An Intuitive Approach behind the Lagrangian

Many concepts in physics are learned early during university studies, accepted, and then used for years without being questioned again. At the time, they feel understood. Often it is only much later - when working with more advanced formalisms, such as path integrals - that one suddenly realizes that this understanding was incomplete, or at least shallow with respect to origins. For me personally, one such concept was the Lagrangian. While its usefulness is beyond doubt, it always felt as if its specific form had to be accepted with surprisingly little justification. In particular, I never saw a truly intuitive argument for why the Lagrangian should take the form it does, rather than being introduced as a convenient postulate.

In this article, I want to take a step back and look at the idea behind the Lagrangian itself. We will see that the expression

[ L = T - V ]

is not merely an arbitrary definition or assumption, but follows naturally from very basic mechanical assumptions. In the same spirit, the variational principle

[ \delta S = 0 ]

will appear not as a mysterious axiom, but as a compact way of expressing these assumptions in a mathematically consistent form.

What is Assumed and What is Derived?

It is worth emphasizing the distinction between assumptions and results in what follows. We assume:

• The basic Newtonian notions of momentum and force
• The existence of conservative forces described by a scalar potential
• The validity of the principle of virtual work expressed through d’Alembert’s formulation

We also assume that the motion can be parametrized by smooth generalized coordinates and that virtual displacements vanish at the temporal boundaries. None of these assumptions go beyond standard classical mechanics as it is usually introduced.

What is not assumed is:

• The form of the Lagrangian
• The principle of least action
• The Euler–Lagrange equations themselves

Instead, these emerge as consequences of applying the virtual-work principle consistently in time. The combination $T-V$ appears as the unique quantity whose variation balances inertial and external forces under arbitrary virtual displacements, and the condition $\delta S = 0$ arises as a compact statement of this balance over an entire trajectory. In this sense, the variational formulation does not replace Newtonian mechanics, but reorganizes it into a form that turns out to be far more general and extensible.

• An Intuitive Derivation
• Euler Lagrange and Recovering Newtons Equations
• Conclusion
• References

An Intuitive Derivation

Starting from the ideas of momentum $p = m\dot{q}$ and a potential $V$. We know that one has to apply force to change the momentum:

[ \begin{aligned} p &= m \frac{\partial q}{\partial t} \\ F &= \frac{\partial p}{\partial t} \\ &= m \frac{\partial^2 q}{\partial t^2} \end{aligned} ]

The idea of the momentum was intuitively based by Newton on assumptions from Descartes, Galilei and Huygens. His idea was based on the observation of pendulums. For clarity, we restrict the discussion to a single generalized coordinate $q$. The extension to multiple degrees of freedom is straightforward.

The second assumption is that objects tend to move towards lower potential regions. This can be imagined as an object being pulled by a force to lower regions of the potential, the force depending on the slope of the potential:

[ \begin{aligned} F &= - \nabla V \end{aligned} ]

The third assumption is the principle of virtual work in the form introduced by d’Alembert: for any externally acting potential, the inertial force balances the applied forces under arbitrary virtual displacements.

[ \begin{aligned} \left(F_\mathrm{ext} - F_\mathrm{inertial}\right) \delta q &= 0 \end{aligned} ]

Note that $F_\mathrm{inertial}$ under variation of $q$ is given by

[ \begin{aligned} \frac{\partial}{\partial t} \left( p \delta q \right) &= \frac{\partial}{\partial t} \left( m \frac{\partial q}{\partial t} \delta q \right) \\ &= m \frac{\partial^2 q}{\partial t^2} \delta q + m \frac{\partial q}{\partial t} \delta \dot{q} \\ \end{aligned} ]

Reordering yields

[ \begin{aligned} m \frac{\partial^2 q}{\partial t^2} \delta q &= \frac{\partial}{\partial t} \left(m \frac{\partial q}{\partial t} \delta q \right) - m \frac{\partial q}{\partial t} \delta \dot{q} \end{aligned} ]

Inserting into our force equation between $F_\mathrm{ext}$ and $F_\mathrm{inertial}$:

[ \begin{aligned} \left(F - \frac{\partial p}{\partial t}\right) \delta q &= 0 \\ F \delta q - \frac{\partial}{\partial t} \left(m \frac{\partial q}{\partial t} \delta q \right) + m \frac{\partial q}{\partial t} \delta \dot{q} &= 0 \end{aligned} ]

We can identify the kinetic and the potential term. The first term is directly identifyable as potential term:

[ \begin{aligned} F = - \nabla V &= - \frac{\partial V}{\partial q} \\ F \delta q = - \frac{\partial V}{\partial q} \delta q &= - \delta V \end{aligned} ]

The last term can be identified as kinetic energy:

[ \begin{aligned} \delta \dot{q} &= \frac{\partial}{\partial t} \delta q \\ \underbrace{m \frac{\partial q}{\partial t}}_{p} \frac{\partial}{\partial t} \delta q \\ T = \frac{m v^2}{2} &= \frac{m}{2} \left(\frac{\partial q}{\partial t} \right)^2 \\ \delta T &= \frac{\partial T}{\partial \dot{q}} \delta \dot{q} = m \dot{q} \delta \dot{q} \end{aligned} ]

At this point it is worth briefly addressing why the kinetic energy takes the familiar quadratic form in the velocities. This is not an arbitrary choice, but follows from very basic structural requirements of classical mechanics. In particular, space and time are assumed to be homogeneous and isotropic and the laws of mechanics should be invariant under Galilean transformations. These requirements exclude any dependence of the kinetic energy on the absolute position or on the sign of the velocity, and they rule out linear or higher odd powers of $\dot{q}$. . Additivity of energy for composite systems further restricts the form. Up to an overall constant factor, the only expression compatible with these assumptions is a quadratic form in the velocities $T \propto \dot{q}^2$. Once this structure is fixed, the identification $\delta T = \left(\frac{\partial T}{\partial \dot{q}}\right) \delta \dot{q}$ follows uniquely.

Inserting those equalities into the equation yields:

[ \begin{aligned} F \delta q = - \frac{\partial V}{\partial q} \delta q &= - \delta V \\ - \delta V + \delta T - \frac{\partial}{\partial t} \left(m \frac{\partial q}{\partial t} \delta q \right) &= 0 \\ \delta T - \delta V &= \frac{\partial}{\partial t} \left(m \frac{\partial q}{\partial t} \delta q \right) \end{aligned} ]

Integrating over time yields

[ \begin{aligned} \int_{t_1}^{t_2} \delta T - \delta V \mathrm{d}t &= m \frac{\partial q}{\partial t} \delta q \mid_{t_1}^{t_2} \end{aligned} ]

Now we can assume that the displacement at the boundaries (initial and end position) is vanishing: $\delta q(t_1) = \delta q(t_2) = 0$:

[ \begin{aligned} \int_{t_1}^{t_2} \delta T - \delta V \mathrm{d}t &= 0 \\ \to \delta \underbrace{\int_{t_1}^{t_2} \overbrace{T - V}^{L} \mathrm{d}t}_{S} &= 0 \end{aligned} ]

This requirements eliminates total time derivatives from the variation, ensuring that only the interior of the trajectory contributes. Here we already see the variation of the action $S = \int L \mathrm{d}t$.

At this point, the structure of the Lagrangian has emerged without invoking any variational principle. Only the requirement that inertial and external forces balance under virtual displacements has been utilized.

Euler Lagrange and Recovering Newtons Equations

First we apply the total derivative for the variation of $\delta S$:

[ \begin{aligned} \delta S &= \int_{t_1}^{t_2} \frac{\partial (T-V)}{\partial q} \delta q + \frac{\partial(T-V)}{\partial \dot{q}} \delta \dot{q} \mathrm{d}t = 0 \\ \end{aligned} ]

Performing integration by parts on the second term

[ \begin{aligned} \int_{t_1}^{t_2} \frac{\partial L}{\partial \dot{q}} \delta \dot{q} \mathrm{d}t &= \frac{\partial L}{\partial \dot{q}} \delta q \mid_{t_1}^{t_2} - \int_{t_1}^{t_2} \frac{\partial}{\partial t} \left(\frac{\partial L}{\partial \dot{q}}\right) \delta q \\ \int_{t_1}^{t_2} \left( \frac{\partial L}{\partial q} - \frac{\partial}{\partial t} \left( \frac{\partial L}{\partial \dot{q}} \right) \right) \delta q \mathrm{d}t &= 0 \end{aligned} ]

Since this has to be valid for any variation $\delta q$ we arive at the Euler-Lagrange equation:

[ \begin{aligned} \frac{\partial L}{\partial q} - \frac{\partial}{\partial t} \left( \frac{\partial L}{\partial \dot{q}} \right) &= 0 \end{aligned} ]

When inserting $L = T - V$ and assuming that $T$ only depends on $\dot{q}$ while $V$ only depends on $q$ we arrive at

[ \begin{aligned} -\frac{\partial V}{\partial q} - \frac{\partial}{\partial t} \left(\frac{\partial T}{\partial \dot{q}} \right) &= 0 \\ \frac{\partial}{\partial t} \left(\frac{\partial T}{\partial \dot{q}} \right) &= -\frac{\partial V}{\partial q} \\ \frac{\partial}{\partial t} \left(\frac{ \frac{\partial m \dot{q}^2}{2} }{\partial \dot{q}} \right) &= -\frac{\partial V}{\partial q} \\ \frac{\partial}{\partial t} m \dot{q} &= -\frac{\partial V}{\partial q} \\ m \frac{\partial^2 q}{\partial t^2} &= -\frac{\partial V}{\partial q} \end{aligned} ]

In the last line we arrived at the standard Newtonian

[ \begin{aligned} F &= - \frac{\partial V}{\partial q} \end{aligned} ]

Conclusion

Seen from this perspective, the Lagrangian does not appear as a clever trick or an inspired guess, but as a natural reorganization of Newtonian mechanics. By insisting that inertial and external forces balance under arbitrary virtual displacements, the combination $T-V$ emerges as the quantity whose variation captures the dynamics of the system in a compact and systematic way. The action principle $\delta S = 0$ is then not an additional postulate, but a concise global statement of this local balance in time.

What makes this reformulation so powerful is not that it replaces Newtons laws, but that it distills them into a form that generalizes effortlessly to constrained systems, to fields and ultimately to quantum mechanics. In particular, when the classical notion of a single trajectory gives way to the sum over all possible paths in Feynmans path-integral formulation, the action no longer merely selects the physical motion but becomes the central object weighting every possible history or realization. From this viewpoint, the variational principle that appeared here as a consequence of classical mechanics reemerges in quantum theory as a structural necessity.

Arriving at the Lagrangian from basic mechanical ideas thus helps demystify both why it works classically and why it remains indispensable far beyond the classical domain. Rather than being an arbitrary starting point, the action turns out to be the natural language in which dynamics, classical or quantum, is most economically expressed.

References

• Goldstein, Herbert, Charles P. Poole, and John Safko. Classical mechanics
• Lanczos, Cornelius. The variational principles of mechanics
• Landau, Lev Davidovich, Evgenii Mikhailovich Lifshits, and Evgenii Mikhailovich Lifshits. Mechanics
• Feynman, Richard Phillips, Albert R. Hibbs, and George H. Weiss. Quantum mechanics and path integrals

This article is tagged: Basics, Physics, Quantum mechanics