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## Maple for Lagrangian Mechanics

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**Maple for Lagrangian Mechanics**Frank Wang**Newton**• Newton’s Second Law: Forces, masses, accelerations**Lagrange**• Variational principle formulation: Kinetic energy minus potential energy is minimum.**1-D Motion under Gravity**x x t t**Fastest Path**lifeguard swimmer**Least Action**• Geometric Optics • Classical Mechanics • General Relativity • Quantum Mechanics (Feynman’s Path Integral)**Euler-Lagrange Equation**• The only thing we need to know! • Euler-Lagrange equations gives equations of motion, which are differential equations.**Calculus of Variations**• Finding derivative of a function w.r.t. another function,**Using Maple**• Substitute x(t) and v(t) with symbols, • Differentiate L w.r.t.var1 and var2 • Substitute var1 and var2 back to x(t) and v(t).**Result**• Lagrangian and Newtonian are identical: ma F**Advantages**• Straightforward • Only simple commands: subs, diff, dsolve • No external library • Treating x(t) and v(t) as two separate dependent variables. Maple 8 has a VariationalCalculuspackage.**Lagrangian in 3 Steps**• Perform coordinate transformation to express KE and PE in generalized coordinates. • Employ the Euler-Lagrange equations to derive equations of motion. • Solve differential equations to find the actual path.**Euler-Lagrange Equation**• The only thing we need to know! • Euler-Lagrange equations gives equations of motion, which are differential equations.**Double Pendulum**q1 q2**Lagrangian for Double Pendulum**Maple produces**Two Degrees of Freedom**• For mass 1, • For mass 2,**Gyroscope**• Many simple things can be deduced mathematically more rapidly than they can be really understood in a fundamental or simple sense. Feynman I-20-6**Kepler Problem**• Kinetic energy in polar coordinates: • Potential energy for inverse square law and a quadrupole term:**Symmetry**• Lagrangian of Kepler problem contains no f, • For rcoordinate,**Planetary Motion**• 18th Century: Lagrange discovered that planetary motion corresponds to least action. • 20th Century: Einstein formulated geodesic equation, i.e., the shortest “distance” in a curved space-time.**General Relativity**• Matter tells geometry how to curve, geometry tells matter how to move. • Motion in gravitational field is none other than finding the shortest connection between two points in a curved space-time.**Geometry**• Flat space • Curved space-time (Schwarzschild)**Shortest Path**• The shortest path corresponds to • Lagrangian is the integrand**Lagrangian for GR**• Flat space • Schwarzschild solution**Lagrangian in 3 Steps**• Perform coordinate transformation to express KE and PE in generalized coordinates. • Employ the Euler-Lagrange equations to derive equations of motion. • Solve differential equations to find the actual path.**Euler-Lagrange Equation**• The only thing we need to know! • Euler-Lagrange equations gives equations of motion, which are differential equations.**Application of Maple**• Perform coordinate transformation which otherwise will be very tedious. • Employ chain rule and rearrange equations. • Solve differential equations (using numerical method in most cases), and graph the results.**Conclusions**• Principle of Least Action is a powerful concept. • Maple is an ideal tool to handle this type of problems. • One can apply simple principle to elementary and sophisticated problems, and Maple does all the calculations.