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1. Fundamentals of Mechanics 1.1 Lagrangian Mechanics View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Almae Matris Studiorum Campus Celestial Mechanics - 2015-16 1 Carlo Nipoti, Dipartimento di Fisica e Astronomia, Universit`adi Bologna 22/3/2016 1. Fundamentals of mechanics 1.1 Lagrangian mechanics [LL; GPS; A] 1.1.1 Generalized coordinates ! Particle: point mass ! Particle position vector r. In Cartesian components r = (x; y; z). ! Particle velocity v = dr=dt = r_. In Cartesian components vx = dx=dt, etc... 2 2 2 2 ! Particle acceleration a = d r=dt = v_ = ¨r. In Cartesian components ax = d x=dt , etc... ! N particles =) s = 3N degrees of freedom ! Generalized coordinates: any s quantities qi that define the positions of the N-body system (q = q1; :::; qs) ! Generalized velocities:q _i (q_ =q _1; :::; q_s) ! We know from experience that, given q and q_ for all particles in the system at a given time, we are able to predict q(t) at any later time t. In other words, if all q and q_ are specified =) q¨ are known. ! Equations of motion are ordinary differential equations for q(t) that relate q¨ with q and q_ . The solution q(t) is the path (orbit). 1.1.2 Principle of least action & Euler-Lagrange equations ! Given a mechanical system, we define the Lagrangian function L(q; q_ ; t). L does not depend on higher derivatives, consistent with the idea that motion is determined if q and q_ are given. ! Given two instants t and t , we define the action, which is the functional S(γ) = R t2 L(q; q_ ; t)dt, where γ 1 2 t1 are paths of the form q(t). 2 Laurea in Astronomia - Universita` di Bologna ! The system occupies positions q1 and q2 at time t1 and t2, respectively. Note that in this formalism instead of fixing position and velocity at the initial time t1, we fix positions at the initial and final times. ! Principle of least action (or Hamilton's principle): from t1 to t2 the system moves in such a way that S is an extremum over all paths (often the extremum is such that S is minimized, hence the name \principle of least action"). So, for 1 degree of freedom Z t2 Z t2 Z t2 @L @L δS = δ L(q; q;_ t)dt = [L(q + δq; q_ + δq;_ t) − L(q; q;_ t)] dt = δq + δq_ dt = 0; t1 t1 t1 @q @q_ Now, we have @L @L d(δq) @L d @L d @L δq + = δq + δq − δq; @q @q_ dt @q dt @q_ dt @q_ so the above equation can be rewritten as @L t2 Z t2 @L d @L δS = δq + − δqdt = 0; @q_ @q dt @q_ t1 t1 which is verified for all δq only when @L d @L − = 0; @q dt @q_ because δq(t1) = δq(t2) = 0, as all possible paths are such that q(t1) = q1 and q(t2) = q2. ! Generalizing to the case of s degrees of freedom we have the Euler-Lagrange (E-L) equations: @L d @L − = 0; i = 1; :::; s @qi dt @q_i ! Transformations like L! AL, with A constant, or L!L + dF=dt, where F = F (q; t) do not affect the particles' motion, because L0 = AL =) δS0 = δAS = AδS = 0 () δS = 0; and Z t2 0 0 dF L = L + dF=dt =) S = S + dt = S + F (q2; t2) − F (q1; t1) = S + C; t1 dt where C is a constant (independent of q, q_ ) =) δS0 = δS, so δS0 = 0 () δS0 = 0. 1.1.3 Inertial frames ! Galileo's relativity principle. Inertial coordinate systems are such that 1. All the laws of nature at all times are the same in all inertial systems. 2. All coordinate systems that move with respect to an inertial sytem with uniform rectilinear motion are inertial. ! In an inertial reference system space is homogeneous and isotropic and time is homogeneous. Celestial Mechanics - 2015-16 3 ! Free particle: particle subject to no force. ! In any inertial reference frame a free particle that is at rest at a given time will remain at rest at all later times. ! Lagrangian of a free particle cannot contain explicitly the position vector r (space is homogeneous) or the time t (time is homogeneous) and cannot depend on the direction of v (space is isotropic) =)L = L(v2) ! More specifically, it can be shown (see LL) that for a free particle 1 L = mv2; 2 where m is particle mass. T = (1=2)mv2 is the particle kinetic energy. ! In Cartesian coordinates q = r = (x; y; z) and q_ = v = (vx; vy; vz), so for a free particle the Lagrangian is 1 2 2 2 L = 2 m(vx + vy + vz ). The E-L equations for a free particle are d @L = 0; dt @v so for the x component d @L dvx = m = 0 =) vx = const; dt @vx dt and similarly for y and z components. =) dv=dt = 0, which is the law of inertia (Newton's first law of motion). 1.1.4 Lagrangian of a free particle in different systems of coordinates ! Let us write the Lagrangian of a free particle in different systems of coordinates. Note that, if dl is the infinitesimal displacement, v2 = (dl=dt)2 = dl2=dt2. ! In Cartesian coordinates dl2 = dx2 + dy2 + dz2, so 1 L = m(_x2 +y _2 +z _2) 2 ! In cylindrical coordinates dl2 = dR2 + R2dφ2 + dz2, so 1 L = m(R_ 2 + R2φ_2 +z _2) 2 ! In spherical coordinates dl2 = dr2 + r2d#2 + r2 sin2 #dφ2, so 1 L = m(_r2 + r2#_2 + r2 sin2 #φ_2) 2 ! The above can be derived also by taking the expression of the position vector r, differentiating and squaring (Problem 1.1; see e.g. BT08 app. B). ! Alternatively, we can derive the same expressions for the cylindrical and spherical coordinates, starting from the Cartesian coordinates (Problem 1.2). 4 Laurea in Astronomia - Universita` di Bologna Problem 1.1 Derive the Lagrangian of a free particle in cylindrical coordinates starting from the expression of the position vector r. r = ReR + zez dr = R_ e + Re_ +z _e ; dt R R z because e_ z = 0. Now, eR = cos φex + sin φey and eφ = − sin φex + cos φey So deR = (− sin φex + cos φey)dφ deR = eφdφ _ e_ R = φeφ so dr = R_ e + Rφ_e +z _e ; dt R φ z 2 2 dr 2 2 2 2 v = = R_ + R φ_ +z _ : dt Problem 1.2 Derive the Lagrangian of a free particle in spherical coordinates starting from the Cartesian coordinates. For instance, in spherical coordinates r, #, φ we have x = r sin # cos φ, y = r sin # sin φ, z = r cos #; so x_ =r _ sin # cos φ + r cos # cos φ#_ − r sin # sin φφ,_ y_ =r _ sin # sin φ + r cos # sin φ#_ + r sin # cos φφ,_ z_ =r _ cos # − r sin ##;_ then v2 =x _ 2 +y _2 +z _2 =r _2 + r2#_2 + r2 sin2 #φ_2: The Lagrangian of a free particle in spherical coordinates is 1 L = m r_2 + r2#_2 + r2 sin2 #φ_2 : 2 1.1.5 Lagrangian of a system of particles ! Additivity of the Lagrangian: take two dynamical systems A and B. If each of them were an isolated system, they would have, respectively, Lagrangians LA and LB. If they are two parts of the same system (but so Celestial Mechanics - 2015-16 5 distant that the interaction is negligible) the total Lagrangian must be L = LA + LB. ! So, for a system of non-interacting particles X 1 L = m v2; 2 a a a=1;:::;N were the subscript a identifies the a-th the particle with velocty va. So L = T , where T is the total kinetic energy. ! Closed system: system of particles that interact, but are not affected by external forces. ! Lagrangian for a closed system of interacting particles: L = T − V where T is the kinetic energy and V is potential energy. ! The potential energy V depends only on the positions of the particles: V = V (q). This is a consequence of the assumption that the interaction is instantaneously propagated: a change in position of one of the particles instantaneously affects the force experienced by the other particles. ! In Cartesian coordinates we have q = ra = (xa; ya; za) positions (a = 1; :::; N) and q_ = va = (vx;a; vy;a; vz;a) (velocities), so the Lagrangian for a closed system of N particles is 1 X L = m v2 − V (r ; r ; :::r ) 2 a a 1 2 N a=1;::;N ! We have seen that in general T = T (q; q_ ): see, for instance, the expression of T in cylindrical or spherical coordinates. ! The transformation between Cartesian and generalized coordinates is given by X @fx;a xa = fx;a(q1; :::qs); x_ a = q_k; @qk k=1;:::;s and similarly for ya and za. By substituting these expressions in the Lagrangian in Cartesian coordinates it follows that in generalized coordinates 1 X L = A (q)_q q_ − V (q); 2 ik i k i;k where i = 1; : : : ; s and k = 1; : : : ; s, where s is the number of degrees of freedom (s = 3N for a system of N particles). So the kinetic energy generally depends also on the generalized coordinates q. ! Newton's second law of motion. Consider the Lagrangian in Cartesian coordinates. Applying Euler-Lagrange equations @L d @L − = 0; @ra dt @va 6 Laurea in Astronomia - Universita` di Bologna we get the equations of motion: @V @V mav_ a = − ; i:e: max¨a = − ; etc: @ra @xa i.e.
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