© Wing Kam Liu, Eduard G. Karpov,Harold Park, David Farrell, Dept. of Mechanical Engineering, Northwestern University,1A Visualization Approach to the Principles of Classical Lagrangian/Hamiltonian Mechanics and its Relations to Molecular Dynamics© Wing Kam Liu, Eduard G. Karpov,Harold Park, David Farrell, Dept. of Mechanical Engineering, Northwestern University,2Reading Assignment• H. Goldstein, Classical mechanics, 2nded., 1980 (3rded., 2002), pages 1-21, 35-45, 339-347.• L. D. Landau, E. M. Lifshitz, Mechanics, 3rded. reprinted since 1976 till 1995, pages 1-10, 131-133.• J.M. Haile, Molecular Dynamics Simulation, Wiley, 2002, pages 46-59, 188-204, 224-234, 277-282.© Wing Kam Liu, Eduard G. Karpov,Harold Park, David Farrell, Dept. of Mechanical Engineering, Northwestern University,3Particles (Material Mass Points)Particle, or material point, or mass point is a mathematical model of a body whose dimensions can be neglected in describing its motion.Particle is a dimensionless object having a non-zero mass.Particle is indestructible; it has no internal structure and no internal degrees of freedoms.Classical mechanics studies “slow” (v<< c) and “heavy” (m >> me) particles. Examples:1) Planets of the solar system in their motion about the sun2) Atoms of a gas in a macroscopic vessel Note: Spherical objects are typically treated as material points, e.g. atoms comprising a molecule. The material point points are associated with the centers of the spheres. Characteristic physical dimensions of the spheres are modeled through particle-particle interaction.© Wing Kam Liu, Eduard G. Karpov,Harold Park, David Farrell, Dept. of Mechanical Engineering, Northwestern University,4Generalized CoordinatesGeneralized coordinates are given by a minimum set of independent parameters (distances and angles) that determine any given state of the system.Standard coordinate systems are Cartesian, polar, elliptic, cylindrical and spherical.Other systems of coordinates can also be chosen. We will look for a basic form of the equation which is invariant for all coordinate systems.12(, ,..., )sqq qExamples: Material point in xy-plane Pendulum Sliding suspension pendulumWe are looking for a basic form of equation motion, which is valid for all coordinate systems.yxjjx12,qxqy==1qϕ=12,qxqϕ==© Wing Kam Liu, Eduard G. Karpov,Harold Park, David Farrell, Dept. of Mechanical Engineering, Northwestern University,5Least Action (Hamilton’s) PrincipleLagrange function (Lagrangian)Action integralTrajectory variation Least action, or Hamilton’s, principle21(,,) 0ttSLqqtdtδδ==∫12 12( , ,..., , , ,..., , )ssLq q q q q q t ,ab acSSSS<<2()qt1()qtbSaScS1tt2t()qt21(,,)ttSLqqtdt=∫12() (), ( ) ( ) 0qt qt qt qtδδδ+==Action is minimum along the true trajectory:The main task of classical dynamics is to find the true trajectories (laws of motion) for all degrees of freedom in the system.© Wing Kam Liu, Eduard G. Karpov,Harold Park, David Farrell, Dept. of Mechanical Engineering, Northwestern University,6Lagrangian Equation of MotionLagrange function (Lagrangian)Least action principle21(,,) 0ttSLqqtdtδδ==∫0, 1,2,...,dL Lsdt q qααα∂∂−= =∂∂12 12( , ,..., ; , ,..., , )ssLqq q qq qt Lagrangian equation is based on the least action principle only, and it is valid for all coordinate systems.Substitution of the Lagrange function into the action integral with further application of the least action principle yields the Lagrangian equation motion:© Wing Kam Liu, Eduard G. Karpov,Harold Park, David Farrell, Dept. of Mechanical Engineering, Northwestern University,7Summary of the Lagrangian Method1. The choice of s generalized coordinates (s – number of degrees of freedom).2. Derivation of the kinetic and potential energy in terms of the generalized coordinates3. The difference between the kinetic and potential energies gives the Lagrangian function.4. Substitution of the Lagrange function into the Lagrangian equation of motion and derivation of a system of s second-order differential equations to be solved.5. Solution of the equations of motion, using a numerical time-integration algorithm.6. Post-processing and visualization.© Wing Kam Liu, Eduard G. Karpov,Harold Park, David Farrell, Dept. of Mechanical Engineering, Northwestern University,8Lagrange Function: ExampleParticle in a circular cavity()2222()2RxymLxyeβα−−+=+−1) Kinetic energy; 2) Potential energy of repulsion between the particle and cavity wall.yx12,qxqy==© Wing Kam Liu, Eduard G. Karpov,Harold Park, David Farrell, Dept. of Mechanical Engineering, Northwestern University,9Particle in a Circular Cavity: Lagrange Function Derivation(, ) (, )LTxy Uxy=−General formKinetic energyPotential energyThe total Lagrangian22()2mTxy=+()()22RxyRrUe eββαα−−+−−==The potential energy grows quickly and becomes larger than the typical kinetic energy, when the distance r between the particle and the center of the cavity approaches value R.R is the effective radius of the cavity. At r < R, U does not alter the trajectory. βis a relative scaling factor: the potential energy growths in eβtimes between r = R and r = R+1.()2222(,,, ) ( )2RxymLxyxy x y eβα−−+=+−  yxrR© Wing Kam Liu, Eduard G. Karpov,Harold Park, David Farrell, Dept. of Mechanical Engineering, Northwestern University,10Particle in a Circular Cavity: Equation of Motion and SolutionLagrangian function and equations: Potential barrier:(0) 2.522nm, (0) 0, (0) 0, (0) 30nm/sxxyy=− ===27 2 210 J, () () ()rt x tytα−==+194nm , 10nm, 10 kgRmβ−−===Parameters:Initial conditions:()22(())22(())() ()/ ()()2() ()/ ()RrtRxyRrtmx t e x t r tmLxyemy t e y t r tβββαβααβ−−−−+−−⎧=−=+−−⇒⎨=⎩© Wing Kam Liu, Eduard G. Karpov,Harold Park, David Farrell, Dept. of Mechanical Engineering, Northwestern University,11Predictable and Chaotic SystemsDivergence of trajectoriesin predictable systems Divergence of trajectoriesin chaotic (unpredictable) systemsTrajectories in MD systems are unpredictable/unstable; they are characterized by a random dependence on initial conditions. ()()(), (0) : , (0) , (0)qqtq qtq qtq Ctδ=++∼()()(), (0) : , (0) , (0)tqqtq qtq qtq eλδδ=++∼Variance of the trajectory depends linearly on time Variance of the trajectory depends exponentially on time (J.M. Haile, Molecular