Book cover for Classical Mechanics

Classical Mechanics

John R. Taylor

ISBN #9781891389221

1st Edition

744 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Lagrange's equations offer a powerful and elegant formulation of mechanics by expressing the laws of motion in a coordinate-independent manner. The method of using a Lagrangian (the difference between kinetic and potential energy) along with the calculus of variations simplifies the treatment of both unconstrained and constrained systems. This approach not only streamlines the analysis by eliminating the need to deal directly with constraint forces but also generalizes to any number of degrees of freedom, enhancing its applicability in complex mechanical systems.

Learning Objectives

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Key Concepts

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Example Problems

Example 1

Write down the Lagrangian for a projectile (subject to no air resistance) in terms of its Cartesian coordinates $(x, y, z),$ with $z$ measured vertically upward. Find the three Lagrange equations and show that they are exactly what you would expect for the equations of motion.

Example 2

Write down the Lagrangian for a one-dimensional particle moving along the $x$ axis and subject to a force $F=-k x$ (with $k$ positive). Find the Lagrange equation of motion and solve it.

Example 3

Consider a mass $m$ moving in two dimensions with potential energy $U(x, y)=\frac{1}{2} k r^{2},$ where $r^{2}=x^{2}+y^{2} .$ Write down the Lagrangian, using coordinates $x$ and $y,$ and find the two Lagrange equations of motion. Describe their solutions. [This is the potential energy of an ion in an "ion trap," which can be used to study the properties of individual atomic ions.]

Example 4

Consider a mass $m$ moving in a frictionless plane that slopes at an angle $\alpha$ with the horizontal. Write down the Lagrangian in terms of coordinates $x,$ measured horizontally across the slope, and $y$, measured down the slope. (Treat the system as two-dimensional, but include the gravitational potential energy.) Find the two Lagrange equations and show that they are what you should have expected.

Example 5

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

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