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Chapter 7: Problem 5

Find the components of \(\nabla f(r, \phi)\) in two-dimensional polar coordinates. [Hint: Remember that the change in the scalar \(f \text { as a result of an infinitesimal displacement } d \mathbf{r} \text { is } d f=\nabla f \cdot d \mathbf{r}.]\)

Short Answer

Expert verified
Components are \( \frac{\partial f}{\partial r} \hat{r} \) and \( \frac{1}{r} \frac{\partial f}{\partial \phi} \hat{\phi} \) in polar coordinates.

Step by step solution

01

Recall the Gradient in Polar Coordinates

The gradient of a scalar field \( f(r, \phi) \) in polar coordinates is given by the vector \( abla f \) such that \( df = abla f \cdot d \mathbf{r} \). Here, \( d\mathbf{r} \) is expressed in terms of infinitesimal changes in \( r \) and \( \phi \).
02

Express Infinitesimal Displacement

In polar coordinates, an infinitesimal displacement \( d \mathbf{r} \) can be expressed as \( d\mathbf{r} = dr \hat{r} + r d\phi \hat{\phi} \), where \( \hat{r} \) and \( \hat{\phi} \) are unit vectors in the directions of increasing \( r \) and \( \phi \), respectively.
03

Formulate the Differential of the Scalar Field

The change in the scalar field \( f(r, \phi) \) due to this displacement is written as \( df = \frac{\partial f}{\partial r} dr + \frac{\partial f}{\partial \phi} d\phi \).
04

Interpret the Dot Product

From \( df = abla f \cdot d \mathbf{r} \), the expression becomes: \[ df = \left( \frac{\partial f}{\partial r} \right) dr + \left( \frac{1}{r} \frac{\partial f}{\partial \phi} \right) r d\phi. \] This implies that the unit vectors' contribution to the gradient is split between the scalar \( \frac{\partial f}{\partial r} \) in the direction of \( \hat{r} \) and \( \frac{1}{r} \frac{\partial f}{\partial \phi} \) in the direction of \( \hat{\phi} \).
05

Determine the Components of \( \nabla f \)

Thus, the components of the gradient \( abla f \) in polar coordinates are \( abla f = \frac{\partial f}{\partial r} \hat{r} + \frac{1}{r} \frac{\partial f}{\partial \phi} \hat{\phi} \). This indicates how the function \( f(r, \phi) \) changes in both radial and angular directions.

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Key Concepts

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

Partial Derivatives
Partial derivatives are a fundamental concept in multivariable calculus, representing how a function changes as one of its input variables is varied, while keeping other variables constant. In the context of polar coordinates, when we have a function \( f(r, \phi) \), its partial derivatives are \( \frac{\partial f}{\partial r} \) and \( \frac{\partial f}{\partial \phi} \). These notations signify the rate of change of the function with respect to \( r \) and \( \phi \), respectively.
When dealing with polar coordinates, partial derivatives must be carefully evaluated because changes in \( r \) and \( \phi \) can affect the function differently compared to Cartesian coordinates. Below are some key points about partial derivatives:
  • The partial derivative \( \frac{\partial f}{\partial r} \) measures how much the function changes as we move radially outward from the origin.
  • The derivative \( \frac{\partial f}{\partial \phi} \) measures the rate of change as the angle \( \phi \) changes.
  • In polar coordinates, these derivatives help build a more nuanced picture about how functions change across a plane.
Understanding partial derivatives helps us determine and analyze the gradient of a function, especially in polar coordinates.
Polar Coordinates
Polar coordinates offer a different way of representing points in a plane from the traditional Cartesian coordinates. Instead of using \( (x, y) \), polar coordinates use \( (r, \phi) \), where \( r \) is the radial distance from the origin, and \( \phi \) is the angular position measured from the positive x-axis.
This system can simplify the equations of curves and make it easier to handle rotation and scaling. Several aspects of polar coordinates are crucial:
  • Polar coordinates are particularly helpful for describing regions and shapes that are naturally circular or involve rotation.
  • The relationship between Cartesian and polar coordinates is given by the transformations: \( x = r\cos\phi \) and \( y = r\sin\phi \).
  • In polar coordinates, the orientation is naturally radial, with angles leading to different expansions opposed to Cartesian which is very linear.
The switch to polar coordinates necessitates a change in how we handle calculus operations, leading to adjusted definitions like those of the gradient.
Scalar Field
A scalar field is a mathematical function that assigns a single scalar value to every point in space. Imagine it as a landscape where each point has a height, corresponding to the scalar value. In the context of our exercise, this scalar field is \( f(r, \phi) \).
Scalar fields can represent various physical quantities, such as:
  • Temperature distribution in a room.
  • Pressure in a gas.
  • Gravitational potential in space.
Our task is to understand how the scalar field, defined by \( f(r, \phi) \), changes over small movements through space. To achieve this understanding, we use gradients, which measure these changes as vectors involving the use of partial derivatives achieved in each coordinate system. The presence of a scalar field makes it possible to explore these spatial variations effectively using calculus and coordinate transformations.

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Most popular questions from this chapter

Noether's theorem asserts a connection between invariance principles and conservation laws. In Section 7.8 we saw that translational invariance of the Lagrangian implies conservation of total linear momentum. Here you will prove that rotational invariance of \(\mathcal{L}\) implies conservation of total angular momentum. Suppose that the Lagrangian of an \(N\) -particle system is unchanged by rotations about a certain symmetry axis. (a) Without loss of generality, take this axis to be the \(z\) axis, and show that the Lagrangian is unchanged when all of the particles are simultaneously moved from \(\left(r_{\alpha}, \theta_{\alpha}, \phi_{\alpha}\right)\) to \(\left(r_{\alpha}, \theta_{\alpha}, \phi_{\alpha}+\epsilon\right)\) (same \(\epsilon\) for all particles). Hence show that $$\sum_{\alpha=1}^{N} \frac{\partial \mathcal{L}}{\partial \phi_{\alpha}}=0.$$ (b) Use Lagrange's equations to show that this implies that the total angular momentum \(L_{z}\) about the symmetry axis is constant. In particular, if the Lagrangian is invariant under rotations about all axes, then all components of \(\mathbf{L}\) are conserved.

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.

The "spherical pendulum" is just a simple pendulum that is free to move in any sideways direction. (By contrast a "simple pendulum"- unqualified - is confined to a single vertical plane.) The bob of a spherical pendulum moves on a sphere, centered on the point of support with radius \(r=R\) the length of the pendulum. A convenient choice of coordinates is spherical polars, \(r, \theta, \phi,\) with the origin at the point of support and the polar axis pointing straight down. The two variables \(\theta\) and \(\phi\) make a good choice of generalized coordinates. (a) Find the Lagrangian and the two Lagrange equations. (b) Explain what the \(\phi\) equation tells us about the \(z\) component of angular momentum \(\ell_{z^{*}}\) (c) For the special case that \(\phi=\) const, describe what the \(\theta\) equation tells us. (d) Use the \(\phi\) equation to replace \(\dot{\phi}\) by \(\ell_{z}\) in the \(\theta\) equation and discuss the existence of an angle \(\theta_{\mathrm{o}}\) at which \(\theta\) can remain constant. Why is this motion called a conical pendulum? (e) Show that if \(\theta=\theta_{0}+\epsilon,\) with \(\epsilon\) small, then \(\theta\) oscillates about \(\theta_{\mathrm{o}}\) in harmonic motion. Describe the motion of the pendulum's bob.

(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.

Using the usual angle \(\phi\) as generalized coordinate, write down the Lagrangian for a simple pendulum of length \(l\) suspended from the ceiling of an elevator that is accelerating upward with constant acceleration \(a\). (Be careful when writing \(T\); it is probably safest to write the bob's velocity in component form.) Find the Lagrange equation of motion and show that it is the same as that for a normal, nonaccelerating pendulum, except that \(g\) has been replaced by \(g+a\). In particular, the angular frequency of small oscillations is \(\sqrt{(g+a) / l}\).
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