This note is about a formalization of classical mechanics due to Gauss. Gauss noticed that, roughly, the constrained motion of masses is as close as possible to their unconstrained motions, while still satisfying the constraints. For example, a pendulum bob would naturally fall straight down, but is constrained to a circle by its string -- so its true acceleration will be as close as possible to straight down, while still remaining on the string. This generalizes to essentially any constrained system. Specifically, to find the true accelerations of masses in a constrained system, we first find the accelerations as if they were unconstrained, and then “project” to the closest acceleration that satisfies the constraints. \[a_{\text{true}} = \argmin_{a \in \text{ConsistentAccelerations}} ||a - a_{\text{unconstrained}} ||^2_M\]
This formulation (which also holds in generalized coordinates) is another nice extremal principle in physics, and is the basis of some recent fast rigid-body physics simulations (MuJoCo), using convex optimization.
In this note, we state, prove, and discuss Gauss's principle, assuming minimal prior knowledge. We use it for a double-pendulum simulation just for fun.
Of course, this is the general problem of classical mechanics, but we are focusing on how to deal with constraints. Directly applying Newtonian mechanics can get very messy when dealing with kinematic constraints, since it requires solving for all the intermediary constraint forces. Often, we are not interested in what forces are required to enforce the constraints, we just want to know the eventual movement of the masses. (For example, a ball rolling along a curved track โ we are not interested in the forces required to keep the ball on the track, just in the eventual motion of the ball). Re-formulations of mechanics (such as Lagrangian mechanics, and Gauss's principle) give us “implicit ”ways of dealing with constraints.
Organization. We first setup the problem and notation, then in Section we state Gauss's Principle and give several examples. In Section we formally define the notion of "consistent accelerations" required for Gauss's Principle. Finally, we give a proof of Gauss's Principle in Section . We conclude with remarks, historical references, and open problems.
A constrained system (to us) is one where the configuration $X$ is constrained to lie on a manifold. For example, we may have the constraint $x_1 = -x_2$, or if two masses are connected by a rod of length $\ell$, the constraint $||\vec x_1 - \vec x_2||_2 = \ell$. Here, we will consider only constraints that do no net work on the system (this is the case for most natural physical constraints).
Given a configuration $X$ on the manifold, let $T_X$ be the set of possible velocities of the system at configuration $X$, which are consistent with the constraints. Notice that $T_X$ is a subspace, and is in fact exactly the tangent space of the manifold at $X$. (We will see examples soon). Our condition that constraint forces do no net work is exactly the condition that (in inertial coordinates) the collection of constraint forces $\Fconst$ is orthogonal to $T_X$.
The state of a system is given by the positions and velocities $(X, \dot{X})$.
The mass matrix is the matrix $M$ such that $\frac{1}{2}||\dot{q}||^2_M = \frac{1}{2}\dot{q}^TM\dot{q}$ is the kinetic energy of the system, where $\dot{q}$ is the generalized velocities. (Such a matrix always exists when the mapping $q_i(r_1, \dots r_n)$ between generalized coordinates $q$ and spatial coordinates $r$ is time-independent).
Our goal is: Given the constraint manifold and the current state $(X, \dot{X})$ of a system, find the accelerations $\ddot{X}$. This determines the entire trajectory $X(t)$.
This goal is the analog of Newton's law ($F=ma$) โ it specifies the local evolution of a system.
Gauss's principle is:
Where the norm $M$ is the mass matrix, and $\consaccel_{X, \dot{X}}$ is the set of accelerations consistent with the constraints at the current state.
The set $\consaccel$ is defined and discussed further below.
For example, if we describe our system in inertial, spatial coordinates, and there is an external (non-constraint) force $\vec F_i$ acting on mass $m_i$, then the true accelerations satisfy:
\[\{\vec a_1, \dots, \vec a_n\} = \argmin_{\{\vec a_1, \dots, \vec a_n\} \in \consaccel} \sum_i m_i ||\vec a_i - \frac{\vec F_i}{m_i}||^2\]
Model the configuration space as the position of the mass $(x, y) \in \R^2$, constrained to lie on the manifold $C = \{(x, y): x^2 + y^2 = 1\} \subset \R^2$. The mass matrix is simply $M = \begin{bmatrix} m &0\\ 0 &m \end{bmatrix}$. The unconstrained acceleration of the mass is simply $g$, straight down by gravity.
At a given state (position and velocity $v$) of the mass, let us consider the space of consistent accelerations. The fixed velocity determines the radial/centripital acceleration ($a_r$, in blue), so the only freedom is in the tangential accelerations ($a_T$). Thus, the consistent accelerations are exactly those with the given radial acceleration, and arbitrary tangential acceleration. (This will be discussed further below.) The projection of $g$ onto the consistent accelerations (as dictated by Gauss's Principle) is equivalent to considering the tangential component of gravity.
Note that we could not have directly used the angle of a pendulum as a generalized coordinate when applying Gauss's Principle, since we can't express the unconstrained acceleration of the pendulum in terms of only this coordinate. Thus, Gauss's Principle requires us to work in an ambient space that includes even "off-shell" trajectories of the constrained system. However, it is still possible to use generalized coordinates in our representation, as we will see below.
In practice, it is still possible to use generalized coordinates in our computations, though we will need to map it back to spatial coordinates. This is useful for simulations, because we can use coordinates that implicitly respect constraints. For example, for the double pendulum, we can represent the configuration space by the two angles $\vec \theta = (\theta_1, \theta_2)$. The set of consistent accelerations in $\theta$-space is simply all of $\R^2$, since the angular accelerations can be arbitrary. Thus, we just need to translate this set back into spatial coordinates. The spatial coordinates $\vec r = (\vec r_1, \vec r_2)$ of the masses are functions of $\theta$: $\vec r_i(\theta_1, \theta_2)$. Now, at a given state $S = (\theta_1, \theta_2, \dot{\theta_1}, \dot{\theta_2})$, we have $$ \ddot{\vec r} = Q_S \ddot{\vec\theta} + \vec b_S $$ for some matrix $Q_S$ and vector $b_S$ depending on the state. This follows simply by differentiating $\vec r(\vec \theta)$. Thus, we can express the set of all kinematically feasible accelerations as the affine space $$\consaccel_S = \{Q_S \ddot{\vec\theta} + \vec b_S : \ddot{\vec \theta} \in \R^2\}$$ This is extremely convenient, because we can now apply Gauss's Principle to directly find $\ddot{\vec\theta}$, the generalized accelerations: $$A = \argmin_{A \in \consaccel_S} ||A - A_\text{unconstrained}||^2 \iff \ddot{\vec\theta}_{true} = \argmin_{\ddot{\vec\theta} \in \R^2} ||Q\ddot{\vec\theta} + b - A_\text{unconstrained}||^2 $$ This is, presumably, something like what MuJoCo does.
Just for fun, here is an example implementation of the above (plus some dampening for stability). Try some random configurations. On the left is a phase-space plot of $\theta$ vs $\dot{\theta}$ ($\theta_1$ in blue, $\theta_2$ in red).
Take the ambient space as the $x$ coordinate of mass $m_1$, and the $x, y$ coordinates of mass $m_2$: $(x_1, x_2, y_2) \in \R^3$. The 2-dimentional constraint manifold (for appropriate choice of orgin) is defined by $\{(x_1, x_2, y_2): \tan \theta = y_2/(x_1-x_2)\}$, expressing the constraint that the block remain on the wedge. The mass matrix is again diagonal, since we are using inertial coordinates: $M = diag(m_1, m_2, m_2)$.
(Left as an exercise for the reader). Note that we can handle internal non-constraint forces, such as springs -- this is just incorporated into the unconstrained accelerations.
We define $\consaccel$ in order to move from constraints on the positions of masses to constraints on their accelerations. This can be done by differentiating the manifold constraint at the current state.
Formally, given the current state $(X, \dot{X})$ of a constrained system, define the “consistent accelerations ”as the following set. Consider all trajectories $\t X(t)$ which are kinematically feasible (lie on the constraint manifold), and agree with the current state at $t = 0$: $(\t X(0) = X, \dot{\t X}(0) = \dot{X})$. The consistent accelerations are the set of all accelerations for these feasible trajectories: \[\consaccel_{X, \dot X} := \{\ddot{\t X}(0)\}\]
The following property is important:
That is, the set $\consaccel_{X, \dot X}$ is an affine shift of $T_X$ (the tangent space of the constraint manifold).
For example, consider the consistent accelerations of a pendulum at its bottommost point: The true acceleration at this point is purely radial (upwards, $a_{true}$ in blue), and the consistent accelerations (dashed green) correspond to further tangential accelerations. Note that these tangential accelerations can be identified with the tangent space of the constraint (tangential displacements).
Here are several equivalent ways of proceeding:
Approach 1: Suppose the true acceleration is $A_{true}$, and consider a consistent acceleration $A$. By Lemma , any consistent acceleration $A$ must have the form $A = A_{true} + \delta$ for some $\delta \in T_X$ (where $T_X$ is the tangent space to the constraint manifold, ie the “consistent velocities”).
We will show that any consistent acceleration $A = A_{true} + \delta$ will have larger norm $||A - A_U||_M^2 \geq ||A_{true} - A_U||_M^2$.
Consider the gradient of the objective \begin{align*} \grad_A ||A - A_U||_M^2 &= (M(A - A_U))^T \end{align*} The key observation is that at the true accelerations $A = A_{true}$, this gradient is exactly the constraint forces $\Fconst$ on the system. That is, \[ \grad_A ||A - A_U||_M^2 |_{A = A_{true}} = 2(M(A_{true} - A_U))^T = 2\Fconst^T \] Because, $MA_{true} - MA_U = F_{net} - F_{\text{external}} = \Fconst$. Further, since constraint forces do no work, we know that $\Fconst \perp T_X$.
Thus, at the true acceleration, the gradient ($=\Fconst$) is orthogonal to all possible consistent perturbations ($\delta \in T$). This is sufficient to show optimality, by convexity of the objective, and the set $\consaccel$. So the true acceleration minimizes the objective among the set of consistent accelerations.
Approach 2: Consider the minimizer $A^*$ in $$ A^* = \argmin_{A \in \consaccel_{X, \dot{X}}} ||A - A_U||^2_M $$ By Lemma , we know the set $\consaccel$ is an affine shift of $T_X$. Thus, this minimization is simply the projection of $A_U$ onto an affine shift of $T_X$, under the $M$-norm. This projection $A^*$ is charecterized uniquely by $M(A^* - A_U) \perp T_X$. But, as we saw above, the true acceleration $A_{true}$ satisfies this, since $M(A_{true} - A_U)$ is the constraint force. Thus, $A^* = A_{true}$.
Approach 3: In fact, we can directly show that any consistent acceleration $A$ must have larger objective: $$ \forall A \in \consaccel_{X, \dot{X}}: ~\|A - A_U\|_M^2 \geq \|A_{true} - A_U\|_M^2. $$
By Lemma , any consistent acceleration $A$ must have the form $A = A_{true} + \delta$ for some $\delta \in T_X$. Then, using the fact that $\Fconst \perp \delta$, $$ \begin{align*} \|A - A_{U}\|_M^2 &= \|A_{\rm true} + \delta - A_{U}\|_M^2\\ &= \|A_{\rm true} - A_{U}\|_M^2 +ย \|\delta\|_M^2 + 2\delta^TM(A_{\rm true} - A_{U})\\ &= \|A_{\rm true} - A_{U}\|_M^2 + \|\delta\|_M^2 + 2\delta^T \Fconst\\ &= \|A_{\rm true} - A_{U}\|_M^2 + \|\delta\|_M^2\\ &\geq \|A_{\rm true} - A_{U}\|_M^2 \end{align*} $$
Note that it was important we used inertial coordinates in the proof, so we could apply $F_{net} = MA$.
Acknowledgements. Thanks to Thibaut Horel for suggestions on the presentation and proof, and Darius Shi for catching various errors.