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1. What is the question?

A student has read Lanczos’s The Variational Principle of Mechanics and encountered the following statement (called Postulate A):

“The virtual work of the forces of reaction is always zero for any virtual displacement which is in harmony with the given kinematic constraints.”

The student wonders:

  • What exactly does this postulate mean?
  • Is it something that can be derived from Newton’s laws, or is it an extra assumption?
  • When Lanczos says that all analytical‑mechanics formulations (Euler–Lagrange, Hamilton, Jacobi, …) follow from Newton’s second law plus Postulate A, is that really true?

In short: Can we obtain the whole machinery of analytical mechanics from Newton’s II law together with the virtual‑work postulate, and why?

We will answer this by (i) spelling out the postulate, (ii) showing step‑by‑step how Newton’s II law + the postulate lead to d’Alembert’s principle, (iii) deriving the Lagrange equations (the cornerstone of analytical mechanics) from there, and (iv) summarising the logical status of the postulate.

2. Detailed reasoning (step‑by‑step)

2.1 Newton’s second law for a system of particles

Consider a system of (N) particles with masses (m_i) and position vectors (\mathbf r_i).
Let the applied (external) forces be (\mathbf F_i) and let the constraint (reaction) forces be (\mathbf R_i).
Newton’s second law for each particle reads

[ m_i\mathbf a_i = \mathbf F_i + \mathbf R_i\qquad (i=1,\dots ,N), \tag{1} ]

where (\mathbf a_i = \ddot{\mathbf r}_i) is the particle’s acceleration.

2.2 Virtual displacements compatible with the constraints

A virtual displacement (\delta\mathbf r_i) is an infinitesimal change of the configuration consistent with the constraints, taken at a fixed instant of time (so (\delta t = 0)).

If the constraints are ideal (the usual situation in analytical mechanics) the constraint forces do no work on any such displacement. This is precisely Postulate A:

[ \boxed{\;\sum_{i=1}^{N}\mathbf R_i!\cdot!\delta\mathbf r_i = 0 \quad\text{for every admissible } \delta\mathbf r_i\;} \tag{2} ]

The adjective “in harmony with the given kinematic constraints’’ simply means “compatible with the constraints”.

2.3 From Newton + Postulate A to d’Alembert’s principle

Take the scalar product of Eq. (1) with the admissible virtual displacement (\delta\mathbf r_i) and sum over all particles:

[ \sum_i (m_i\mathbf a_i - \mathbf F_i - \mathbf R_i)!\cdot!\delta\mathbf r_i = 0 . \tag{3} ]

Insert the postulate (2) to eliminate the term containing (\mathbf R_i):

[ \sum_{i=1}^{N}\bigl(m_i\mathbf a_i - \mathbf F_i\bigr)!\cdot!\delta\mathbf r_i = 0 . \tag{4} ]

Equation (4) is d’Alembert’s principle: the total virtual work of the applied forces minus the inertial forces (m_i\mathbf a_i) vanishes for any admissible virtual displacement.

Thus d’Alembert’s principle is exactly Newton’s second law supplemented by the ideal‑constraint assumption (Postulate A). No further hypothesis is required.

2.4 Introducing generalized coordinates

Suppose the constraints can be expressed (locally) by (k) independent generalized coordinates (q_\alpha) ((\alpha = 1,\dots ,k)), with

[ \mathbf r_i = \mathbf r_i(q_1,\dots ,q_k,t). \tag{5} ]

A virtual displacement consistent with the constraints is then

[ \delta\mathbf r_i = \sum_{\alpha=1}^{k}\frac{\partial\mathbf r_i}{\partial q_\alpha}\,\delta q_\alpha . \tag{6} ]

Insert (6) into d’Alembert’s principle (4):

[ \sum_i\bigl(m_i\mathbf a_i - \mathbf F_i\bigr)!\cdot! \sum_{\alpha}\frac{\partial\mathbf r_i}{\partial q_\alpha}\,\delta q_\alpha = 0 . \tag{7} ]

Because the virtual variations (\delta q_\alpha) are independent (they can be chosen arbitrarily), the coefficients of each (\delta q_\alpha) must vanish:

[ \sum_{i=1}^{N}\bigl(m_i\mathbf a_i - \mathbf F_i\bigr)!\cdot! \frac{\partial\mathbf r_i}{\partial q_\alpha}=0, \qquad \alpha = 1,\dots ,k . \tag{8} ]

2.5 Introducing the kinetic energy

Define the kinetic energy

[ T(q,\dot q,t)=\frac12\sum_i m_i\,\dot{\mathbf r}i^{\,2}, \qquad \dot{\mathbf r}_i =\sum\beta \frac{\partial\mathbf r_i}{\partial q_\beta}\dot q_\beta +\frac{\partial\mathbf r_i}{\partial t}. \tag{9} ]

A straightforward differentiation (using the chain rule) yields the identity

[ \frac{d}{dt}!\left(\frac{\partial T}{\partial\dot q_\alpha}\right) -\frac{\partial T}{\partial q_\alpha} = \sum_i m_i\mathbf a_i!\cdot!\frac{\partial\mathbf r_i}{\partial q_\alpha}. \tag{10} ]

Insert (10) into (8) and define the generalized forces

[ Q_\alpha \equiv \sum_i \mathbf F_i!\cdot!\frac{\partial\mathbf r_i}{\partial q_\alpha}. \tag{11} ]

Equation (8) becomes the celebrated Euler–Lagrange (Lagrange) equations:

[ \boxed{\; \frac{d}{dt}!\left(\frac{\partial T}{\partial\dot q_\alpha}\right) -\frac{\partial T}{\partial q_\alpha}=Q_\alpha,\qquad \alpha=1,\dots ,k . \;} \tag{12} ]

If the applied forces are conservative, i.e. (\mathbf F_i = -\nabla_i V) for a potential (V(q,t)), then

[ Q_\alpha = -\frac{\partial V}{\partial q_\alpha}, ]

and defining the Lagrangian (L = T - V) we obtain the compact form

[ \boxed{\; \frac{d}{dt}!\left(\frac{\partial L}{\partial\dot q_\alpha}\right) -\frac{\partial L}{\partial q_\alpha}=0 . \;} \tag{13} ]

Equations (13) are the core of analytical mechanics.

All the other variational formulations (Hamilton’s principle, Hamiltonian equations, Jacobi’s principle, etc.) are mathematically equivalent rewritings of (13). Hence every formulation of analytical mechanics follows from:

  • Newton’s second law for each particle (1), and
  • The ideal‑constraint assumption that the virtual work of reaction forces vanishes (2).

2.6 Is Postulate A deducible from Newton’s laws alone?

No. Newton’s second law (1) tells us how each particle accelerates under the total force (applied + constraint). It says nothing about the direction or magnitude of the constraint forces themselves.

The statement “constraint forces do no virtual work” is an additional physical hypothesis about the nature of the constraints (they are ideal). It is not a theorem that follows from Newton’s three laws; rather, it is an empirical observation that holds for many everyday constraints (smooth surfaces, rigid rods, inextensible strings, etc.).

One can justify it microscopically (e.g. by modeling a rigid rod as a collection of tightly bound atoms whose internal forces are equal and opposite, producing zero net work for admissible deformations), but such a justification still rests on extra assumptions about the internal constitution of the material. Consequently, analytical mechanics needs both Newton’s second law and Postulate A (or an equivalent statement about ideal constraints).

3. Final answer

  • Postulate A is the assumption that the reaction (constraint) forces are ideal—they do no work on any virtual displacement compatible with the constraints.

  • When this assumption is combined with Newton’s second law for each particle, one obtains d’Alembert’s principle (Eq. 4).

  • From d’Alembert’s principle, by introducing generalized coordinates and the kinetic energy, the Euler–Lagrange equations (Eq. 12) are derived.

  • The Euler–Lagrange equations are the starting point for all the variational formulations of analytical mechanics (Hamilton’s principle, Hamiltonian formalism, Jacobi’s principle, etc.).

Therefore, yes: Lanczos’s claim is correct—the whole edifice of analytical mechanics can be built from Newton’s second law together with Postulate A. However, Postulate A is not a consequence of Newton’s three laws; it is an independent physical postulate (or definition) concerning the nature of the constraints.

4. Common mistakes & how to avoid them

Mistake Why it’s wrong How to correct it
Thinking the virtual work of any forces is always zero. The zero‑work statement applies only to the constraint (reaction) forces, not to applied forces. Remember the distinction: (\sum \mathbf R_i!\cdot!\delta\mathbf r_i =0) (postulate) vs. (\sum \mathbf F_i!\cdot!\delta\mathbf r_i) which is generally non‑zero.
Assuming d’Alembert’s principle is a separate law. It is just Newton’s second law rewritten after eliminating the constraint forces via the virtual‑work postulate. Derive Eq. 4 from (1) + (2) explicitly; see the steps above.
Confusing “ideal constraints’’ with “no forces”. Ideal constraints do exert forces; they simply do no work on admissible virtual displacements. Emphasise the definition of an ideal constraint: (\mathbf R_i) orthogonal (in the work sense) to every (\delta\mathbf r_i) allowed by the constraints.
Trying to treat non‑holonomic constraints the same way. For non‑integrable (velocity‑dependent) constraints  

Original question: The role of the virtual work principle on Physics Stack Exchange, licensed CC BY-SA.