imagine a space of 2 n dimensions spanned by the 2 n variables q 1 , q 2 ,…, q n , p 1 , p 2 ,…, p n .
eBook - PDF- Michael A. Parker(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
Example 5.1.3 The momentum P x describes the momentum of a particle along the x direction. Consider the pulley system shown in Figure 5.1.4. The momentum conjugate to the generalized coordinate q ¼ is the total angular momentum along the axis of the pulley. The Hamilton formulation of dynamics uses phase-space coordinates. q 1 , q 2 , . . . , q k , p 1 , p 2 , . . . , p k ð 5 : 1 : 4 Þ Each member of the set of the phase-space coordinates in Equation (5.1.4) has the same level of importance as any other member so that one cannot be more fundamental than another. For example, a point particle can be independently given position coordinates x , y , z and momentum coordinates p x , p y , p z . This means that the particle can be assigned a random position and a random velocity. Given that the phase space coordinates are all independent, we can also vary the coordinates in an independent manner; that is, the variations q , p must be independent of one another. The term configuration space applies to the coordinates q 1 , q 2 , . . . , q k and the term ‘‘phase space’’ applies to the full set of coordinates q 1 , q 2 , . . . , q k , p 1 , p 2 , . . . , p k . Essentially, in the absence of dynamics, position and momentum can be arbitrarily assigned to each particle. 260 Physics of Optoelectronics 5.2 Introduction to the Lagrangian and the Hamiltonian The notion that nature follows a ‘‘law of least action’’ has a long history starting around 200 BC. The optical laws of reflection and refraction can be derived from the principle that light follows a path that minimizes the transit time. In the 1700s, the law was reformulated to require the dynamics of mechanical systems to minimize the action defined as Energy Time. In the 1800s, Hamilton stated the most general form. A dynamical system will follow a path that minimizes the action defined as the time integral of the Lagrangian.
eBook - PDF- Harold Josephs, Ronald Huston(Authors)
- 2002(Publication Date)
- CRC Press(Publisher)
Generalized Dynamics: Kinematics and Kinetics 367 the moment exerted on B is zero when B is in the vertical static equilibrium position. As with the two previous examples, this system also has only one degree of freedom, represented by the angle θ . Let G be the mass center of B . Then, the velocity and partial velocity of G are: (11.5.12) The angular velocity and partial angular velocity of B are: (11.5.13) Consider a free-body diagram showing the applied forces on B as in Figure 11.5.9 where O x and O y represent hori z ontal and vertical components of the pin reaction forces. From Eq. (11.5.7), the generalized force is: (11.5.14) Observe that because the pin O has zero velocity and as a consequence, zero partial velocity, the pin reaction forces do not contribute to the generalized force. 11.6 Generalized Forces: Gravity and Spring Forces The simple examples of the foregoing section demonstrate the ease of determining gen-eralized forces. Indeed, all that is required is to project the forces and moments along the partial velocity and partial angular velocity vectors. In this section, we will see that it is possible to obtain general expressions for the contributions to generalized forces from gravity and spring forces. That is, for gravity and spring forces we will see that we can obtain their contributions to the generalized forces without computing the projections onto the partial velocity and partial angular velocity vectors. To this end, consider first a particle P of a mechanical system S where S has n degrees of freedom in an inertial frame R represented by the coordinates q r ( r = 1,…, n ). Let P have FIGURE 11.5.9 A free-body diagram of the pendulum rod. O k θ mg n O k n n y x θ r z v n v n G G = ( ) = ( ) l l 2 2 ˙ ˙ θ θ θ θ and ω ω = = ˙ ˙ θ θ n n z z and F k v n n n θ θ θ θ θ θ θ = − ⋅ + + ( ) ⋅ − ⋅ = − ( ) + − mg O O v k mg k G x x y y O z ˙ ˙ ˙ sin ω l 2 0 368 Dynamics of Mechanical Systems mass m . Then, from Eq.
eBook - PDF- Jerry Ginsberg(Author)
- 2007(Publication Date)
- Cambridge University Press(Publisher)
The only generalized coordinate required here is the nutation angle θ because the precession rate is specified as ˙ ψ = . Let pin A be the origin of xyz , which is attached to bar AB such that its x axis is aligned with the bar and the z axis is situated in the vertical plane. Then the angular velocity of the bar is ¯ ω = − cos θ ¯ i − ˙ θ j + sin θ ¯ k . 9.2 Generalized Momentum Principles 581 With the pin defined to be the datum for gravitational potential energy, the corre-sponding energy functions are T = 1 2 1 3 mL 2 ˙ θ 2 + 2 (sin θ ) 2 , V = − mg L 2 cos θ. (1) Time does not occur explicitly in these expressions, so Eq. ( 9.2.25 ) is applicable. The generalized momentum corresponding to the sole generalized coordinate is p 1 = ∂ T ∂ ˙ θ = 1 3 mL 2 ˙ θ, (2) so M 1 , 1 = (1 / 3) mL 2 . Inspection of T shows that it does not contain a term that is linear in ˙ θ, so the kinetic energy matches the standard form, Eqs. ( 7.6.4 ), with N 1 = 0 , T 0 = 1 2 1 3 mL 2 2 (sin θ ) 2 . These expressions enable us to employ Eq. ( 9.2.13 ) to derive the Hamiltonian, which leads to H = 1 2 1 M 1 , 1 p 2 1 − T 0 + V = 3 2 mL 2 p 2 1 − 1 6 mL 2 2 (sin θ ) 2 − mg L 2 cos θ. (3) A virtual displacement increments θ by δθ, but the only force that does work in such a movement is gravity, which is included in V . Therefore Q 1 = 0 , from which it follows that Jacobi’s integral, which is the conservation form of Eq. ( 9.2.25 ), applies. We may establish the constant value H 0 of the Hamiltonian by evaluating it at the initial state. Hence any motion of the system must be such that the Hamiltonian in Eq. (3) remains constant: 3 2 mL 2 p 2 1 − 1 6 mL 2 2 (sin θ ) 2 − mg L 2 cos θ = H 0 . We may express this conservation equation in terms of θ by substituting Eq. (2), which yields 1 6 mL 2 ˙ θ 2 − 2 (sin θ ) 2 − mg L 2 cos θ = H 0 . (4) This relation is an integral of the equation of motion, just like the principle of con-servation of energy for a conservative, time-independent system.
eBook - ePub- L Cveticanin(Author)
- 2022(Publication Date)
- Routledge(Publisher)
2 General Priniciples of Dynamics of Systems with Variable Mass2.0 Introduction
In this chapter, which has four sections, the general principles of dynamics of systems with variable mass are considered. The first section deals with the dynamics of a single particle with variable mass, and the second of a system of particles with variable mass. The principles of momentum and angular momentum and the energy principle are considered for variable mass systems. In the third section the dynamics of bodies with variable mass are analyzed. Differential equations for rigid bodies with variable mass are formed. Lagrange’s differential equations are adopted for systems with variable mass. Using the theoretical results from the previous sections, in the fourth section mathematical models of motion of a special group of mechanisms are obtained. The differential equations are derived of motion of vibrations of one-degree-of-freedom mechanisms with variable mass, of a symmetrical rotor with band winding, and of a clamped free rotor with variable mass.2.1 Dynamics of Particles with Variable Mass
The derivation of equations of motion for systems with variable mass is connected with difficulties which come from the fact that the basic principles of classical dynamics (Newton-Euler equations, Lagrange’s equations, etc) are only valid for systems comprising definite sets of objects with constant masses. Yet, if attention is to be restricted to B in Fig. 2.1.1 , as practical considerations demand, then the equations of motion cannot be obtained by direct application of any of formulations of classical dynamics described above, because the mass of B varies with time.FIGURE 2.1.1 Model of variable mass system.To circumvent this difficulty, a different strategy was developed. The system at any given instant consists of three parts, B, B’ and B" (see Fig. 2.1.1 ). Here B is the main part of the system, B" represents the part of B that will be lost at a later time, and ’ is material that will coalesce with B
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Mass in general relativity The concept of mass in general relativity (GR) is more complex than the concept of mass in special relativity. In fact, general relativity does not offer a single definition for the term mass, but offers several different definitions which are applicable under different circumstances. Under some circumstances, the mass of a system in general relativity may not even be defined. Review of mass in special relativity In special relativity, the invariant mass ( hereafter simply mass ) of an isolated system, can be defined in terms of the energy and momentum of the system by the relativistic energy-momentum equation: Where E is the total energy of the system, p is the total momentum of the system and c is the speed of light. Concisely, the mass of a system in special relativity is the norm of its energy-momentum four vector. Defining mass in general relativity: concepts and obstacles Generalizing this definition to general relativity, however, is problematic; in fact, it turns out to be impossible to find a general definition for a system's total mass (or energy). The main reason for this is that gravitational field energy is not a part of the energy-momen-tum tensor; instead, what might be identified as the contribution of the gravitational field to a total energy is part of the Einstein tensor on the other side of Einstein's equation (and, as such, a consequence of these equations' non-linearity). While in certain situation it is possible to rewrite the equations so that part of the gravitational energy now stands alongside the other source terms in the form of the Stress-energy-momentum pseu-dotensor, this separation is not true for all observers, and there is no general definition for obtaining it.
eBook - PDFMechanics
Lectures on Theoretical Physics
- Arnold Sommerfeld(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
Incidentally, the mechanical system which is next in simplicity to that of a single mass point, namely, the rotating rigid body, leads us to an equation of motion along the lines of (3), in the form rate of change of moment of momentum (angular momentum) = moment of force (torque) ; a description in terms of angular acceleration, similar to (3a), is not possible. An effect similar to the non-constancy of mass in relativity must be taken into account: the moment of inertia, here replacing the mass, changes with changing location of the axis of rotation in the body. We must now seek to get a clear idea of the concept of force. Kirchhoff 5 wanted to degrade it to a quantity defined by the product of mass and acceleration. Hertz 6 , too, tried to eliminate and replace it by coupling the system under consideration with other, generally hidden systems interacting with the former. Hertz carried out this program with admirable consistency. His method, however, hardly produced fruitful results; and it is especially unsuitable for the beginner. We are of the opinion that we have at least a qualitative notion of force which we acquire quite directly through the feeling we experience when using our muscles. In addition the earth has provided us with the comparison standard of gravity, with which we can measure all other forces quantitatively. For this purpose we need merely balance the effect 5 Gustav Kirchhoff, Vol. I of his Vorlesungen über mathematische Physik, p. 22. β Heinrich Hertz, Miscellaneous Papers, Vol. Ill, Principles of Mechanics, Macmillan, New York, 1896. 6 Mechanics of a Particle 1.1 of a given force by a suitable weight. (By means of a pulley and string we can let the vertical force of gravity act in a direction opposed to the given force.) If, in addition, we procure a number of equally heavy bodies, a set of weights, we obtain a tentative scale with which to measure forces quantitatively.
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