TY - JOUR
T1 - Finding extremals of Lagrangian actions
AU - Noakes, Lyle
AU - Zhang, Erchuan
PY - 2023/1
Y1 - 2023/1
N2 - x0,xT is an element of M, we look for an extremal x : [0, T] -> M of the action fT Given a smooth m-manifold M, a smooth Lagrangian L :TM -> R and endpoints 0 L(x(t), ?x(t))dt satisfying x(0) = x(0) and x(T) = x(T). When interpolating between endpoints, this amounts to a 2-point boundary value problem for the Euler-Lagrange equation. Single or multiple shooting is one of the most popular methods to solve boundary value problems, but the efficiency of shooting and the quality of solutions depends heavily on initial guesses. In the present paper, by dividing the interval [0, T] into several sub-intervals, on which extremals can be found efficiently by shooting when good initial guesses are available from the geometry of a variational problem, we then adjust all junctions by finding zeros of vector fields associated with the velocities at junctions with Newton's method. We discuss the cases where L is the difference between kinetic energy and potential, M is a hypersurface in Euclidean space, or M is a Lie group. We make some comparisons in numerical experiments for a double pendulum, for obstacle avoidance by a moving particle on the 2-sphere, and for obstacle avoidance by a planar rigid body. (C) 2022 Elsevier B.V. All rights reserved.
AB - x0,xT is an element of M, we look for an extremal x : [0, T] -> M of the action fT Given a smooth m-manifold M, a smooth Lagrangian L :TM -> R and endpoints 0 L(x(t), ?x(t))dt satisfying x(0) = x(0) and x(T) = x(T). When interpolating between endpoints, this amounts to a 2-point boundary value problem for the Euler-Lagrange equation. Single or multiple shooting is one of the most popular methods to solve boundary value problems, but the efficiency of shooting and the quality of solutions depends heavily on initial guesses. In the present paper, by dividing the interval [0, T] into several sub-intervals, on which extremals can be found efficiently by shooting when good initial guesses are available from the geometry of a variational problem, we then adjust all junctions by finding zeros of vector fields associated with the velocities at junctions with Newton's method. We discuss the cases where L is the difference between kinetic energy and potential, M is a hypersurface in Euclidean space, or M is a Lie group. We make some comparisons in numerical experiments for a double pendulum, for obstacle avoidance by a moving particle on the 2-sphere, and for obstacle avoidance by a planar rigid body. (C) 2022 Elsevier B.V. All rights reserved.
KW - Euler-Lagrange equation
KW - Jacobi equation
KW - Shooting
KW - Newton?s method
KW - Leapfrog
KW - NEWTONS METHOD
KW - CONVERGENCE
KW - AVOIDANCE
KW - ALGORITHM
KW - PLANAR
UR - http://www.scopus.com/inward/record.url?scp=85140143346&partnerID=8YFLogxK
U2 - 10.1016/j.cnsns.2022.106826
DO - 10.1016/j.cnsns.2022.106826
M3 - Article
SN - 1007-5704
VL - 116
JO - Communications in Nonlinear Science and Numerical Simulation
JF - Communications in Nonlinear Science and Numerical Simulation
M1 - 106826
ER -