Question

(a) Write down the Lagrangian for a particle moving in three dimensions under the influence of a conservative central force with potential energy $U(r),$ using spherical polar coordinates $(r,\theta ,\varphi )$. (b) Write down the three Lagrange equations and explain their significance in terms of radial acceleration, angular momentum, and so forth. (The $\theta$ equation is the tricky one, since you will find it implies that the $\varphi$ component of $\ell$ varies with time, which seems to contradict conservation of angular momentum. Remember, however, that ${\ell }_{\varphi }$ is the component of $\ell$ in a variable direction.) (c) Suppose that initially the motion is in the equatorial plane (that is, ${\theta }_0=\pi /2$ and ${\dot{\theta }}_0=0$ ). Describe the subsequent motion. (d) Suppose instead that the initial motion is along a line of longitude (that is, ${\dot{\varphi }}_0=0$ ). Describe the subsequent motion.

Short answer

The Lagrangian describes the system's dynamics; Lagrange's equations express conservation laws and radial processes. The motion continues in the equatorial plane or precesses along a longitude, depending on initial conditions.

Step-by-step solution

Write the Lagrangian

The Lagrangian for a particle moving in three dimensions with position represented in spherical polar coordinates $(r,\theta ,\varphi )$ is given by the kinetic energy minus the potential energy. The kinetic energy in spherical coordinates is:$$T=\frac{1}{2}m({\dot{r}}^2+r^2{\dot{\theta }}^2+r^2{\sin }^2\theta {\dot{\varphi }}^2)$$where $m$ is the mass of the particle and $\dot{r},\dot{\theta },\dot{\varphi }$ are the time derivatives of the coordinates. The potential energy due to the conservative central force is $U(r)$. Therefore, the Lagrangian is:$$L=T-U=\frac{1}{2}m({\dot{r}}^2+r^2{\dot{\theta }}^2+r^2{\sin }^2\theta {\dot{\varphi }}^2)-U(r)$$

Derive Lagrange's Equations

Lagrange's equations are derived from the Lagrangian $L$, using the Euler-Lagrange equation $\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{q}}_i}\right)-\frac{\partial L}{\partial q_i}=0$ for each generalized coordinate $q_i=r,\theta ,\varphi$:1. **Radial equation $r$:**$$\frac{d}{dt}(m\dot{r})-mr({\dot{\theta }}^2+{\sin }^2\theta {\dot{\varphi }}^2)+\frac{\partial U}{\partial r}=0$$2. **Polar angle equation $\theta$:**$$\frac{d}{dt}(m{r}^2\dot{\theta })-m{r}^2\sin \theta \cos \theta {\dot{\varphi }}^2=0$$3. **Azimuthal angle equation $\varphi$:**$$\frac{d}{dt}(m{r}^2{\sin }^2\theta \dot{\varphi })=0$$These equations describe the radial acceleration, conservation of angular momentum, and variations in angular dynamics.

Analyze Motion in the Equatorial Plane

When the motion is in the equatorial plane, ${\theta }_0=\frac{\pi }{2}$ and ${\dot{\theta }}_0=0$, implying $\theta =\frac{\pi }{2}$ remains constant. In this case:- The radial equation determines radial acceleration based on $\dot{r}$.- The $\varphi$-equation $$\frac{d}{dt}(m{r}^2\dot{\varphi })=0$$ shows that the angular momentum around the $z$-axis is conserved. The motion remains purely in the $\frac{\pi }{2}$ plane.

Analyze Motion Along Line of Longitude

If the motion initially is along a line of longitude with ${\dot{\varphi }}_0=0$:- Initially, there is no azimuthal motion ($\dot{\varphi }=0$).- The $\theta$-equation requires $\theta$ to change as the $\varphi$-component of angular momentum can vary; this leads to a precessional motion.- Radial and $\theta$ dynamics depend on the potential shape and the initial values of $r$ and $\dot{r}$. The system effectively behaves in two coupled dimensions.

Key concepts

Understanding Radial Acceleration in Spherical Coordinates

In spherical coordinates, radial acceleration is a crucial concept that describes how the speed of a particle changes as it moves towards or away from the center of a potential field. This concept is essential when dealing with central force problems in Lagrangian Mechanics. To grasp radial acceleration, we must start with the radial component of the force acting on the particle.

• Radial acceleration is essentially the second derivative of the radial position coordinate, often denoted as $\ddot{r}$.
• It is derived from the radial component of Lagrange's equations, which for a particle subject to potential energy $U(r)$ is expressed as:
  $$m\ddot{r}=mr({\dot{\theta }}^2+{\sin }^2\theta {\dot{\varphi }}^2)-\frac{\partial U}{\partial r}$$ This equation effectively balances the radial acceleration with centripetal contributions from angular motion, noted by $mr({\dot{\theta }}^2+{\sin }^2\theta {\dot{\varphi }}^2)$.
• Radial acceleration is influenced by how the potential energy changes with radial distance. If the derivative of the potential with respect to $r$, denoted $\frac{\partial U}{\partial r}$ is positive, it suggests a restoring force, akin to a spring trying to return to equilibrium.
  

Understanding these dynamics is key to predicting a particle’s path in a field governed by a central potential.

Exploring Angular Momentum in Lagrangian Mechanics

Angular momentum is a fundamental concept describing rotational motion, and it's deeply linked to the symmetry of circular or spherical systems. In Lagrangian Mechanics, angular momentum appears naturally in the formulation of the equations of motion and provides insights into the conservation laws.

• In spherical coordinates, angular momentum has components associated with the angles $\theta$ (polar angle) and $\varphi$ (azimuthal angle).
• The azimuthal angle component, often more intuitive, is given by:
  $$m{r}^2{\sin }^2\theta \dot{\varphi }$$ The Azimuthal equation $\frac{d}{dt}(m{r}^2{\sin }^2\theta \dot{\varphi })=0$ asserts that this component is conserved, meaning there's no change over time, frequently leading to circular or elliptical paths.
  
• Angular momentum in the $\theta$ component reveals more complexity. It manifests in the equation
  $$m{r}^2\ddot{\theta }+2mr\dot{r}\dot{\theta }-m{r}^2\sin \theta \cos \theta {\dot{\varphi }}^2=0$$ This implies variations especially when initially ${\theta }_{0}eq\frac{\pi }{2}$ (not in the equatorial plane), where the angle $\theta$ changes dynamically, showing potential precession or wobbling in motion.
  

Angular momentum is thereby much more than a static quantity; it helps poviding a map of dynamic rotational movements in constrained motion scenarios.

Navigating Spherical Coordinates in Lagrangian Mechanics

Spherical coordinates are a natural choice when dealing with problems involving central forces and rotational symmetry, such as planets orbiting a star or electrons around a nucleus. Understanding how to work with spherical coordinates in Lagrangian mechanics can simplify complex three-dimensional problems.

• The coordinates $(r,\theta ,\varphi )$ correspond to the radial distance, polar angle, and azimuthal angle, respectively, changing how we perceive movement in space compared to Cartesian coordinates.
• The kinetic energy in spherical coordinates is a sum of its radial, polar, and azimuthal components:
  $$T=\frac{1}{2}m({\dot{r}}^2+r^2{\dot{\theta }}^2+r^2{\sin }^2\theta {\dot{\varphi }}^2)$$ Each component can independently describe motion influences on the particle, allowing targeted analysis and simplification depending on the initial conditions.
  
• Based on how the particle's movement is initiated, different coordinate equations like the one for $\theta$,$\varphi$, or $r$, can take precedence, altering the motion's characteristics profoundly.
  

By breaking down motion into these key components, spherical coordinates help manage the chaos of three-dimensional dynamics, guiding us toward clear solutions in seemingly intractable problems.