diff options
Diffstat (limited to 'content/know/concept')
| -rw-r--r-- | content/know/concept/calculus-of-variations/index.pdc | 6 | ||||
| -rw-r--r-- | content/know/concept/pulay-mixing/index.pdc | 2 | ||||
| -rw-r--r-- | content/know/concept/rayleigh-plesset-equation/index.pdc | 2 |
3 files changed, 6 insertions, 4 deletions
diff --git a/content/know/concept/calculus-of-variations/index.pdc b/content/know/concept/calculus-of-variations/index.pdc index a8861cb..26c5753 100644 --- a/content/know/concept/calculus-of-variations/index.pdc +++ b/content/know/concept/calculus-of-variations/index.pdc @@ -14,7 +14,7 @@ markup: pandoc # Calculus of variations The **calculus of variations** lays the mathematical groundwork -for Lagrangian mechanics. +for [Lagrangian mechanics](/know/concept/lagrangian-mechanics/). Consider a **functional** $J$, mapping a function $f(x)$ to a scalar value by integrating over the so-called **Lagrangian** $L$, @@ -28,6 +28,8 @@ If $J$ in some way measures the physical "cost" (e.g. energy) of the path $f(x)$ taken by a physical system, the **principle of least action** states that $f$ will be a minimum of $J[f]$, so for example the expended energy will be minimized. +In practice, various cost metrics may be used, +so maxima of $J[f]$ are also interesting to us. If $f(x, \varepsilon\!=\!0)$ is the optimal route, then a slightly different (and therefore worse) path between the same two points can be expressed @@ -238,7 +240,7 @@ $$\begin{aligned} ## Constraints -So far, for multiple functions $f_1, ... f_N$, +So far, for multiple functions $f_1, ..., f_N$, we have been assuming that all $f_n$ are independent, and by extension all $\eta_n$. Suppose that we now have $M < N$ constraints $\phi_m$ that all $f_n$ need to obey, introducing implicit dependencies between them. diff --git a/content/know/concept/pulay-mixing/index.pdc b/content/know/concept/pulay-mixing/index.pdc index 4e7a411..476932d 100644 --- a/content/know/concept/pulay-mixing/index.pdc +++ b/content/know/concept/pulay-mixing/index.pdc @@ -125,7 +125,7 @@ $$\begin{aligned} You might be confused by the absence of any $\rho_m^\mathrm{new}$ in the creation of $\rho_{n+1}$, as if the iteration's outputs are being ignored. This is due to the first assumption, -which states that $\rho_n^\mathrm{new}$ are $\rho_n$ are already similar, +which states that $\rho_n^\mathrm{new}$ and $\rho_n$ are already similar, such that they are basically interchangeable. Speaking of which, about those assumptions: diff --git a/content/know/concept/rayleigh-plesset-equation/index.pdc b/content/know/concept/rayleigh-plesset-equation/index.pdc index 9325f3f..75feb13 100644 --- a/content/know/concept/rayleigh-plesset-equation/index.pdc +++ b/content/know/concept/rayleigh-plesset-equation/index.pdc @@ -20,7 +20,7 @@ inside an incompressible liquid. Notably, it leads to [cavitation](/know/concept/cavitation/). Consider the main -[Navier-Stokes equations](/know/concept/navier-stokes-equations/) +[Navier-Stokes equation](/know/concept/navier-stokes-equations/) for the velocity field $\va{v}$: $$\begin{aligned} |