Lagrangian and Hamiltonian Mechanics

Transcription

1 Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying classical mechanics as formulate by Sir Isaac Newton; this is calle Newtonian mechanics. Newtonian mechanics is mathematically fairly straightforwar, an can be applie to a wie variety of problems. It is not a unique formulation of mechanics, however; other formulations are possible. Here we will look at two common alternative formulations of classical mechanics: Lagrangian mechanics an Hamiltonian mechanics. It is important to unerstan that all of these formulations of mechanics equivalent. In principle, any of them coul be use to solve any problem in classical mechanics. The reason they re important is that in some problems one of the alternative formulations of mechanics may lea to equations that are much easier to solve than the equations that arise from Newtonian mechanics. Unlike Newtonian mechanics, neither Lagrangian nor Hamiltonian mechanics requires the concept of force; instea, these systems are expresse in terms of energy. Although we will be looking at the equations of mechanics in one imension, all these formulations of mechanics may be generalize to two or three imensions. Newtonian Mechanics We begin by reviewing Newtonian mechanics in one imension. In this formulation, we begin by writing Newton s secon law, which gives the force F require to give an acceleration a to a mass m: F D ma: (1) Generally the force is a function of x. Since the acceleration a D x=, Eq. (1) may be written F.x/ D m x : () This is a secon-orer orinary ifferential equation, which we solve for x.t/ to fin the position x at any time t. Solving a problem in Newtonian mechanics then consists of these steps: 1. Write own Newton s secon law (Eq. );. Substitutefor F.x/ the specific force present in the problem; 3. Solve the resulting ifferential equation for x.t/. 1

2 Partial Derivatives The equations of Lagrangian an Hamiltonian mechanics are expresse in the language of partial ifferential equations. We will leave the methos for solving such equations to a more avance course, but we can still write own the equations an explore some of their consequences. First, in orer to unerstan these equations, we ll first nee to unerstan the concept of partial erivatives. You ve alreay learne in a calculus course how to take the erivative of a function of one variable. For example, if f.x/d 3x C 7x 5 then f x D 6x C 35x4 : But what if f is a function of more that one variable? For example, if (3) (4) f.x;y/d 5x 3 y 5 C 4y 7xy 6 (5) then how o we take the erivative of f? In this case, there are two possible first erivatives you can take: one with respect to x, an one with respect to y. These are calle partial erivatives, an are inicate using the backwar-6 in place of the symbol use for orinary erivatives. To compute a partial erivative with respect to x, you simply treat all variables except x as constants. Similarly, for the partial erivative with respect to y, you treat all variables except y as constants. For example, if g.x; y/ D 3x 4 y 7, then the partial erivative of g with respect to x D 1x 3 y 7,since both 3 an y 7 are consiere constants with respect to x. As another example, the partial erivatives of Eq. D 15x y 5 7y D 5x3 y 4 C 8y 4xy 5 (7) Notice that in Eq. (6), the erivative of the term 4y with respect to x is 0,since4y is treate as a constant. Lagrangian Mechanics The first alternative to Newtonian mechanics we will look at is Lagrangian mechanics. Using Lagrangian mechanics instea of Newtonian mechanics is sometimes avantageous in certain problems, where the equations of Newtonian mechanics woul be quite ifficult to solve. In Lagrangian mechanics, we begin by efining a quantity calle the Lagrangian (L), which is efine as the ifference between the kinetic energy K an the potential energy U : L K U (8) Since the kinetic energy is a function of velocity v an potential energy will typically be a function of position x, the Lagrangian will (in one imension) be a a function of both x an v: L.x; v/. The motion of a particle is then foun by solving Lagrange s equation; in one imension D 0 (9)

3 Example: Simple Harmonic Oscillator As an example of the use of Lagrange s equation, consier a one-imensional simple harmonic oscillator. We wish to fin the position x of the oscillator at any time t. We begin by writing the usual expression for the kinetic energy K: K D 1 mv (10) The potential energy U of a simple harmonic oscillator is given by U D 1 kx (11) The Lagrangian in this case is then L.x; v/ D K U (1) D 1 mv 1 kx (13) Lagrange s equation in one D 0 (14) Substituting for L from Eq. (13), we mv 1 mv 1 kx D 0 (15) Evaluating the parital erivatives, we get.mv/ C kx D 0 or, since v D x=, (16) m x C kx D 0; (17) which is a secon-orer orinary ifferential equation that one can solve for x.t/. Note that the first term on the left is ma D F, so this equation is equivalent to F D kx (Hooke s Law). The solution to the ifferential equation (17) turns out to be x.t/ D A cos.!t C ı/; (18) where A is the amplitue of the motion,! D p k=m is the angular frequency of the oscillator, an ı is a phase constant that epens on where the oscillator is at t D 0. Example: Plane Penulum Part of the power of the Lagrangian formulation of mechanics is that one may efine any coorinates that are convenient for solving the problem; those coorinates an their corresponing velocities are then use in place of x an v in Lagrange s equation. For example, consier a simple plane penulum of length ` with a bob of mass m, where the penulum makes an angle with the vertical. The goal is to fin the angle at any time t. In this case we replace 3

4 x with the angle, an we replace v with the penulum s angular velocity!. The kinetic energy K of the penulum is the rotational kinetic energy K D 1 I! D 1 m`! ; (19) where I is the moment of inertia of the penulum, I D m`. The potential energy of the penulum is the gravitational potential energy U D mg`.1 cos / (0) The Lagrangian in this case is then L.;!/ D K U (1) D 1 m`! mg`.1 cos / () Lagrange s D 0 (3) Substituting for 1 m`! mg`.1 cos Computing the partial erivatives, we 1 m`! mg`.1 cos / D 0 m`! C mg` sin D 0: (5) Since! D =,thisgives m` C mg` sin D 0; (6) which is a secon-orer orinary ifferential equation that one may solve for the motion.t/.thefirst term on the left-han sie is the torque on the penulum, so this equation is equivalent to D mg` sin. The solution to the ifferential equation (6) is quite complicate, but we can simplify it if the penulum only makes small oscillations. In that case, we can approximate sin, an the ifferential equation (6) becomes a simple harmonic oscillator equation with solution.t/ 0 cos.!t C ı/; (7) where 0 is the (angular) amplitue of the penulum,! D p g=` is the angular frequency, an ı is a phase constant that epens on where the penulum is at t D 0. Hamiltonian Mechanics The secon formulation we will look at is Hamiltonian mechanics. In this system, in place of the Lagrangian we efine a quantity calle the Hamiltonian, to which Hamilton s equations of motion are applie. While Lagrange s equation escribes the motion of a particle as a single secon-orer ifferential equation, Hamilton s equations escribe the motion as a couple system of two first-orer ifferential equations. 4

5 One of the avantages of Hamiltonian mechanics is that it is similar in form to quantum mechanics, the theory that escribes the motion of particles at very tiny (subatomic) istance scales. An unerstaning of Hamiltonian mechanics provies a goo introuction to the mathematics of quantum mechanics. The Hamiltonian H is efine to be the sum of the kinetic an potential energies: H K C U Here the Hamiltonian shoul be expresse as a function of position x an momentum p (rather than x an v, as in the Lagrangian), so that H D H.x;p/. This means that the kinetic energy shoul be written as K D p =m, rather than K D mv =. Hamilton s equations in one imension have the elegant nearly-symmetrical form x p Example: Simple Harmonic Oscillator (9) (30) As an example, we may again solve the simple harmonic oscillator problem, this time using Hamiltonian mechanics. We first write own the kinetic energy K, expresse in terms of momentum p: (8) K D p m As before, the potential energy of a simple harmonic oscillator is U D 1 kx The Hamiltonian in this case is then H.x;p/ D K C U (31) (3) (33) D p m C 1 kx (34) Substituting this expression for H into the first of Hamilton s equations, we fin x m C 1 kx (36) D p (37) m Substituting for H into the secon of Hamilton s equations, we get p m C 1 kx (39) D kx (40) Equations (35) an (38) are two couple first-orer orinary ifferential equations, which may be solve simultaneously to fin x.t/ an p.t/. Note that for this example, Eq. (35) is equivalent to p D mv, aneq. (38) is just Hooke s Law, F D kx. 5

6 Example: Plane Penulum As with Lagrangian mechanics, more general coorinates (an their corresponing momenta) may be use in place of x an p. For example, in fining the motion of the simple plane penulum, we may replace the position x with angle from the vertical, an the linear momentum p with the angular momentum L. To solve the plane penulum problem using Hamiltonian mechanics, we first write own the kinetic energy K, expresse in terms of angular momentum L: K D L I D L ; (41) m` where I D m` is the moment of inertia of the penulum. As before, the gravitational potential energy of a plane penulum is U D mg`.1 cos /: (4) The Hamiltonian in this case is then H.;L/ D K C U (43) L D C mg`.1 cos / (44) m` Substituting this expression for H into the first of Hamilton s equations, we L C mg`.1 cos m` D L m` Substituting for H into the secon of Hamilton s equations, we get (45) (46) (47) m` D mg` sin C mg`.1 cos / (48) (49) (50) Equations (45) an (48) are two couple first-orer orinary ifferential equations, which may be solve simultaneously to fin.t/ an L.t/. Note that for this example, Eq. (45) is equivalent to L D I!,anEq. (48) is the torque D mg` sin. 6