When can we add a total time derivative of $f(q, \dot{q}, t)$ to a Lagrangian?

Question

The other day, I was listening to this lecture on the Lagrangian for a charged particle in an electromagnetic field, and at one point in the video, the lecturer mentions that we can add any total time derivative of a function $f(q, t)$ to the Lagrangian without altering its equations of motion.

This is nothing new to me, and I understand it fully, but shortly afterwards (approximately two minutes after the linked starting point), he goes on to say that you can, in fact, add a total time derivative of a function $f(q, \dot{q}, t)$, given certain conditions. This definitely surprised me, and I would love to know more about it, but the lecturer quickly moves on, so my question is as follows: under what conditions can one add the total time derivative of a function which depends on the particle's generalized velocities in addition to its generalized coordinates and time without affecting the particle's equations of motion?

Answer 1

I) In general, it is true that if we plug a local Lagrangian

$$\tag{1} L\quad \longrightarrow \quad \tilde{L}~=~L+\frac{df}{dt}$$

modified with a total derivative term into the Euler-Lagrange expression

$$\tag{2} \sum_{n} \left(-\frac{d}{dt}\right)^n \frac{\partial \tilde{L}}{\partial q^{(n)}}~=~\sum_{n} \left(-\frac{d}{dt}\right)^n \frac{\partial L}{\partial q^{(n)}}, $$

it would lead to identically the same Euler-Lagrange expression without any restrictions on $L$ and $f$.

II) The caveat is that the Euler-Lagrange expression (2) is only$^1$ physically legitimate, if it has a physical interpretation as a variational/functional derivative of an action principle. However, existence of a variational/functional derivative is a non-trivial issue, which relies on well-posed boundary conditions for the variational problem. In plain English: Boundary conditions are needed in order to justify integration by parts. See also e.g. my related Phys.SE answers here & here.

III) A Lagrangian $L(q,\dot{q},\ldots, q^{(N)},t)$ of order $N$ leads to equation of motion of order $\leq 2N$. Typically we require the Lagrangian $L(q,\dot{q},t)$ to be of first order $N=1$. See e.g. this and this Phys.SE posts.

IV) Concretely, let us assume that we are given a first-order Lagrangian $L(q,\dot{q},t)$. If one redefines the Lagrangian with a total derivative

$$\tag{3} \tilde{L}(q, \dot{q}, \ddot{q}, t)~=~L(q, \dot{q}, t)+\frac{d}{dt}f(q, \dot{q}, t), $$

where $f(q, \dot{q}, t)$ depends on velocity $\dot{q}$, then the new Lagrangian $\tilde{L}(q, \dot{q}, \ddot{q}, t)$ may also depend on acceleration $\ddot{q}$, i.e. be of higher order.

V) With a higher-order $\tilde{L}(q, \dot{q}, \ddot{q}, t)$, we might have to impose additional boundary conditions in order to derive Euler-Lagrange equations from the principle of a stationary action by use of repeated integrations by parts.

VI) It seems that Prof. V. Balakrishnan in the video has the issues IV and V in mind when he spoke of 'putting further conditions' on the system. Finally, OP may also find this Phys.SE post interesting.

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$^1$ Here we ignore derivations of Lagrange equations directly from Newton's laws, i.e. without the use of the principle of a stationary action, such as e.g. this Phys.SE post, because they usually don't involve redefinitions (3).

Answer 2

It is trivial to show that any $\frac{df}{dt}$ can be added to the lagrangian under the condition that $f$ vanishes on the boundary. Indeed, the action is $$S[q] = \int_{t_1}^{t_2} L(q,\dot{q},t) + \frac{d f}{dt}(q,\dot{q},t) dt = \int_{t_1}^{t_2} L(q,\dot{q},t) dt + f(q(t_2),\dot{q}(t_2), t_2) - f(q(t_1), \dot{q}(t_1), t_1),$$ which yields the usual Euler-Lagrange eqs. for $f$ vanishing at $t_1$, $t_2$.