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1.2 Lagrangian Mechanics

Table of Contents​

d'Alembert's Principle​

In my own studies, I have seen many derivations of Lagrangian mechanics. There are a few approaches: we can start from Maupertuis' Principle of Least Action; we can simply posit that plugging into the Euler-Lagrange equations gives the correct equations of motion; we can show from Feynman's path integral formulation that the classical path extremizes the action; or we can see how a worldline minimizes the proper time and derive Lagrangian mechanics from there.

Another approach is to start from Newtonian mechanics and derive Lagrangian mechanics using d'Alembert's Principle. The key idea is that in addition to the equations of motion, we also have constraint equations that restrict the motion of the system. For example, if we have a bead sliding on a frictionless wire, the bead's motion is constrained to lie along the wire. A rigid body is constrained such that the distances between its constituent particles remain constant. These constraints can be holonomic or non-holonomic, and they can be time-dependent or time-independent.

When we abstract away the physical system, we use a set of quantities to describe the configuration of the system.

Definition 1.2.1 (Holonomic Constraint)

A holonomic constraint is a constraint that can be expressed as an equation relating the coordinates and time:

where describe the configuration of the system.

Holonomic constraints reduce the number of degrees of freedom of a system. A set of holonomic constraints are independent if and only if the Jacobian matrix

has rank . Meaning, the gradients are linearly independent.

Example 1.2.2 (Examples of Constraints)

  • A rigid body is a system of particles where the distances between all pairs of particles are constant. For a system of particles, the rigid body constraints can be expressed as

    where is the constant distance between particles and . This is a holonomic constraint with .

  • A double pendulum consists of two masses connected by rigid rods. The lengths of the rods impose constraints on the motion of the masses. If and are the position vectors of the two masses, and and are the lengths of the rods, the constraints can be expressed as

    These are holonomic constraints.

  • Suppose a cylinder of radius rolls without slipping on a flat plane. Let be the position of the center of the cylinder. Let be the angle the cylinder makes with the horizontal axis, and let be the angle of rotation of the cylinder about its center. The no-slip condition imposes a constraint on the motion of the cylinder, which can be expressed as

  • For a set of gas particles confined in a box, the walls of the box impose constraints on the motion of the particles. These constraints can be expressed as inequalities, such as

    where are the coordinates of particle and are the dimensions of the box. These are non-holonomic constraints.

Non-holonomic constraints cannot be expressed solely in terms of coordinates and time; they often involve inequalities or differential relationships. If we include differential relationships, we have a general form

A differential relationship can be written as such because we can always integrate it to obtain a relationship between coordinates and time. If we cannot integrate the relationship to obtain such a form, then the constraint is non-holonomic. Recall from the Frobenius Theorem that a set of differential constraints is integrable (and thus holonomic) if and only if the distribution defined by the constraints is involutive. To take an example, let's prove that the no-slip condition for a rolling cylinder is non-holonomic.

Example 1.2.3 (Non-holonomic Constraint: Rolling Cylinder)

Consider a cylinder of radius rolling without slipping on a flat plane, as described in Example 1.2.2. The no-slip condition imposes the following constraints on the motion of the cylinder:

We need to show that these constraints are non-holonomic.

Proof. Define the one-forms

The constraints can be expressed as and . Next, define the 2-form

Calculating , we have

since . The exterior derivatives of the one-forms are

Therefore,

Since neither nor is zero, by the Frobenius Theorem, the distribution defined by the constraints is not involutive. Thus, the constraints are non-holonomic.

Generalized Forces and Virtual Work​

Recall that we describe the state of a mechanical system using its configuration, which is specified by a configuration . Suppose we have a set of coordinates that describe a system at some time . At the same instant , we can apply some infinitesimal displacements that do not violate the constraints of the system. Let's return to physical coordinates for a moment. Let be the position vector of particle in the system, and let be an infinitesimal displacement of particle that does not violate the constraints. Such displacements are called virtual displacements. (Also, don't confuse the notation with variations.)

Definition 1.2.4 (Virtual Displacement)

A virtual displacement of particle in a mechanical system is an infinitesimal change in the position of the particle that is consistent with the constraints of the system at a fixed time .

Example 1.2.5 (Virtual Displacement: Particle on a Surface)

Consider a particle constrained to move on some surface defined by . At a fixed time , a virtual displacement must satisfy

If we perform a first-order Taylor expansion (perfectly valid since is infinitesimal), we have

Since by the constraint, we have

This shows that the virtual displacement is orthogonal to the gradient of the constraint function at the point .

By the previous example, any smooth surface leads to virtual displacements that are tangent to the surface. The constraining forces that keep the particle on the surface act normal to the surface. This means that constraining forces do no work as a result of virtual displacements.

Definition 1.2.6 (Ideal Constraint)

An ideal constraint is a constraint for which the total work done by the constraining forces during any virtual displacement is zero.

Why make this definition? Friction is an example of a non-ideal constraint, since frictional forces can do work during virtual displacements. However, friction is an emergent phenomenon that arises from microscopic interactions between molecules. As we are studying theoretical physics at a fundamental level, we are less concerned with emergent phenomena and more concerned with fundamental interactions. Thus, we will assume that all constraints we consider are ideal constraints.

d'Alembert's Principle​

We know that sometimes, we need to introduce more coordinates than the number of degrees of freedom in order to describe a system while accounting for constraints. For example, a bead sliding on a wire in three-dimensional space has three coordinates , but only one degree of freedom (the position along the wire). The other two coordinates are constrained by the shape of the wire.

When there are constraining forces present, we can write Newton's Second Law for each particle in the system as

where is the applied (non-constraining) force on particle , and is the constraining force on particle . The problem is that we cannot know a priori because the constraining forces depend on the specific constraints of the system.

So we seek a new formulation of classical mechanics that eliminates the constraining forces from consideration. If we have Newton's Second Law written as , we can think of a virtual "reversed force" that, when added to the real force , gives equilibrium. To proceed, we need a theorem.

Theorem 1.2.7 (Principle of Virtual Work)

Systems in equilibrium satisfy the Principle of Virtual Work:

where are the forces acting on particle , and are the virtual displacements of particle .

Proof. For a system with equilibrium, the forces are zero, so the virtual work is automatically zero:

By the definition of ideal constraints, the second term is zero, so we have

Using the Principle of Virtual Work, we can now plug into the theorem to obtain

This is known as d'Alembert's Principle.

Theorem 1.2.8 (d'Alembert's Principle)

A system of particles subject to ideal constraints satisfies d'Alembert's Principle:

Proof. Substitute into the Principle of Virtual Work.

d'Alembert's Principle allows us to eliminate the constraining forces from consideration, as they do no work during virtual displacements.

Example 1.2.9 (d'Alembert's Principle: Atwood's Machine)

A classic example encountered in introductory physics courses is the Atwood's machine, which consists of two masses and connected by a massless, inextensible string that passes over a frictionless pulley. Let and be the vertical positions of masses and , respectively, measured from some reference point. Our masses have positions and , where points downwards. The constraint imposed by the inextensible string is

where is the length of the string.

We will analyze the system with d'Alembert's Principle. But first, let's derive the equations of motion using Newtonian mechanics to make sure we get the same result. The net forces on the masses are

Here, is the tension in the string, which acts as a constraining force. To eliminate , we can take an additional step by differentiating the constraint twice to obtain

Substituting into the second equation of motion, we have

Adding this to the first equation of motion, we get

Finally, we can solve for the equation of motion for mass :

The second equation of motion can be obtained by substituting back into the constraint equation.

Now, let's use d'Alembert's Principle to analyze the same system. Any virtual displacement of mass must be accompanied by a virtual displacement of mass such that the constraint is satisfied. Thus we have

In vector form, the virtual displacements are

Now we are ready to apply d'Alembert's Principle. The forces acting on the masses are just gravity (as the tension is a constraining force). Thus we have

Substituting into d'Alembert's Principle, we have

Since is arbitrary, we have

Substituting from the constraint, we get

which simplifies to the same equation of motion we derived earlier using Newtonian mechanics.

Generalized Coordinates​

Given a system with particles and holonomic constraints, the number of degrees of freedom of the system is . Geometrically speaking, if the configuration space of the unconstrained system is , then the constraints restrict the motion of the system to an -dimensional submanifold of . If we choose a set of parameters that act as local coordinates on this submanifold, then we call these parameters the generalized coordinates of the system.

Theorem 1.2.10 (Generalized Coordinates)

A system with particles and independent holonomic constraints has degrees of freedom. The configuration of the system can be described by a set of parameters , called the generalized coordinates. The position of any particle is uniquely determined by the generalized coordinates and time, and constraints can be expressed as functions of the generalized coordinates and time.

Proof. Let be the position vector of particle in the system, where . As the Jacobian matrix of the constraints has rank , we can partition the coordinates into two sets: a set of dependent coordinates and a set of independent coordinates . The constraints can then be written as

By the Implicit Function Theorem, there exists a unique smooth function in the neighborhood of any point (satisfying the constraints) such that

So the configuration of the system can be described by the independent generalized coordinates . Let's call the generalized coordinates of the system. If so, then we can express the position of any particle using the generalized coordinates and time as discussed above.

Also, as the original constraints are expressed as

we can substitute into the constraints to express them solely in terms of the generalized coordinates and time:

By the definition of from the Implicit Function Theorem, the expression on the left is identically zero for all in its domain. As such, both claims are proven.

Example 1.2.11 (Examples of Generalized Coordinates)

  • For a simple pendulum consisting of a mass attached to a massless rod of length swinging in a plane, the position of the mass can be described using a single generalized coordinate: the angle the rod makes with the vertical. The Cartesian coordinates of the mass can be expressed in terms of as

    The constraint can be expressed in terms of the generalized coordinate :

    This is automatically satisfied for all .

  • For a double pendulum consisting of two masses and connected by massless rods of lengths and , respectively, the positions of the masses can be described using two generalized coordinates: the angles and that the rods make with the vertical. The Cartesian coordinates of mass (x_2, y_2)m_2\theta_1\theta_2$ as

    The constraints and can be expressed in terms of the generalized coordinates and :

  • The Atwood's machine described in Example 1.2.9 has two masses connected by an inextensible string. The positions of the masses can be described using a single generalized coordinate: the vertical position of mass . The vertical position of mass can be expressed in terms of using the constraint imposed by the string:

    The constraint can be expressed in terms of the generalized coordinate :

    This is automatically satisfied for all .

  • A particle constrained to move on the surface of a sphere of radius can have its position described using two generalized coordinates: the polar angle and the azimuthal angle . The Cartesian coordinates of the particle can be expressed in terms of and as

    The constraint can be expressed in terms of the generalized coordinates and :

    This is automatically satisfied for all and .

Definition 1.2.12 (Configuration Space, Configuration Manifold)

The configuration space of a mechanical system is defined as the set of all possible configurations of the system, represented by the generalized coordinates. If the system has degrees of freedom, the configuration space is an -dimensional manifold.

Virtual displacements can also be expressed in terms of generalized coordinates via the chain rule. If we have a virtual displacement of particle , we can write

where are the virtual displacements in the generalized coordinates. Mathematically speaking, letting the point at time represent the configuration of the system, the virtual displacements represent a tangent vector in the tangent space of the configuration manifold at the point .

Velocities can be expressed as

The virtual work of the forces acting on the system can now be expressed as

Comparing this to the definition of virtual work in physical coordinates , we can identify the quantity in parentheses as the generalized force associated with the generalized coordinate . This allows us to define generalized forces in a similar manner as virtual displacements:

Thus, the generalized force represents the component of the applied forces in the direction of the generalized coordinate .

Definition 1.2.13 (Generalized Force)

The generalized force associated with the generalized coordinate is defined as

where are the applied (non-constraining) forces acting on particle , and is the position vector of particle expressed in terms of the generalized coordinates.

Theorem 1.2.14 (Equations from d'Alembert's Principle)

In generalized coordinates, d'Alembert's Principle can be expressed as

where is the kinetic energy of the system, and is the generalized force associated with the generalized coordinate .

Proof. Starting from d'Alembert's Principle in physical coordinates:

we have already expressed the work done by the applied forces in terms of generalized coordinates. Now consider the term involving the time derivative of momentum. We have

Let's rewrite the term in parentheses using integration by parts:

To expand the last term , we use the chain rule:

Turning to the term , we can use to write

Substituting these results back into the expression for , we have

where we have identified the kinetic energy of the system as . d'Alembert's Principle now reads

Since the virtual displacements are independent and arbitrary, the expression in brackets must be zero for each . Thus, we have

which completes the proof.

Example 1.2.15 (Examples of Using Equations from d'Alembert's Principle)
Theorem 1.2.16 (Euler-Lagrange Equations)

In the case where all forces are conservative (i.e., can be derived from a potential energy function ), the equations from d'Alembert's Principle reduce to the Euler-Lagrange Equations:

where is the Lagrangian of the system.

It is interesting that the Euler-Lagrange equations appear in classical mechanics, because it implies that the dynamics of a mechanical system can be derived from a variational principle. More specifically, physical paths must be extremal paths of a certain variational problem.