Physlib.ClassicalMechanics.EulerLagrange

Given a Lagrangian function $L : \text{Time} \to X \to X \to \mathbb{R}$ and a trajectory $q : \text{Time} \to X$, the Euler-Lagrange operator is a function from $\text{Time}$ to $X$ defined at each time $t$ by: $$\frac{\partial L}{\partial q}(t, q(t), \dot{q}(t)) - \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}}(t, q(t), \dot{q}(t)) \right)$$ where $\dot{q}(t)$ is the time derivative of the trajectory $q$ at time $t$. Here, the partial derivative $\frac{\partial L}{\partial q}$ denotes the gradient of $L$ with respect to its second argument (position), and $\frac{\partial L}{\partial \dot{q}}$ denotes the gradient of $L$ with respect to its third argument (velocity).

For any Lagrangian function $L : \text{Time} \to X \to X \to \mathbb{R}$ and any trajectory $q : \text{Time} \to X$, the Euler-Lagrange operator $(\text{eulerLagrangeOp } L \ q)$ evaluated at time $t$ is given by: $$\nabla_2 L(t, q(t), \dot{q}(t)) - \frac{d}{dt} \left( \nabla_3 L(t, q(t), \dot{q}(t)) \right)$$ where $\dot{q}(t) = \partial_t q(t)$ is the time derivative of the trajectory, $\nabla_2 L$ denotes the gradient of $L$ with respect to its second argument (position), and $\nabla_3 L$ denotes the gradient of $L$ with respect to its third argument (velocity).

For any trajectory $q : \text{Time} \to X$, the Euler-Lagrange operator applied to a Lagrangian function $L$ that is identically zero (i.e., $L(t, q, \dot{q}) = 0$ for all $t, q, \dot{q}$) results in the zero function.

Let $X$ be a finite-dimensional real inner product space. For a smooth ($C^\infty$) Lagrangian function $L : \text{Time} \to X \to X \to \mathbb{R}$ and a smooth trajectory $q : \text{Time} \to X$, the variational derivative (functional derivative) of the action functional $$S[q] = \int_{\text{Time}} L(t, q(t), \dot{q}(t)) \, dt$$ with respect to the trajectory $q$ is equal to the Euler-Lagrange operator applied to $q$. That is, $$\frac{\delta S}{\delta q}(t) = \frac{\partial L}{\partial q}(t, q(t), \dot{q}(t)) - \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}}(t, q(t), \dot{q}(t)) \right)$$ where $\dot{q}(t)$ is the time derivative of the trajectory.