Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0, the initial positions an velocities r i t 0 ) an ṙ i t 0 ) for all point masses forming the system. Another formulation for the problem coul be to only give the positions of all point masses in the system, at the initial time t an final time t of the motion uner consieration. To see why this is mathematically equivalent, consier first t such that t = t t 0. Specifying r i t ) an r i t ) is then tantamount to specifying r i t ) an ṙ i t ). By continuity, for t sufficiently small, the motion remains unequivocally etermine if one specifies r i t ) an r i t ). As t grows arbitrarily, however, one can fin several possible motions with the same en points. The principle of stationary action, sometimes also calle less accurately the principle of least action, says that among all possible paths from r i t ) an r i t ), the physically realizable paths are the paths that extremize a functional calle the action S efine as follows Sq,..., q N ] = t Lq t),..., q N t), q t,..., q N, t) ) where L is calle the Lagrangian of the system. Note that the functions q i t) are generalize coorinates; they o not necessarily have to be Cartesian coorinates or other stanar coorinates. q i oes not have to have units of meter, an q i / oes not have to have the units of velocity. What the principle of stationary action is The principle of stationary action can be seen as a reformulation of the question of fining the proper evolution of a system in configuration space from a ifferential formulation to a variational/integral formulation. This has several avantages, which we briefly mentione in the last lecture, an which we will highlight as we start to use the variational formulation in practical examples. What the principle of stationary action is not The principle of stationary action oes not contain any new physics. All the physics is in fining appropriate Lagrangians o escribe a system of interest. We will soon learn a simple metho to construct Lagrangians for a large class of physical systems. However, there is no general metho to construct a Lagrangian for every system. In fact, in new fiels of physics, fining an appropriate Lagrangian to escribe the ynamics of a system can be a major accomplishment in its own right. Examples Before we learn how to construct Lagrangians for a certain class of physical systems, let us first see explicit variational formulations for the first two simple examples we looke at in the previous lecture. Newton s apple We saw that Newton s equations for Newton s apple are z = g ) The Newtonian ynamics for the apple can be written in a variational form as follows. A Lagrangian for the free-fall of an object suject to gravity is Lz, z, t) = ) z gz 3) where the mass of the object has been set to m = without any loss of generality. The action for the motion between times t an t thus is t Sz] = Lz, z t ) t, t) = z gz] t
As we have seen in our review of the calculus of variation, extremizing the action with fixe enpoints zt ) an zt ) leas to the following Euler-Lagrange equation for zt): Lz, z ), t) = Lz, z, t) 4) We have So Equation 4) can be rewritten as as expecte. The simple penulum A Lagrangian for the simple penulum is Minimizing the action Lz, z z, t) =, Lz, z, t) = g z = g Lθ, θ/, t) = ) θ l + gl cos θ 5) Sθ] = t t l ) θ + gl cos θ] leas to the E-L equations Lθ, θ/, t)) = Lθ, θ/, t) 6) We have Lθ, θ/, t) = l θ, Lθ, θ/, t) = gl sin θ so Equation 6) takes the form l g θ = gl sin θ θ + l sin θ = 0 which agrees with the secon-orer ODE we ha previously obtaine for that system. 3 Arbitrariness in the efinition of the Lagrangian Suppose a Lagrangian L escribes the ynamics of a system of point masses. Then, given the form of the E-L equation L) = L it is clear that the Lagrangian L = AL + B, where A an B are two constants, leas to the same equations of motion for the system. More interestingly, the equations of motion are also recovere if one replaces L with the Lagrangian L efine by L = L + ft, qt)) 7) Inee, if S is the action associate with the Lagrangian L an S is the action associate with the Lagrangian L, then Equation 7) implies t S q] = Sq] + t ft, qt)) = Sq] + ft, qt )) ft, qt )) 8) The principle of stationary action states that the physically relevant trajectory in configuration space is obtaine by extremals of the action holing the initial an final times an positions fixe. Since t, t, qt ) an qt ) are fixe, the extremals of S are the same as those of S, so both L an L can be use to escribe the motion of the system.
4 Hamilton s principle 4. Statement of the principle An important class of problems in classical mechanics is the class of problems that involves conservative forces only. Conservative forces are forces that can be expresse as the graient of a scalar function one often calls a potential, that epens only on time an the positions of the particles: F = V t, rt)). For these situations, the construction of a Lagrangian is straightforwar, an given by Hamilton s principle, which we state below: A system of point masses for which the forces are erive from a potential energy that is inepenent of velocity evolves along a path q for which the action Sq] = t t Lt, qt), q ) is stationary with respect to variations of the path q that leave the enpoints fixe, where L = T V is the ifference between the kinetic energy T an the potential energy V. 4. Illustration The examples provie in section are illustrations of Hamilton s principle. Let us stuy one more important example. Newton s law of gravitation tells us that two objects with masses m an M separate by a istance R exert a force with magnitue F G = G mm R on each other, an the force is an attractive force, irecte along the line joining the centers of gravity of the two objects. G 6.67 0 N.m.kg is calle the gravitational constant. We will consier the particular case in which M m, so the object of mass M is essentially at rest, an the lighter object, with mass m, orbits aroun the larger object. This is a very goo approximation for the motion of the Earth aroun the sun, or satellites, natural the Moon) or not, orbiting aroun Earth. It easy to show that the motion of the object of mass m is containe in a plane. If we take the center of gravity of the massive object as the origin of our coorinate system, we can write that the force acting on the light object is F G = GmM r r 3 Let us compute the quantity r r ṙ) = r r = r GM r 3 = 0 where the secon equality was obtaine by applying Newton s law. We conclue that Lt) = r ṙ = Cst = L0) = r0) ṙ0) 9) This means that the motion of the object of mass m is at all times in the plane perpenicular to L0). The analysis of the motion of the object can thus be reuce to stuying a system with two egrees of freeom. Let us consier the polar coorinate system r, θ) in the plane perpenicular to L0), with the origin fixe at the center of gravity of the massive object. In this coorinate system, the force F G can be written as F G = V where V = GmM 0) r is the gravitational potential. We can therefore use Hamilton s principle to write a Lagrangian for the object of mass m. The kinetic energy of the object is T = m r + r θ ) ] 3
so a Lagrangian for the object is Lr, θ, r, θ, t) = T V = m r + r θ ) ] + GmM r ) The equations for the motion of the object are given by the E-L equations. For the r coorinate, the E-L equation is 3Lr, θ, r, θ, t) = Lr, θ, r, θ ), t) r θ = r GM r ) The E-L equation for the θ coorinate is 4Lr, θ, r, θ, t) = Lr, θ, r, θ, t) r θ ) = 0 3) Equation 3) tells us that the quantity r θ is a conserve quantity of the motion. This quantity is calle the angular momentum. It is the magnitue of the vector L introuce in Equation 9). We will soon see that this quantity is conserve for all potentials V such that V = V r ), sometimes calle central potentials the associate forces are calle central forces). Let us write r θ = Γ = Cst Plugging Γ into Equation ), we have r = Γ r 3 GM r Multiplying this equation by r/, one fins ) r + Γ r GM r ] = 0 We conclue that ) r + Γ r GM r = Cst = E 4) The total energy of the system is conserve: the first two terms correspon to the kinetic energy, the last term is the potential energy. Equation 4) can be seen as the energy equation for a system with one egree of freeom, with an effective potential energy V eff r) = Γ r GM r This potential is ominate by the attractive gravitational part at large istances, but ominate by the repulsive angular momentum part at short istances. V eff r) is shown in Figure for the particular case Γ = GM =. Since 4) can be rewritten as ) r = E V eff r) point masses with E < 0 have boune orbits: there exists r in the attractive part of the potential such that E = V eff r ), an this r is a turning point of the trajectory. On the other han, point masses with E > 0 have unboune orbits. We will soon see that there are very efficient ways of fining out from the outset, from the functional epenence of the Lagrangian, that energy an angular momentum are conserve in the system stuie here. We coul have save a bit of algebra. Still, it oes not hurt too much to get practice with the umb ways at first. 4
.5 V eff r) 0.5 0 0.5 0 4 6 8 0 r Figure : V eff r) as a function of r for Γ = GM = A little igression regaring the classical -boy problem It turns out that any system consisting of two boies with masses m an m, at positions r t) an r t) an each exerting the force F = ±G m m r r 3 r r ) on the other boy can be stuie within the framework we presente above, even if the masses m an m are comparable an none of the boies can be consiere static. Here is the reason why. Newton s equations for the boies are Let us efine the position vectors m r = G m m r r 3 r r ) 5) m r = G m m r r 3 r r ) 6) R = m r + m r m + m r = r r R is the position of the center of mass of the system. Aing Equations 5) an 6), one fins the following equation for R: R = 0 7) This is expecte. The system of the two boies is isolate, so in the absence of external forces, the center of mass of the whole system has a constant velocity. Subtracting 6)/m from 5)/m, one fins a secon orer ODE for r: µ r = G m m r 3 r 8) with µ = m m /m + m ), which is often calle the reuce mass. Equation 8) escribes the motion of a particle of mass µ an position vector r with respect to a fixe center of force. This is precisely the situation we have stuie with the Lagrangian formulation above. 5
The separate motion of the two boies is obtaine by solving for r using either Newton s approach or a variational approach), an reconstructing r t) an r t) with the relations m r = R + r m + m m r = R r m + m 6