Non-Conservative Forces

Suppose \(y = y(x)\). If we want the stationary value, i.e. minimize the value for integration between two fixed values, \(x_1,x_2\): \(I = \int_{x_1}^{x_2} f(y,y',x)dx\)

Then, \(I = \int_{x_1}^{x_2} \sqrt{(dx)^2 + (dy)^2} = \int_{x_1}^{x_2}\sqrt{1 + (y'(x))^2}dx\). Now we can ask what we want \(y(x)\) to be to minimize \(I\).

Let \(x\) be an independent variable on the interval \([x_1,x_2]\). Let \(y(x)\) be some differentiable function defined on \([x_1,x_2]\) with fixed values at \(x_1\) and \(x_2\). Find \(y(x)\) such that \(I = \int_{x_1}^{x_2}\phi(y,y',x)dx\) has a stationary value. Let \(\overline{y}(x) = y(x) + \varepsilon\eta(x)\) be a neighboring path, \(\varepsilon>0\) be small, and \(\eta(x_1)=\eta(x_2)=0\).

So,

\begin{align*} I(\overline{y}, \overline{y}', x) &= I(\varepsilon) \\ &= \int_{x_1}^{x_2}\phi(\overline{y}, \overline{y}', x)dx \\ &= \int_{x_1}^{x_2} \phi(y(x) + \varepsilon\eta(x), y'(x), \varepsilon\eta'(x), x)dx \\ &= \int_{x_1}^{x_2} \phi(y(x), y'(x), x) + \frac{\partial\phi}{\partial y}\varepsilon\eta(x) + \frac{\partial\phi}{\partial y'}\varepsilon\eta'(x)dx \\ &= I + \varepsilon\int_{x_1}^{x_2}\frac{\partial\phi}{\partial y}\eta(x) + \frac{\partial\phi}{\partial y'}\eta'(x)dx, \\ 0 = \left[\frac{dI(\varepsilon)}{d\varepsilon}\right]_{\varepsilon=0} &= \int_{x_1}^{x_2}\frac{\partial\phi}{\partial y}\eta(x) + \frac{\partial\phi}{\partial y'}\eta'(x)dx \\ &= \left.\frac{\partial\phi}{\partial y'}\eta(x)\right|_{x_1}^{x_2} + \int_{x_1}^{x_2}\frac{\partial\phi}{\partial y}\eta(x) - \frac{d}{dx}\left(\frac{\partial\phi}{\partial y'}\right)\eta(x)dx \\ &= \int_{x_1}^{x_2}\eta(x)\left(\frac{\partial\phi}{\partial y} - \frac{d}{dx}\left(\frac{\partial\phi}{\partial y'}\right)\right)dx \\ \end{align*}

Since it is zero for all \(\eta(x)\) then, \(\frac{\partial\phi}{\partial y} - \frac{d}{dx}\left(\frac{\partial\phi}{\partial y'}\right) = 0\), our condition on \(y\).

\(\frac{d}{dx}\left(\frac{\partial}{\partial y'}\sqrt{1+y'(x)^2}\right) - \frac{\partial\phi}{\partial y} = \frac{d}{dx}(\frac{2y'(x)}{2\sqrt{1+y'(x)^2}}) = 0\) Thus, \(y'(x)/\sqrt{1+y'^2}\) is a constant so \(y''=0\) hence a straight line.

Let \(I[a,b;\phi;y,y',x]\) be the definite integral between two fixed points: \(\int_a^b\phi(y,y',x)dx\). Written \(I[a,b]\) when \(\phi,y,y',x\) are understood. The stationary value of \(I[a,b]\) occurs when \(\delta I[a,b] = 0\).

• \(\delta y(x)|_{x=a} = 0\)
• \(\delta y(x)|_{x=b} = 0\)
• \(\frac{d}{dx}\delta y = \delta \frac{dy}{dx}\)
• \(\delta\int_a^b y(x)dx = \int_a^b\delta y(x)dx\)
• \(\delta x = 0\)

\(\delta I[a,b] = \delta \int_a^b\phi dx = \int_a^b \delta\phi dx = \int_a^b \frac{\partial\phi}{\partial y}\delta y + \frac{\partial\phi}{\partial y'}\delta y' + \frac{\partial \phi}{\partial x}\delta x dx = \int_a^b \frac{\partial\phi}{\partial y}\delta y + \frac{\partial\phi}{\partial y'}\delta y' dx = \int_a^b \frac{\partial\phi}{\partial y}\delta y + \frac{\partial\phi}{\partial y'}\frac{d}{dx}\delta y dx = \left.\frac{\partial \phi}{\partial y'}\delta y\right|_a^b + \int_a^b \left(\frac{\partial\phi}{\partial y} - \frac{d}{dx}\frac{\partial\phi}{\partial y'}\right)\delta y dx = \int_a^b \left(\frac{\partial\phi}{\partial y} - \frac{d}{dx}\frac{\partial\phi}{\partial y'}\right)\delta y dx\) This arrives at the same Euler-Lagrange equation of the second kind.

What if we have holonomic constraints: \(f_\ell(y_1,\cdots,y_n,x) = 0\)? Ansatz: Then, we get: \(\delta I[a,b] = \int_a^b \sum_{j}^{n-m}\left(\frac{\partial\phi}{\partial y_j} - \frac{d}{dx}\frac{\partial\phi}{\partial y_j'} - \sum_\ell\lambda_\ell \frac{\partial f_\ell}{\partial y_i}\right)\delta y_j dx = 0\) Suppose we have \(\phi' = \phi + \sum_\ell \lambda_\ell f_\ell\). We get,

\begin{align*} 0=\delta I[a,b] &= \int_a^b \delta\phi(y,y',x)dx + \sum_\ell\lambda_\ell\int_a^b\delta f_\ell(y,x)dx \\ &= \sum_j\int_a^b \left(\frac{\partial\phi}{\partial y_j}-\frac{d}{dx}\frac{\partial\phi}{\partial y_j'}\right)\delta y_jdx + \sum_\ell\lambda_\ell\int_a^b \sum_j\frac{\partial f_\ell}{\partial y_j}\delta y_jdx \\ &= \left[\int_a^b \sum_j\left(\frac{\partial\phi}{\partial y_j}-\frac{d}{dx}\frac{\partial\phi}{\partial y_j'} + \sum_\ell\lambda_\ell\frac{\partial f_\ell}{\partial y_j}\right)\delta y_jdx\right] \end{align*}

Hence, we get an Euler-Lagrange Equation of the First Kind, for the covariate coordinates.