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I just want someone to confirm if whatever I'm writing down is correct or wrong.

  1. If the Lagrangian is given by $L(x,v,t)$, then we say that it explicitly depends on time.

Here, $x$ and $v$ are independent variables as they can be independently specified to a system initially (as an initial condition). It is only later after $t=0$ that $v$ and $x$ starts to depend on time.

  1. If the Lagrangian is given by $L(x,v)$, then it implicitly depends on time.

Here, again, $v$ and $x$ can be independently specified initially and then after $t=0$, $x$ and $v$ start to depend on time, so we basically have $L(x(t),v(t))$ and this is what is meant by the Lagrangian being 'implicitly' dependent on time as there is no time independent term in the Lagrangian. So, in a way, the system does evolve with time (and isn't a snapshot) but the Lagrangian is just explicitly time-independent. If this is true, it would also mean that if one says Lagrangian is explicitly and implicitly independent of time, it would basically represent a system that doesn't evolve with time at all; basically a snapshot.

Could anyone go through this and let me know if I'm wrong anywhere or if there's any conceptual misunderstanding? I've been having a few issues with understanding what it means to be explicitly and implicitly dependent, and why $v$ and $x$ are independent variables etc.

Qmechanic
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Pratham Hullamballi
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    I feel like you're falling into a linguistic trap, which isn't entirely your fault because in Physics we often refer to several things by the same name (for example the use of $x,v$ as independent variables and then suddenly also being referred to as the trajectory taken by a system). Anyway, the resolution is to be crystal clear on the definitions of functions, and the role played by compositions of functions. FOr that, I would suggest you take a look at my MSE answer About the "ambiguity" of explicit and implicit function of a variable. – peek-a-boo Feb 25 '22 at 11:03
  • But overall, you're not wrong, though this is a slippery slope, because if you're unclear with these basic definitions, things can easily get confusing (which is why it's always good to learn the precise mathematical definitions and only afterwards take the liberty to be less precise/pedantic with the language). – peek-a-boo Feb 25 '22 at 11:12

1 Answers1

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It is improper to think of $L(x,v)$ as an implicitly time-dependent quantity when you find solutions $x(t)$ and $v(t)$ because $x$ and $v$ are functions of time only along solutions. Otherwise, they're general "phase space" coordinates.

On the other hand, when you have $L(x,v,t)$, that means that the value of the Lagrangian at a fixed point $(x_0,v_0)$ in phase space may vary in time.

Attaching terms like "explicit" and "implicit" can get you stuck with terminology; it's more constructive to think about the system's nature to figure out what the partial derivatives and total time derivatives of the Langrangian are, and that allows you to consider when coordinates like $x$, $v$, and $t$ can be considered independent coordinates, and when you're considering a map between them that represents a solution.

A related misconception is addressed here: Calculus of variations -- how does it make sense to vary the position and the velocity independently?

Rishi
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