book:chap6:6.2_lagrange_formalism
Differences
This shows you the differences between two versions of the page.
Next revision | Previous revision | ||
book:chap6:6.2_lagrange_formalism [2022/02/14 12:10] – created abril | book:chap6:6.2_lagrange_formalism [2022/02/14 14:06] (current) – [6.2.2 Mathematical background: variational calculus] abril | ||
---|---|---|---|
Line 299: | Line 299: | ||
[[# | [[# | ||
where one minimizes the surface //area// of a soap film, rather than a feature of a one-dimensional object. | where one minimizes the surface //area// of a soap film, rather than a feature of a one-dimensional object. | ||
- | [[book: | + | [[book: |
Unless two points lie exactly on opposite sides of the sphere (like North and South pole) there are exactly two trajectories of extremal length. | Unless two points lie exactly on opposite sides of the sphere (like North and South pole) there are exactly two trajectories of extremal length. | ||
One of them is the shortest trajectory. | One of them is the shortest trajectory. | ||
Line 463: | Line 463: | ||
expressed in terms of the generalized coordinates $\mathbf q$ and their time derivatives $\dot{\boldsymbol q}$.\\ | expressed in terms of the generalized coordinates $\mathbf q$ and their time derivatives $\dot{\boldsymbol q}$.\\ | ||
**d)** Determine the EOM for the component $q_i$ of $\mathbf q$ by evaluating the // | **d)** Determine the EOM for the component $q_i$ of $\mathbf q$ by evaluating the // | ||
+ | <wrap # | ||
\begin{align} | \begin{align} | ||
- | \label{eq: | ||
\frac{\mathrm{d}}{\mathrm{d} t} \; \frac{\partial \mathcal L}{\partial \dot q_i} | \frac{\mathrm{d}}{\mathrm{d} t} \; \frac{\partial \mathcal L}{\partial \dot q_i} | ||
= \frac{\partial \mathcal L}{\partial q_i} \tag{6.2.6} | = \frac{\partial \mathcal L}{\partial q_i} \tag{6.2.6} |
book/chap6/6.2_lagrange_formalism.1644837058.txt.gz · Last modified: 2022/02/14 12:10 by abril