classical mechanics | diff × weyl

Despite its simplicity, classical mechanics is not taught well in the typical physics curriculum. This is unfortunate because the general philosophy of constructing Lagrangians based on symmetries underlies all of modern physics. In this article, I explain basic Lagrangian mechanics in a systematic way starting from fundamental physical principles. It basically follows Landau and Lifshitz Vol. 1 but ties up some loose ends.

Principle of stationary action

Classical mechanics describes the motion of objects modeled as point particles. First, consider a single particle in empty space. At any given time, it has a position $\vec x(t)$ and velocity $\vec v(t)=\frac{d\vec{x}}{dt}$ .

Define a quantity $S_{if}\{\vec x(t)\}$ that depends on the path of the particle $\vec x(t)$ from time $t_i$ to $t_f$ . The principle of stationary action, or action principle, states that the path the particle actually takes is one where the action is stable to small perturbations in the path $\vec x(t) \rightarrow \vec x(t) + \vec{\delta x}(t)$ .

To elaborate, consider dividing the time interval from $t_i$ to $t_f$ into $N$ segments, and take $N\rightarrow \infty$ in the end. You may think of $S_{if}$ as a function of many variables $\{\vec{x}(t_i),t_i,\vec{x}(t_i+\Delta t),t_i+\Delta t,\cdots, \vec{x}(t_f), t_f\}$ , where $\Delta t = (t_f-t_i)/N$ . (Note that the velocity $\vec{v}(t) = \frac{\vec{x}(t+\Delta t)-\vec{x}(t)}{\Delta t}$ , so it is not an independent variable here.) Such a “function of a function” is called a functional. The principle of stationary action is then $\frac{\delta S_{12}}{\delta x_i(t)}=0$ , i.e. the partial derivative of $S_{12}$ with respect to any component of the position $x_i$ at any time $t$ is zero. The $\delta$ symbol is generally used instead of $\partial$ for functional derivatives.

Finally, the action principle only applies to perturbations that are zero at the boundaries: $\vec{\delta x}(t_i) = \vec{\delta x}(t_f) = 0$ . This will become important later.

The Lagrangian

Consider the action $S_{12}$ for time $t_1$ to $t_2$ , and the action $S_{34}$ for time $t_3$ to $t_4$ , with $t_1 < t_2 < t_3 < t_4$ . We require locality in time, meaning that a perturbation in the first interval only affects $S_{12}$ and not $S_{34}$ . Also, we assume additivity of the action: $S_{12}+S_{23}=S_{13}$ . These conditions imply that $S_{12}$ can be written as an integral from $t_1$ to $t_2$ of some quantity: $S_{12}=\int_{t_1}^{t_2} \mathcal{L}(\vec{x}(t),\vec{v}(t), t)$ . $\mathcal{L}(\vec{x}(t),\vec{v}(t), t)$ is known as the Lagrangian. In general, it may depend on the position and velocity at time $t$ , as well as the time $t$ itself¹.

Note that we may add a total time derivative $\frac{df}{dt}(\vec{x},t)$ to the Lagrangian without affecting the principle of stationary action. Such a term produces the action:

$\displaystyle\int_{t_i}^{t_f} dt\frac{df}{dt}(\vec{x},t) = f(\vec{x}(t_f), t_f)-f(\vec{x}(t_i), t_i)$

by the fundamental theorem of calculus. The perturbation $\vec{\delta x}(t)$ is zero at the boundaries by definition, so does not affect this action.

Let us now derive the form of the Lagrangian based on some other fundamental principles:

Homogeneity of space and time. No point in space or time is any different from any other, so the Lagrangian cannot depend on $\vec{x}$ or $t$ explicitly.

Isotropy of space. No direction in space is different from any other, so the Lagrangian can only depend on the magnitude (squared) of the velocity $\vec{v}(t)^2$ .

Galilean invariance. The theory should be invariant under shifts by a constant velocity, $\vec{x}\rightarrow \vec{x}+\vec{v}_0 t$ . In other words, there is no universal stationary frame of reference. Taking the time derivative, this is $\vec{v}\rightarrow \vec{v}+\vec{v}_0$ . To first order in $\vec{v}_0$ , the Lagrangian changes as

$\displaystyle\mathcal{L}(\vec{v}^2)\rightarrow \mathcal{L}(\vec{v}^2+2\vec{v}\cdot \vec{v}_0) = \mathcal{L}(\vec{v}^2)+2\frac{\delta \mathcal{L}}{\delta \vec{v}^2}(\vec{v}^2) \vec{v}\cdot \vec{v}_0$

The term $2\frac{\delta \mathcal{L}}{\delta \vec{v}^2}(\vec{v}^2) \vec{v}\cdot \vec{v}_0$ will not affect the physics if it is a total time derivative of the form above. This only occurs if $\frac{\delta \mathcal{L}}{\delta \vec{v}^2}(\vec{v}^2)$ is a constant. Call this constant $\frac{1}{2} m$ . Thus, the Lagrangian for a single particle in free space is: $\mathcal{L} = \frac{1}{2} m \vec{v}^2$ . The constant $m$ is, of course, the mass.

To summarize, we derived the unique action and Lagrangian (up to a total time derivative) for a single particle from the following postulates:

Locality in time
Additivity of the action
Homogeneity of space and time
Isotropy of space
Galilean invariance

Multiple particles

Now consider the $n$ -particle case. The Lagrangian may generally depend on all the positions and velocities $\vec{x}_1, \vec{v}_1, \cdots, \vec{x}_n, \vec{v}_n$ . Following the postulates above, it must take the form²:

$\displaystyle \mathcal{L} = \left(\sum_{i=1}^n \frac{1}{2} m_i \vec{v}_i^2\right) - U(\Delta \vec{x}_{ij})$

where the function $U(\Delta \vec{x}_{ij})$ depends on all the separations between the particles $\{\Delta\vec{x}_{12} = \vec{x}_1-\vec{x}_2, \Delta\vec{x}_{13} =\vec{x}_1-\vec{x}_3, \cdots\}$ .

Euler-Lagrange equations

Let us now apply the principle of stationary action to the action:

$\displaystyle S=\int dt\left(\sum_{i=1}^n \frac{1}{2} m_i \vec{v}_i^2\right) - U(\Delta \vec{x}_{ij})$

Plugging in the variation $\vec{x}_i\rightarrow \vec{x}_i+\vec{\delta x}_i$ for particle $i$ , and expanding to first order in $\vec{\delta x}_i$ , we get:

$\displaystyle S\rightarrow S+ \int dt\left(m_i \vec{v}_i\cdot \vec{\delta v}_i - \nabla_i U \cdot \vec{\delta x}_i\right)$

where $\nabla_i U$ is the gradient of $U$ with respect to $\vec{x}_i$ . Using $\vec{\delta v}=\frac{d}{dt}\vec{\delta x}$ , we can integrate the first term by parts, discarding the boundary term $m_i \vec{v}_i\cdot \vec{\delta x}_i$ since $\vec{\delta x}_i= 0$ at the boundaries. We obtain:

$\displaystyle \frac{\delta S}{\delta \vec{x}_i(t)}=-m_i \vec{a}_i(t)-\nabla_i U(t) = 0$

where $\vec{a} = \frac{d\vec{v}}{dt}$ . The equations obtained using the action principle are known as Euler-Lagrange equations or equations of motion. In this case, we have found Newton’s law for a conservative potential:

$\displaystyle \vec{F} = -\nabla_i U=m_i \vec{a}_i$

Beyond classical mechanics

Finally, it is interesting to see how the postulates above are modified in quantum and relativistic theories.

Principle of stationary action. In quantum physics, the particle takes all paths instead of only the classical one! The quantum amplitude is given by summing up $e^{i S\{x\}}$ over all paths. This is known as a path integral.
Locality in time gets promoted to locality in space and time in field theory.
Additivity of the action remains the same.
Homogeneity of space and time remains the same.
Isotropy of space remains the same.
Galilean invariance is promoted to Lorentz invariance in relativity. Lorentz transformations relate space and time.

In modern theories, there are often additional symmetry principles that constrain the Lagrangian, such as gauge invariance and conformal invariance.

¹ It also cannot depend on higher time derivatives due to the Ostrogradsky instability.

² A term like $\vec{v}_i\cdot \vec{v}_j$ with $i \neq j$ is possible, but would imply that particles infinitely far away can affect each other, violating common sense (or, if you like, the cluster decomposition principle).

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