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While I was studying the Euler-Lagrange equation, Hamilton's principle of least action and geodesics, I started to wonder how to find the equations of motion of a particle restricted to a particular surface (e.g. the sphere). I know that the Lagrangian is defined as $L=K-U$ where $K$ is the kinetic energy and $U$ is the potential energy, and that the Euler-Lagrange equations give the stationary points of a functional (e.g the action). Is it true that if a particle was constrained to move on a particular surface, it would follow the path of a geodesic on that surface? For example, on a sphere, would it follow great circles in its motion?

The more general question I seek to answer is this: is it possible to model the path $\gamma(t)$ a particle will take as a geodesic on a manifold $M$(or simply surface)?

To be a little more concrete, take the examples of projectile motion without drag, where the particle is launched with initial velocity $v_0$ at an angle of $\theta$ from the horizontal at an initial height $H_0$. Here are the functions of time that describe their position on a Cartesian coordinate system: $$y(t)=H_0+v_0\sin(\theta)t-\frac{1}{2}gt^2$$ $$x(t)=v_0\cos(\theta)t.$$ Here, $g$ is the acceleration due to gravity that is approximated as constant throughout the trajectory of flight. Going back to my question, is there a surface $S\subset\mathbb{R}^3$ on which the path of projectile motion this particle takes is a geodesic? How should I find such a path?


Update:

Ok. Let's say that a particle is constrained a to move on a frictionless surface $S(x, y, z) \subset \mathbb{R}^3$ under the influence of gravity, where $F_g=-mg\hat{k}$ where $\hat{k}$ is the unit vector in the upwards z direction. Our Lagrangian is then $$L=\frac{1}{2}m\left(\dot{x}^2+\dot{y}^2+\dot{z}^2\right)-mgz$$ We then have 3 Euler-Lagrange differential equations to solve to find the stationary points and minimize the action. However, this is not taking into account that the particle has to stay on the surface $S$. We can add the following restrictions (?): $$x(t)=S_x$$ $$y(t)=S_y$$ $$z(t)=S_z$$ where $x, y, z$ are function of time derived form the Euler-Lagrange equation that tell us where the particle will be located, and $S_i$ represents the $i^{th}$ coordinate (e.g x-coordinate). One idea that I had in order to find the functions of time that conform to these restrictions is the use of Lagrange multipliers. However, I am not sure how to apply this technique to this problem.

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  • $\begingroup$ In the case of holonomic constraints, simply use the surface equation $S(x(t),y(t),z(t))=0$ $\endgroup$ – Alex Trounev Feb 19 at 17:20

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