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Synopsis
This compendium of physics covers the key equations and fundamental principles that are taught in graduate programs. It offers a succinct yet systematic treatment of all areas of physics, including mathematical physics, solid state physics, particle physics, statistical mechanics, and optics. In one complete, self-contained volume, author Charles P. Poole provides both review material for students preparing for PhD qualifying examinations and a quick reference for physicists who need to brush up on basic topics or delve into areas outside their expertise.
In this second edition the author devotes two chapters to such regularly needed information as trigonometric and vector identities and special functions. The remaining chapters incorporate less frequently summoned concepts, including Lagrangians, parity, dispersion relations, chaos, free energies, statistical mechanical ensembles, and elementary particle classification. A brand new chapter on entanglement and quantum computing has been added, making this an indispensable resource for graduate students and physicists in both industry and academia.
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About the Author
Charles P. Poole, Jr., Ph.D., is professor emeritus in the Department of Physics and Astronomy at the University of South Carolina. He is the author or coauthor of a dozen books, including 'Superconductivity and Introduction to Renormalization Group Methods'. He has taught qualifying examination preparation courses for more than two decades.
From the Back Cover
This compendium of physics covers the key equations and fundamental principles that are taught in graduate programs. It offers a succinct yet systematic treatment of all areas of physics, including mathematical physics, quantum mechanics, solid state physics, particle physics, statistical mechanics, and optics. In one complete, self-contained volume, author Charles P.n Poole, Jr. provides both review material for students preparing for Ph.D. qualifying examinations and a quick reference for physicists who need to brush up on basic topics or delve into areas outside their expertise.<br /> The author devotes two chapters to regularly needed information such as trigonometric and vector identities, infinite series, Fourier Transforms, and special functions. The remaining chapters incorporate less frequently looked-up concepts, including, e.g., Lagrangians, parity, dispersion relations, chaos, free energies, statistical mechanical ensembles, and elementary particle classification. A brand new chapter on entanglement and quantum computing written by Professor Horacio A. Farach has been added to this second edition. The book is an indispensable resource for graduate students and physicists in industry and academia.
From the Inside Flap
This compendium of physics covers the key equations and fundamental principles that are taught in graduate programs. It offers a succinct yet systematic treatment of all areas of physics, including mathematical physics, quantum mechanics, solid state physics, particle physics, statistical mechanics, and optics. In one complete, self-contained volume, author Charles P.n Poole, Jr. provides both review material for students preparing for Ph.D. qualifying examinations and a quick reference for physicists who need to brush up on basic topics or delve into areas outside their expertise.
The author devotes two chapters to regularly needed information such as trigonometric and vector identities, infinite series, Fourier Transforms, and special functions. The remaining chapters incorporate less frequently looked-up concepts, including, e.g., Lagrangians, parity, dispersion relations, chaos, free energies, statistical mechanical ensembles, and elementary particle classification. A brand new chapter on entanglement and quantum computing written by Professor Horacio A. Farach has been added to this second edition. The book is an indispensable resource for graduate students and physicists in industry and academia.
Excerpt. © Reprinted by permission. All rights reserved.
The Physics Handbook
Fundamentals and Key EquationsBy Charles P. Poole, Jr.John Wiley & Sons
Copyright © 2007 Charles P. Poole, Jr.All right reserved.
ISBN: 978-3-527-40699-9
Chapter One
LAGRANGIAN AND HAMILTONIAN FORMULATIONS
1 Introduction / 1
2 Newton's Law Approach / 2
3 Lagrangian Formulation / 3
4 Hamiltonian Formulation / 6
5 Variational Principles and Virtual Displacements / 8
1 INTRODUCTION
In this chapter we outline several approaches to treating problems of dynamics in classical mechanics, i.e., problems concerning motion in which forces are involved. We examine the force and the energy approaches, then discuss the Lagrangian and Hamiltonian formulations, and finally point out some variational approaches. Explicit expressions are given for the Lagrangians, Hamiltonians, and canonical momenta of various commonly encountered systems.
2 NEWTON'S LAW APPROACH
The force approach to non-relativistic dynamics makes use of Newton's second law which states that the force F on a body equals its rate of change of momentum [??]
F = [??] = dp/dt (1)
Any reference frame in which this law holds is called an inertial frame. In one dimension the momentum for a particle of mass m and velocity v is p = mv, and Eq. (1) becomes
F = ma = m dv/dt = m [d.sup.2]]x/d[t.sup.2] (2)
where x is the position coordinate and a is the acceleration. The more general expression (1) must be used when the mass changes, as in nuclear decay problems involving the creation and annihilation of particles.
In dynamics we encounter several types of forces, such as:
F = mg gravity near the earth's surface (3a)
F = Gmm']??]/[r.sup.2] Newton's law of gravity (3b)
F = qq']??]/4[pi][[epsilon].sub.0] Coulomb's law (3c) [r.sup.2]
F = -k(x - [x.sub.0]) harmonic restoring force from the (3d) equilibrium position [x.sub.0]
F = -N static or kinetic friction (3e)
F = -k [|v|.sup.n] frictional force, often n = 1 as in Stokes' law F = 6[pi][eta]rv for streamline flow, or n = 2 with turbulence (3f)
F = q(E + v B) Lorentz force on the charge q (3g)
Ordinarily it is easier to solve non-relativistic elementary mechanics problems by balancing the energy at the beginning (b) and at the end (e) of the interaction. For a many-particle system in which each particle i has the kinetic energy 1/2 [m.sub.i]v.sup.2.sub.ib and the potential energy [V.sub.i], we can write
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where the summations are over the particles of the system. If the particles interact with each other, then we can add double summations over the initial and final interaction energies [V.sup.b.sub.ij] and [V.sup.e.sub.ij] to this equation. This energy approach is valid for conservative systems for which there is no dissipation.
3 LAGRANGIAN FORMULATION
In the Langrangian approach the kinetic and potential energies are expressed in terms of generalized coordinates [q.sub.i] and the velocities [[??].sub.i] of each particle i. The Lagrangian itself, L([q.sub.1],..., [q.sub.N], [[??].sub.1],..., [[??].sub.n], t), is the difference between the total kinetic energy T and the total potential energy V of the system
L = T - V (5)
Examples of some Lagrangians are:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] harmonic oscillator (6a)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] charge q in electromagnetic fields [gamma] = (1 - [.sup.2]).sup.-1/2] (6b)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Kepler problem of the earth (6c) in orbit around the sun
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6d)
Symmetric top with one point fixed using Euler angles
[psi] = rotation about top axis [empty set] = precession about vertical [theta] = angle of inclination from vertical
In writing Lagrangians it is good to keep in mind the following details about three commonly used coordinate systems with their respective differential lengths d[s.sub.i] = [h.sub.i]d[q.sub.i].
cartesian: x, y, z (7a) dx, dy, dz
cylindrical: [rho], [empty set], z (7b) d[rho], [rho]d[empty set], dz
spherical: r, [theta], [empty set] (7c) dr, rd]theta], r sin [theta]d[empty set]
Lagrangians are important because, by Hamilton's principle, the motion from time [t.sub.1] to time [t.sub.2] follows a path that makes the line integral
I = [[integral].sup.[t.sub.2].sub.[t.sub.1]] Ldt (8)
a stationary value. Using this principle we can show that Lagrange's equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
is satisfied for each of the N coordinates [q.sub.i] and their velocities [[??].sub.i].
The motion is often limited by constraints. Of particular interest is a holonomic constraint, which can be expressed in terms of equations involving the positions [r.sub.i] of the particles, but not the velocities
f([r.sub.1], ...,[r.sub.N], [t) = 0 (10)
If the constraint contains an inequality rather than an equal sign, such as the condition [r.sup.2] - [a.sup.2] [greater than or equal to] 0 for being outside a sphere of radius a, then it is not holonomic. The differential df of the function f ([q.sub.i],t)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
where the summation is over the N particles, may be written in the form
[summation] [a.sub.i]d[q.sub.i] + [a.sub.t][d.sub.t] = 0 (12)
where [a.sub.i = [partial derivative]f/[partial derivative][q.sub.i]. If there is more than one constraint equation, then we add an index, writing [a.sub.ji] and [a.sub.jt]. The coefficients [a.sub.ji] of the n constraint equations written as first-order differential equations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
enter the N Lagrangian equations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
through the n Lagrange multipliers [[lambda].sub.j], where the summation over j is from 1 to n. These are now N + n equations to solve and n unknown Lagrange multipliers to be determined. Each [[lambda].sub.j] usually has the dimension of force or torque, and in a typical case it is a reaction force.
Sometimes the constraint can be written in the form of Eq. (12) or (13) even though no function f(q, t) exists. Such a constraint might be called pseudoholonomic; it is not holonomic but it is adequate for the application of the Lagrange multiplier method (14).
Sometimes friction can be taken into account with the aid of a velocity dependent Rayleigh dissipation function D
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
associated with the frictional force [F.sub.f]
[F.sub.f] = -[[nabla].sub.v]D (16)
where the gradient is with respect to the velocities. The associated Lagrange equations are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
The Rayleigh dissipation function corresponds to the force of Eq. (3f) with n = 1.
Each coordinate [q.sub.j] has a conjugate momentum [p.sub.j] defined by the expression
[p.sub.j] = [partial derivative]L/[partial derivative][[??].sub.j] (18)
Sometimes the conjugate momentum is a linear momentum, as m]??], or an angular momentum, as I]??], where I is the moment of inertia, but it can also be more complicated than that, as in the case of a charge q moving in the presence of a magnetic field B = [nabla] A. Several conjugate momenta are:
[p.sub.i] = [gamma]m[v.sub.i] + q[A.sub.i] charge q in an electromagnetic field (19a)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Kepler problem (19b)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] symmetric top with one (19c) one point fixed in Euler angles
4 HAMILTONIAN FORMULATION
We have discussed the properties of the Lagrangian L([q.sub.i], [[??].sub.i], t), which is a function of the generalized coordinates and velocities. There is another function, called the Hamiltonian H([q.sub.i], [p.sub.i], t), which depends on the coordinates [q.sub.i] and their associated conjugate momenta [p.sub.i] defined by Eq. (18), and it is formed from the Lagrangian through the following Legendre transformation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
where the summation is over the coordinates. To carry out this transformation each conjugate momentum [p.sub.i] is determined by differentiating the Lagrangian using Eq. (18). Then the resulting equations are solved for the generalized velocities [[??].sub.i] and the latter are put into Eq. (20). This provides the Hamiltonian H expressed as a function of the [q.sub.i], [p.sub.i] and t variables. Sometimes this procedure is easy, as with the angle of inclination [theta] of the symmetric top (19c) for which we have
[??] = [p.sub.[theta]]/[I.sub.[parallel]] (21)
It is more complicated to determine the other two symmetric top angles [psi] and [empty set] which require the solution of two simultaneous equations.
The equations of motion are found from the canonical equations of Hamilton
[[??].sub.i] = [partial derivative]H/[partial derivative][p.sub.i] (22a)
[[??].sub.i] = [partial derivative]H/[partial derivative][q.sub.i] (22b)
[partial derivative]L/[partial derivative]t = -[partial derivative]H/[partial derivative]t (22c)
These 2N first-order Hamilton differential equations replace the N second-order Lagrange equations (8).
Examples of some Hamiltonians are:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] harmonic oscillator (23a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] charge q in an electromagnetic field (23b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Kepler problem (23c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] symmetric top (23d)
Coordinates that do not appear explicitly in the Lagrangian and Hamiltonian are called cyclic or ignorable. We see from Eqs. (22b) that the momentum [p.sub.i] conjugate to such a coordinate is a constant of the motion, independent of the time, which we will denote by [[alpha].sub.i]
[[??].sub.i] = 0 (24)
[[??].sub.i] = [[alpha].sub.i] (25)
Because of this it is ordinarily easier to use the Hamiltonian formulation for writing the equations of motion for cyclic coordinates, and the Lagrangian approach for the non-cyclic coordinates.
When some coordinates [q.sub.i],..., [q.sub.c] are present explicitly in the Lagrangian and the remainder [q.sub.c+1],..., [q.sub.N] are cyclic, then it is convenient to restrict the term [summation] [q.sub.i][p.sub.i] of Eq. (20) to a summation over the cyclic coordinates and thereby form the Routhian R
R = R([q.sub.1], ..., [q.sub.N], [[??].sub.i], ..., [[??].sub.c], [p.sub.c+1], ..., [p.sub.N]) (26)
which is a function of the non-cyclic velocities [[??].sub.i] and the constant momenta [p.sub.i]. This permits Hamilton's equations to be written for the cyclic coordinates and Lagrange's equations for the non-cyclic ones. For example, the Kepler problem has the cyclic coordinate [theta] and hence from Eqs. (6c) and (19b) the Routhian R(r, [theta], [??], [p.sub.[theta]]) is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)
where, since [theta] is cyclic, the angular momentum [p.sub.[theta]] = m[r.sup.2]]??] is constant in time.
5 VARIATIONAL PRINCIPLES AND VIRTUAL DISPLACEMENTS
There are several general principles associated with the subject of mechanics. We have already encountered Hamilton's principle (8), which states that the line integral of the Lagrangian over time has a stationary value for the correct path of motion. This can be expressed differently by saying that the variation of the line integral I of Eq. (8) with fixed end points [t.sub.1] and [t.sub.2] is zero
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)
In other words, as infinitesimal deviation in the path of integration about its stationary trajectory does not change the value of the integral.
Now let us consider infinitesimal displacements [delta][r.sub.i] in the presence of an applied force [F.sub.app] which is holding a system in equilibrium. The principle of virtual work states that the work done by the applied force, called the virtual work, is zero
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)
and this provides the condition for equilibrium in statics. In dynamics when the applied forces can cause accelerations [??] = [[??].sub.x]/m, we have D'Alembert's principle
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)
which reduces to the principle of virtual work for static conditions.
(Continues...)
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