24 May 2025
Coordinate Systems, Virtual Work, and D'Alembert's Principle
Coordinate transformations, degrees of freedom, generalized velocities, virtual work, and D'Alembert's equation.
Coordinate systems
Cartesian coordinates $(x,y,z)$ use fixed, mutually perpendicular unit vectors. Two common curvilinear transformations are cylindrical coordinates $(\rho,\phi,z)$,
\[x=\rho\cos\phi, \qquad y=\rho\sin\phi, \qquad z=z,\]and spherical coordinates $(r,\theta,\phi)$,
\[x=r\sin\theta\cos\phi, \qquad y=r\sin\theta\sin\phi, \qquad z=r\cos\theta.\]Their standard coordinate ranges are
\[\rho\geq0,\quad 0\leq\phi<2\pi; \qquad r\geq0,\quad 0\leq\theta\leq\pi,\quad 0\leq\phi<2\pi.\]The inverse relations include
\[\rho=\sqrt{x^2+y^2}, \qquad \phi=\operatorname{atan2}(y,x),\]and
\[r=\sqrt{x^2+y^2+z^2}, \qquad \theta=\cos^{-1}\!\left(\frac zr\right).\]At $\rho=0$ the azimuth $\phi$ is undefined; at $r=0$ both spherical angles are undefined. These are coordinate singularities, not physical singularities. Differentiating the position vector gives
\[\mathbf v =\dot\rho\,\hat{\boldsymbol\rho} +\rho\dot\phi\,\hat{\boldsymbol\phi} +\dot z\,\hat{\mathbf z}\]in cylindrical coordinates, and
\[\mathbf v =\dot r\,\hat{\mathbf r} +r\dot\theta\,\hat{\boldsymbol\theta} +r\sin\theta\dot\phi\,\hat{\boldsymbol\phi}\]in spherical coordinates. The factors $1$, $r$, and $r\sin\theta$ arise because the curvilinear basis vectors and arc lengths depend on position.
Degrees of freedom and generalized velocities
A system of $N$ point particles requires $3N$ Cartesian coordinates before constraints are imposed. If $k$ independent holonomic constraints
\[f_\alpha(\mathbf r_1,\ldots,\mathbf r_N,t)=0, \qquad \alpha=1,\ldots,k,\]have rank $k$, the system has
\[\boxed{s=3N-k}\]degrees of freedom. Choose independent generalized coordinates $q_1,\ldots,q_s$ and write
\[\mathbf r_i=\mathbf r_i(q_1,\ldots,q_s,t).\]The chain rule gives the actual velocity,
\[\boxed{ \mathbf v_i =\sum_{j=1}^{s}\frac{\partial\mathbf r_i}{\partial q_j}\dot q_j +\frac{\partial\mathbf r_i}{\partial t}}.\]The $q_j$ need not be lengths: they may be angles or other independent parameters. Their time derivatives $\dot q_j$ are generalized velocities.
Virtual displacement and virtual work
A virtual displacement compares neighbouring allowed configurations at the same instant, so $\delta t=0$. Hence
\[\boxed{ \delta\mathbf r_i =\sum_{j=1}^{s}\frac{\partial\mathbf r_i}{\partial q_j}\delta q_j}.\]It must also satisfy the linearized constraints,
\[\sum_i\nabla_i f_\alpha\cdot\delta\mathbf r_i=0.\]This differs from an actual displacement during a time $dt$, for which an additional $(\partial f_\alpha/\partial t)dt$ term can occur.
At static equilibrium, ideal constraint forces do no virtual work. If $\mathbf F_i$ denotes the applied force, the principle of virtual work is
\[\delta W=\sum_i\mathbf F_i\cdot\delta\mathbf r_i=0.\]Substitution of the generalized displacement gives
\[\delta W=\sum_jQ_j\delta q_j, \qquad \boxed{Q_j=\sum_i\mathbf F_i\cdot \frac{\partial\mathbf r_i}{\partial q_j}}.\]$Q_j$ has the units of energy divided by the units of $q_j$. Thus it is a force for a length coordinate and a torque for an angular coordinate.
D’Alembert’s principle
Let $\mathbf R_i$ be ideal constraint forces. Newton’s equations are
\[\mathbf F_i+\mathbf R_i=m_i\mathbf a_i.\]Multiplication by allowed virtual displacements and summation eliminates the constraint forces because $\sum_i\mathbf R_i\cdot\delta\mathbf r_i=0$. Therefore
\[\boxed{ \sum_i(\mathbf F_i-m_i\mathbf a_i)\cdot\delta\mathbf r_i=0}.\]To express the inertial term in generalized coordinates, start from
\[T=\frac12\sum_i m_i\mathbf v_i^2.\]The coordinate transformation implies
\[\frac{\partial\mathbf v_i}{\partial\dot q_j} =\frac{\partial\mathbf r_i}{\partial q_j}, \qquad \frac{d}{dt}\left(\frac{\partial\mathbf r_i}{\partial q_j}\right) =\frac{\partial\mathbf v_i}{\partial q_j}.\]Consequently,
\[\frac{\partial T}{\partial\dot q_j} =\sum_i m_i\mathbf v_i\cdot\frac{\partial\mathbf r_i}{\partial q_j},\]and differentiation with respect to time gives
\[\frac{d}{dt}\frac{\partial T}{\partial\dot q_j} -\frac{\partial T}{\partial q_j} =\sum_i m_i\mathbf a_i\cdot\frac{\partial\mathbf r_i}{\partial q_j}.\]D’Alembert’s principle therefore becomes
\[\sum_j\left[ Q_j-\frac{d}{dt}\frac{\partial T}{\partial\dot q_j} +\frac{\partial T}{\partial q_j} \right]\delta q_j=0.\]Because the independent $\delta q_j$ are arbitrary,
\[\boxed{ \frac{d}{dt}\frac{\partial T}{\partial\dot q_j} -\frac{\partial T}{\partial q_j}=Q_j}.\]This is D’Alembert’s equation in generalized coordinates and is the direct bridge from Newton’s laws to Lagrange’s equations.
Discussion