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In beginner classical mechanics, which I've just started learning, a particle with coordinates $mathbf qinmathbb R^n$ has its equation of motion specified by $F(mathbf q,dot{mathbf q},t)=mddot{mathbf q}$. Force is a function of all the coordinates necessary to describe the (rigid) body, and should cover all the degrees of freedom in the system. However, it seems to me that since $dot{mathbf q}=frac{dmathbf q}{dt}$, it's only necessary to specify $F(mathbf q,t)$ for a complete description. I'm not sure I understand why this isn't the case, but my best guess is as follows.

If $mathbf q,dot{mathbf q}$ are given in a differential equation, such as $dot{mathbf q}=t^mathbf qmathbf q^dot{mathbf q}$, then it's necessary to specify all of $mathbf q,dot{mathbf q},t$ in order to locate its position and velocity at any given time, unless the differential equation has a solution and we use it.

But this explanation is strange to me. Since we can have a $k$th-order differential equation which specifies $mathbf q,frac{d}{dt}mathbf q,ldots,frac{d^k}{dt^k}mathbf q$ with no obvious solution, wouldn't that mean our equation of motion is actually $F(mathbf q,frac{d}{dt}mathbf q,ldots,frac{d^k}{dt^k}mathbf q,t)=mfrac{d^2}{dt^2}mathbf q$?

Edit The differential equation above which has a vector exponentiated by a vector is just a poorly thought-out attempt at an example of a diff eq with no obvious solution to me, it doesn't really matter what it is. Or if you want to consider that case, treat it as a 1D system, I guess.

However, it seems to me that since $dot{mathbf q}=frac{dmathbf q}{dt}$, it's only necessary to specify $F(mathbf q,t)$ for a complete description.

While it's true that the function $q$ determines its derivative $frac{dq}{dt}$, it's not true that the value of $q$ at a particular value $t_0$ of $t$ determines the value of the derivative $frac{dq}{dt}$ at that same value. The Lagrangian has a value at a particular time $t_0$ which is a function of the three numbers $q(t_0), q'(t_0)$, and $t_0$. In particular, it depends on more information than $q(t_0)$, but on less information than all of the higher derivatives of $q$ at $t_0$.

Lagrangian mechanics is based on Newtonian mechanics (specifically d'Alembert's principle of virtual works), i.e. sets of second order ODEs. Under some regularity conditions, these second order ODEs can be turned into a system of first order ODEs, where the $q$s and the $dot q$s are independent variables. Hence if you want to retrieve the equations of motion out of a Lagrangian (or an action principle), this must depend on the $q$s and the $dot q$s. Observe that the action integral itself is assumed to depend on the trajectory alone, i.e.

$$S[q] = int_a^b L(q(t),dot q(t), t)text dt.$$

Hamilton noticed that there is a standard procedure for passing from Euler-Lagrange equations, which are the second order ODEs of Newtonian mechanics, to the equivalent system of first order ODEs. For this to work, the condition is that the Hessian of $L$ w.r.t. the $dot q$ should not vanish (so it must be positive definite on the domain of interest, i.e $L$ is concave w.r.t. the $dot q$s). Such equations are the Hamilton equation coming from the Hamiltonian, which is the Legendre transform of $L$ w.r.t. the generalised velocities.