Given a smooth $m$-manifold $M$, a smooth Lagrangian $L:TM\to \mathbb{R}$ and endpoints $x_0,x_T\in M$, we look for an extremal $x:[0,T]\to M$ of the action ${\int }_0^{T}L(x\left(t\right),\dot{x}\left(t\right))\mathrm{d}t$ satisfying $x\left(0\right)=x_0$ and $x\left(T\right)=x_T$. 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.

给定一个光滑的$米$-歧管$米$, 一个光滑的拉格朗日$\mathrm{大号}：吨米\to \mathbb{R}$和端点$X_0,X_{吨}\in 米$，我们寻找一个极值$X：[0,吨]\to 米$行动的${\int }_0^{吨}\mathrm{大号}(X\left(吨\right),\dot{X}\left(吨\right))\mathrm{d}吨$令人满意的$X\left(0\right)=X_0$和$X\left(吨\right)=X_{吨}$. 在端点之间进行插值时，这相当于欧拉-拉格朗日方程的2 点边界值问题。单次或多次射击是解决边值问题最流行的方法之一，但射击的效率和解决方案的质量在很大程度上取决于初始猜测。在本文中，通过划分区间$[0,吨]$分成几个子区间，当可以从变分问题的几何中获得良好的初始猜测时，可以通过射击有效地找到极值，然后我们通过使用牛顿方法找到与连接处的速度相关的向量场的零点来调整所有连接处。我们讨论的情况$\mathrm{大号}$是动能和势能之差，$米$是欧几里得空间中的超曲面，或$米$是李群。我们在数值实验中对双摆、2-球体上的运动粒子避障和平面刚体避障进行了一些比较。