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I was reading the lecture notes titled: 'An introduction to Lagrangian and Hamiltonian mechanics'.

In these notes, he writes at one place:

We consider mechanical systems that are holonomic and conservative (or for which the applied forces have a generalized potential). For such a system we can construct a Lagrangian $L(q, \dot q,t)$, where $q = (q_1, . . . , q_n) ^T$, which is the difference of the total kinetic $T$ and potential $V$ energies. These mechanical systems evolve according to the n Lagrange equations. These are each second-order ordinary differential equations and so the system is determined for all time once $2n$ initial conditions $(q(t_0), \dot q(t_0))$ are specified (or $n$ conditions at two different times). The state of the system is represented by a point $\textbf{q} = (q_1, . . . , q_n)^T$ in configuration space.

The last line confused me since as far as I know and even as Wikipedia says, a point in the phase space gives the state of the system while the above gives only the configuration. So, is the above line wrong?

Also, it seems that when $q$ is known $p=\frac{\partial L}{\partial\dot q} $ can be calculated. So, shouldn't specifying $q$ itself specify the state completely since that is uniquely related to conjugate momenta $p$?

Lost
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2 Answers2

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Usually, by the "state" of a system at an instant of time we mean all information necessary to determine uniquely a solution to the equations of motion that then determines all future (and past) states of the system.

For the Lagrangian formalism, such states are tuples of numbers $(q,\dot{q})$, i.e. the generalized coordinates and velocities at an instant of time, for the Hamiltonian formalism, such states are tuples of numbers $(q,p)$, i.e the generalized coordinates and momenta at an instant of time. Note that a function $q(t)$ determines tuples $(q(t),\dot{q}(t))$ and $(q(t),p(t))$ at all instants of time, since the functions $\dot{q}(t)$ and $p(t)$ can be determined from $q(t)$ either by differentiation or use of $p(t) = \partial_{\dot{q}} L(q(t),\dot{q}(t),t)$. See also this answer of mine for more discussion on how the abstract (and time-independent!) coordinates $q,\dot{q},p$ relate to functions $q(t)$

That is, by solving the equations of motion (and assuming they indeed have unique solutions for fully-specified initial conditions, unlike e.g. Norton's dome), the state of a mechanical system at all instants of time can in principle be specified by any of the three things above: A tuple of numbers $(q,\dot{q})$ given at an instant of time $t_0$, a tuple of numbers $(q,p)$ or a function of time $q(t)$.

That your text claims the "state" of a system is given just by a tuple of numbers $q$ means it is using "state" to mean something different, but there is no telling what exactly. The configuration $q$ at an instant of time is of course all you would need to say what one would see on a (idealized) photograph of the system at that instant, but photographs do not capture velocity - the configuration at an instant of time alone is not enough to tell how the system will evolve in time (see e.g. this question for an explicit example).

ACuriousMind
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Configuration space is basically analogous to the 3-dimensional space we know and love. On the other hand, phase space is a mathematical construct to represent the state of a given system by specifying its position and momentum, classically at least.

The need for phase space arises from the fact that configuration space fails to account for the velocities of the particles.

So, yes, the last statement has an error. However, it might have been used lightly but not to define a rigorous physical concept.

However, in the question you have made a mistake in equating $p=\frac{\delta L}{\delta q} $, in fact $p=\frac{\delta L}{\delta \dot{q}}$

We need to know both the position, $q$ and velocity, $\dot{q}$ to be able to determine the trajectory of a particle.

Anagh Venneti
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