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(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 \((r, \theta, \phi)\). (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 \(\phi\) component of \(\ell\) varies with time, which seems to contradict conservation of angular momentum. Remember, however, that \(\ell_{\phi}\) 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, \(\dot{\phi}_{0}=0\) ). Describe the subsequent motion.

Short Answer

Expert verified
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

01

Write the Lagrangian

The Lagrangian for a particle moving in three dimensions with position represented in spherical polar coordinates \((r, \theta, \phi)\) is given by the kinetic energy minus the potential energy. The kinetic energy in spherical coordinates is:\[ T = \frac{1}{2} m \left(\dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2 \theta \dot{\phi}^2 \right) \]where \(m\) is the mass of the particle and \(\dot{r}, \dot{\theta}, \dot{\phi}\) are the time derivatives of the coordinates. The potential energy due to the conservative central force is \( U(r) \). Therefore, the Lagrangian is:\[ L = T - U = \frac{1}{2} m \left(\dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2 \theta \dot{\phi}^2 \right) - U(r) \]
02

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 \( q_i = r, \theta, \phi \):1. **Radial equation \( r \):**\[ \frac{d}{dt}(m \dot{r}) - m r (\dot{\theta}^2 + \sin^2 \theta \dot{\phi}^2) + \frac{\partial U}{\partial r} = 0 \]2. **Polar angle equation \( \theta \):**\[ \frac{d}{dt}(m r^2 \dot{\theta}) - m r^2 \sin \theta \cos \theta \dot{\phi}^2 = 0 \]3. **Azimuthal angle equation \( \phi \):**\[ \frac{d}{dt}(m r^2 \sin^2 \theta \dot{\phi}) = 0 \]These equations describe the radial acceleration, conservation of angular momentum, and variations in angular dynamics.
03

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 \( \phi \)-equation \[ \frac{d}{dt}(m r^2 \dot{\phi}) = 0 \] shows that the angular momentum around the \( z \)-axis is conserved. The motion remains purely in the \( \frac{\pi}{2} \) plane.
04

Analyze Motion Along Line of Longitude

If the motion initially is along a line of longitude with \( \dot{\phi}_0 = 0 \):- Initially, there is no azimuthal motion (\( \dot{\phi} = 0 \)).- The \( \theta \)-equation requires \( \theta \) to change as the \( \phi \)-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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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:
    \[m \ddot{r} = m r (\dot{\theta}^2 + \sin^2 \theta \dot{\phi}^2) - \frac{\partial U}{\partial r}\] This equation effectively balances the radial acceleration with centripetal contributions from angular motion, noted by \( m r (\dot{\theta}^2 + \sin^2 \theta \dot{\phi}^2) \).
  • 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 \( \phi \) (azimuthal angle).
  • The azimuthal angle component, often more intuitive, is given by:
    \[m r^2 \sin^2 \theta \dot{\phi}\] The Azimuthal equation \( \frac{d}{dt}(m r^2 \sin^2 \theta \dot{\phi}) = 0 \) 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
    \[m r^2 \ddot{\theta} + 2m r \dot{r} \dot{\theta} - m r^2 \sin \theta \cos \theta \dot{\phi}^2 = 0\] 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 \( (r, \theta, \phi) \) 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:
    \[T = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2 \theta \dot{\phi}^2)\] 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 \( \theta \),\( \phi \), 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.

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Most popular questions from this chapter

Two equal masses, \(m_{1}=m_{2}=m,\) are joined by a massless string of length \(L\) that passes through a hole in a frictionless horizontal table. The first mass slides on the table while the second hangs below the table and moves up and down in a vertical line. (a) Assuming the string remains taut, write down the Lagrangian for the system in terms of the polar coordinates \((r, \phi)\) of the mass on the table. (b) Find the two Lagrange equations and interpret the \(\phi\) equation in terms of the angular momentum \(\ell\) of the first mass. (c) Express \(\dot{\phi}\) in terms of \(\ell\) and eliminate \(\dot{\phi}\) from the \(r\) equation. Now use the \(r\) equation to find the value \(r=r_{0}\) at which the first mass can move in a circular path. Interpret your answer in Newtonian terms. (d) Suppose the first mass is moving in this circular path and is given a small radial nudge. Write \(r(t)=r_{0}+\epsilon(t)\) and rewrite the \(r\) equation in terms of \(\epsilon(t)\) dropping all powers of \(\epsilon(t)\) higher than linear. Show that the circular path is stable and that \(r(t)\) oscillates sinusoidally about \(r_{\mathrm{o}}\) What is the frequency of its oscillations?

A small cart (mass \(m\) ) is mounted on rails inside a large cart. The two are attached by a spring (force constant \(k\) ) in such a way that the small cart is in equilibrium at the midpoint of the large. The distance of the small cart from its equilibrium is denoted \(x\) and that of the large one from a fixed point on the ground is \(X,\) as shown in Figure \(7.13 .\) The large cart is now forced to oscillate such that \(X=A \cos \omega t,\) with both \(A\) and \(\omega\) fixed. Set up the Lagrangian for the motion of the small cart and show that the Lagrange equation has the form $$\ddot{x}+\omega_{0}^{2} x=B \cos \omega t$$ where \(\omega_{\mathrm{o}}\) is the natural frequency \(\omega_{\mathrm{o}}=\sqrt{k / m}\) and \(B\) is a constant. This is the form assumed in Section 5.5, Equation (5.57), for driven oscillations (except that we are here ignoring damping). Thus the system described here would be one way to realize the motion discussed there. (We could fill the large cart with molasses to provide some damping.)

Write down the Lagrangian for a cylinder (mass \(m\), radius \(R\), and moment of inertia \(I\) ) that rolls without slipping straight down an inclined plane which is at an angle \(\alpha\) from the horizontal. Use as your generalized coordinate the cylinder's distance \(x\) measured down the plane from its starting point. Write down the Lagrange equation and solve it for the cylinder's acceleration \(\ddot{x}\). Remember that \(T=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2},\) where \(v\) is the velocity of the center of mass and \(\omega\) is the angular velocity.

Prove that the potential energy of a central force \(\mathbf{F}=-k r^{n} \hat{\mathbf{r}}(\text { with } n \neq-1)\) is \(U=k r^{n+1} /(n+1)\). In particular, if \(n=1,\) then \(\mathbf{F}=-k \mathbf{r}\) and \(U=\frac{1}{2} k r^{2}\).

Consider two particles moving unconstrained in three dimensions, with potential energy \(U\left(\mathbf{r}_{1}, \mathbf{r}_{2}\right) .\) (a) Write down the six equations of motion obtained by applying Newton's second law to each particle. (b) Write down the Lagrangian \(\mathcal{L}\left(\mathbf{r}_{1}, \mathbf{r}_{2}, \dot{\mathbf{r}}_{1}, \dot{\mathbf{r}}_{2}\right)=T-U\) and show that the six Lagrange equations are the same as the six Newtonian equations of part (a). This establishes the validity of Lagrange's equations in rectangular coordinates, which in turn establishes Hamilton's principle. since the latter is independent of coordinates, this proves Lagrange's equations in any coordinate system.

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