16.61 Aerospace Dynamics Spring 2003 Lecture #7 Lagrange's equations Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 1 Lecture #7 Lagrange's Equations
16.61 Aerospace Dynamics Spring 2003 Lagrange's equations Joseph-Louis lagrange 1736-1813 http://www-groups.dcs.st-and.ac.uk/-history/mathematicians/lagranGe.html Born in Italy. later lived in berlin and paris Originally studied to be a lawyer Interest in math from reading halleys 1693 work on algebra in optics If I had been rich, I probably would not have devoted myself to mathematics Contemporary of euler Bernoulli Leibniz. D Alembert Laplace, Legendre (Newton 1643-1727) e Contributions o Calculus of variations o Calculus of probabilities o Propagation of sound o Vibrating strings o Integration of differential equations Orbits o Number theory whatever this great man says, deserves the highest degree of consideration, but he is too abstract for youth student at ecole polytechnique Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 1 Lagrange’s Equations Joseph-Louis Lagrange 1736-1813 • http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html • Born in Italy, later lived in Berlin and Paris. • Originally studied to be a lawyer • Interest in math from reading Halley’s 1693 work on algebra in optics • “If I had been rich, I probably would not have devoted myself to mathematics.” • Contemporary of Euler, Bernoulli, Leibniz, D’Alembert, Laplace, Legendre (Newton 1643-1727) • Contributions o Calculus of variations o Calculus of probabilities o Propagation of sound o Vibrating strings o Integration of differential equations o Orbits o Number theory o … • “… whatever this great man says, deserves the highest degree of consideration, but he is too abstract for youth” -- student at Ecole Polytechnique
16.61 Aerospace Dynamics Spring 2003 Why Lagrange(or why NOT Newton) Newton -Given motion deduce forces Rotating launcher FBD of projectile N g Or given forces-solve for motion Spring mass system F m Great for simple systems Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 2 Why Lagrange (or why NOT Newton) • Newton – Given motion, deduce forces ω Rotating Launcher N mg FBD of projectile • Or given forces – solve for motion Spring mass system m1 m2 x1 x2 F x2 t Spring mass system m1 m2 x1 x2 F x2 t Great for “simple systems
16.61 Aerospace Dynamics Spring 2003 What about real systems? Complexity increased by Vectoral equations-difficult to manage Constraints- what holds the system together? No general procedures Lagrange provides: Avoiding some constraints Equations presented in a standard form Termed analytic Mechanics Originated by leibnitz(1646-1716) Motion(or equilibrium)is determined by scalar equations Big picture Use kinetic and potential energy to solve for the motion No need to solve for accelerations(Ke is a velocity term) Do need to solve for inertial velocities Lets start with the answer, and then explain how we get there Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 3 What about “real” systems? Complexity increased by: • Vectoral equations – difficult to manage • Constraints – what holds the system together? • No general procedures Lagrange provides: • Avoiding some constraints • Equations presented in a standard form Termed Analytic Mechanics • Originated by Leibnitz (1646-1716) • Motion (or equilibrium) is determined by scalar equations Big Picture • Use kinetic and potential energy to solve for the motion • No need to solve for accelerations (KE is a velocity term) • Do need to solve for inertial velocities Let’s start with the answer, and then explain how we get there
16.61 Aerospace Dynamics Spring 2003 Define: Lagrangian Function .L=T-V(Kinetic-Potential energies) Lagrange’ s Equation For conservative systems d/aLdL 0)g Results in the differential equations that describe the equations of motion of the system Key point: Newton approach requires that you find accelerations in all 3 directions, equate F=ma, solve for the constraint forces and then eliminate these to reduce the problem to characteristic size Lagrangian approach enables us to immediately reduce the problem to this"characteristic size, we only have to solve for that many equations in the first place The ease of handling external constraints really differentiates the two approaches Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 4 Define: Lagrangian Function • L = T – V (Kinetic – Potential energies) Lagrange’s Equation • For conservative systems 0 i i dL L dt q q ∂ ∂ − = ∂ ∂ • Results in the differential equations that describe the equations of motion of the system Key point: • Newton approach requires that you find accelerations in all 3 directions, equate F=ma, solve for the constraint forces, and then eliminate these to reduce the problem to “characteristic size” • Lagrangian approach enables us to immediately reduce the problem to this “characteristic size” we only have to solve for that many equations in the first place. The ease of handling external constraints really differentiates the two approaches