Lagrangian mechanics of continuous systems

[adriank]

I'm thinking about generalizations of Lagrangian mechanics to systems with infinitely many degrees of freedom, but what I've got uses some extremely sketchy math that still appears to give a correct result. I only consider conservative systems that do not explicitly depend on time.

Of course, if our system is described by finitely many coordinates $q_i$ and a Lagrangian $L = L(\mathbf{q}, \dot{\mathbf{q}})$, then the motion obeys Lagrange's equation
$$\frac{\partial L}{\partial q_i} - \frac{d}{dt} \frac{\partial L}{\partial \dot q_i} = 0,$$
or equivalently,
$$\frac{\partial L}{\partial \mathbf{q}} - \frac{d}{dt} \frac{\partial L}{\partial \dot{\mathbf{q}}} = 0,$$
where $\partial L / \partial \mathbf{q}$ is a directional derivative
$$\frac{\partial L}{\partial \mathbf{q}}(\mathbf{v}) = \left. \frac{d}{d\tau} L(\mathbf{q} + \tau \vect{v}, \dot{\mathbf{q}}) \right\rvert_{\tau = 0},$$
and similarly for $\partial L / \partial \dot{\mathbf{q}}$.

Now say our system is described by some continuous parameter $s$, so we again have $L = L(q, \dot q)$, but this time $q$ is a function of both $s$ and $t$. I make the bold claim that Lagrange's equation still holds in this sense:
$$\frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot q} = 0,$$
where the derivatives of the Lagrangian are some functional directional derivative:
$$\frac{\partial L}{\partial q}(f) = \left. \frac{d}{d\tau} L(q + \tau f, \dot q) \right\rvert_{\tau = 0},$$
where $f$ is some function of $s$.

As an example, suppose we have a continuous spring, with an unextended length of 0, confined to the $y$ axis. Its position is described by a function $y(s, t)$, where $s$ ranges from $0$ to $\ell$; the parameter $s$ perhaps represents length along the coil from one end. Its mass and spring constant per unit length are $\mu$ and $\kappa$. I write $y'$ to mean $\partial y / \partial s$. The potential energy and kinetic energy are
$$V = \int_0^\ell \mu g y(s) + \frac12 \kappa \left( \frac{\partial y(s)}{\partial s} \right)^2 \,ds, \quad T = \int_0^\ell \frac12 \mu \dot y(s)^2 \,ds,$$
so the Lagrangian is
$$L = \int_0^\ell \frac12 \mu \dot y(s)^2 - \mu gy(s) - \frac12 \kappa y'(s)^2 \,ds.$$
The directional derivatives with respect to $y$ and $y'$ of the Lagrangian satisfy
$$\frac{\partial L}{\partial y}(f) = \left. \frac{d}{d\tau} \int_0^\ell \frac12 \mu \dot y(s)^2 - \mu g (y(s) + \tau f(s)) - \frac12 \kappa (y'(s) + \tau f'(s))^2 \,ds \right\rvert_{\tau = 0} = -\int_0^\ell \mu g f(s) + \kappa y'(s) f'(s) \,ds$$
$$\frac{\partial L}{\partial \dot y}(f) = \left. \frac{d}{d\tau} \int_0^\ell \frac12 \mu (\dot y(s) + \tau f(s))^2 - \mu gy(s) - \frac12 \kappa y'(s)^2 \right\rvert_{\tau = 0} = \int_0^\ell \mu \dot y(s) f(s) \,ds$$
Thus, Lagrange's equation gives
$$\int_0^\ell \mu g f(s) + \kappa y'(s) f'(s) + \frac{d}{dt} \mu \dot y(s) f(s) \,ds = 0.$$
Integration by parts on the second term and simplifying gives
$$\kappa (y'(\ell) f(\ell) - y'(0) f(0)) - \int_0^\ell \left[ \mu g - \kappa y''(s) + \mu \ddot y(s) \right] f(s) \,ds = 0.$$
I claim that somehow the term on the left disappears. Since this equation must be true for any function $f$, the expression in brackets must be zero; we then get the equation of motion
$$\ddot y(s) = \frac{\kappa}{\mu} y''(s) - g.$$
It seems like a physically reasonable equation; solutions oscillate over time, and if we say fix $y(\ell) = 0$, then in an equilibrium solution $y(s) = (\mu g/2\kappa) (s^2 - \ell^2)$ the tension $\kappa y'(s)$ at any point is equal to the weight $\mu gs$ of the spring below that point.

Has anyone seen anything that looks somewhat like this? I looked around and couldn't really find anything that looked like this.

 

[saibod]

cool, so you've reinvented Lagrangian field theory! Since your Lagrangian is an integral over s, is better to consider its integrand and call it a "Lagrangian density". Can you derive the Euler-Lagrange equations for a Lagrangian density? In any case, I hope you have enough buzzwords now to read up on this, there's a huge literature on classical field theory.

 

[astrokreng]

I am intrigued by your exploration of generalizing Lagrangian mechanics to systems with infinitely many degrees of freedom. The use of functional directional derivatives in your proposed equation for Lagrange's equation is certainly interesting and warrants further investigation.

I have not personally come across a similar approach in my own research, but that does not mean it has not been explored by others. It would be beneficial to search through literature and see if there are any prior studies or discussions on this topic.

Your example of a continuous spring is a good starting point for applying your proposed equation of motion. The resulting equation does seem physically reasonable and it would be interesting to see how it compares to other methods of solving for the motion of a continuous spring.

I encourage you to continue exploring this idea and see where it leads. Perhaps with further refinement and analysis, it could potentially offer new insights or applications in the field of Lagrangian mechanics for systems with infinite degrees of freedom.