Generalised Coordinates

In classical mechanics, especially when dealing with complex systems or systems with constraints, it becomes convenient to move away from the standard Cartesian coordinates (x, y, z). Generalised coordinates are a set of independent variables that completely describe the configuration of a mechanical system. These coordinates do not necessarily have to be lengths; they can be angles, areas, or any other quantity that uniquely defines the system's position. The number of generalised coordinates required is equal to the number of degrees of freedom of the system.

For a system of N particles in three dimensions, there are 3N Cartesian coordinates. If there are 'k' independent constraints on the system, the number of degrees of freedom is (3N - k). We then need (3N - k) independent generalised coordinates, denoted as q₁, q₂, ..., q_n, where n = 3N - k.

Examples of Generalised Coordinates

Consider a simple pendulum. In Cartesian coordinates, we would need the x and y positions of the bob. However, the length of the pendulum is fixed, imposing a constraint. The system has only one degree of freedom. We can describe its configuration using a single generalised coordinate: the angle θ the pendulum makes with the vertical. So, q₁ = θ.

For a particle moving on the surface of a sphere of radius R, we can use spherical coordinates (R, θ, φ). Since R is fixed, the degrees of freedom are two, and we can use θ and φ as our generalised coordinates (q₁ = θ, q₂ = φ).

The transformation from Cartesian coordinates (x_i) to generalised coordinates (q_j) is given by: x_i = x_i(q₁, q₂, ..., q_n, t) y_i = y_i(q₁, q₂, ..., q_n, t) z_i = z_i(q₁, q₂, ..., q_n, t) where 't' is explicitly present if the constraints are time-dependent.

The velocities in terms of generalised coordinates are obtained by differentiating these expressions with respect to time: ẋ_i = Σ_j (∂x_i/∂q_j) q̇_j + ∂x_i/∂t where q̇_j = dq_j/dt are the generalised velocities.

D'Alembert's Principle

D'Alembert's principle is a fundamental principle in classical mechanics that extends Newton's second law of motion to systems with constraints. It reformulates the law of motion for a dynamic system into a statement about static equilibrium. The principle states that the sum of the applied forces on a system, plus the "inertial forces" acting on each particle, is zero.

Mathematically, for a system of N particles, Newton's second law for the i-th particle is: F_i = m_i a_i where F_i is the net force acting on the i-th particle, m_i is its mass, and a_i is its acceleration.

Rearranging this equation, we get: F_i - m_i a_i = 0

D'Alembert's principle interprets the term -m_i a_i as an "inertial force" acting on the particle. The principle then states that the system is in "dynamic equilibrium" if the sum of all applied forces and all inertial forces is zero.

For a system of N particles, D'Alembert's principle is expressed as: Σ_i=1^N (F_i - m_i a_i) = 0

This principle is particularly useful when dealing with systems subject to constraints. The constraint forces (forces that maintain the constraints) do not need to be explicitly considered if they do no work. D'Alembert's principle allows us to handle these systems by transforming a problem in dynamics into a problem that resembles statics.

Application to Constrained Systems

Consider a particle constrained to move on a surface. The constraint force is perpendicular to the surface and does no work if the particle moves along the surface. By including the inertial force, we can write the equation of motion for the particle as: F_{applied, i} + F_{constraint, i} - m_i a_i = 0 If the constraint forces do no work, their contribution to the virtual work is zero. D'Alembert's principle, when combined with the principle of virtual work, leads to the Lagrange equations.

Lagrange's Equations of Motion

Lagrange's equations of motion are a set of second-order ordinary differential equations that describe the motion of a mechanical system. They are derived from D'Alembert's principle and the principle of virtual work, using generalised coordinates. The Lagrangian, denoted by L, is a scalar function defined as the difference between the kinetic energy (T) and the potential energy (V) of the system: L = T - V

The kinetic energy T is expressed in terms of generalised coordinates (q_j) and generalised velocities (q̇_j). The potential energy V is typically expressed in terms of generalised coordinates only, assuming conservative forces.

Lagrange's equations of motion for a system with n degrees of freedom are given by: d/dt (∂L/∂q̇_j) - ∂L/∂q_j = 0 for j = 1, 2, ..., n.

These equations provide a powerful alternative to Newton's laws, especially for systems with constraints or when using generalised coordinates. They automatically handle constraint forces that do no work.

Derivation Outline (Conceptual)

1. Express the kinetic energy (T) and potential energy (V) of the system in terms of generalised coordinates (q_j) and generalised velocities (q̇_j).
2. Form the Lagrangian: L = T - V.
3. Apply D'Alembert's principle in its generalised form, considering virtual displacements (δq_j) and virtual work.
4. The principle of virtual work states that for a system in equilibrium (or dynamic equilibrium under D'Alembert's principle), the total virtual work done by all forces is zero: Σ_j (Q_j - ∂V/∂q_j) δq_j = 0, where Q_j is the generalised force.
5. The generalised force Q_j is related to the time derivative of ∂L/∂q̇_j. Specifically, Q_j = d/dt (∂L/∂q̇_j) - ∂L/∂q_j.
6. Setting the total virtual work to zero leads directly to Lagrange's equations.

Example: Simple Pendulum

For a simple pendulum of length l and mass m, the generalised coordinate is the angle θ. The kinetic energy is T = 1/2 m (lθ̇)² = 1/2 ml²θ̇². The potential energy (taking the lowest point as V=0) is V = mgl(1 - cosθ). The Lagrangian is L = T - V = 1/2 ml²θ̇² - mgl(1 - cosθ).

Now we apply Lagrange's equation for θ: ∂L/∂θ̇ = ml²θ̇ d/dt (∂L/∂θ̇) = ml²θ̈ ∂L/∂θ = -mgl sinθ

Lagrange's equation becomes: ml²θ̈ - (-mgl sinθ) = 0 ml²θ̈ + mgl sinθ = 0 θ̈ + (g/l) sinθ = 0 This is the well-known equation of motion for a simple pendulum.

Hamiltonian Equations of Motion

Hamiltonian mechanics is another reformulation of classical mechanics, developed by William Rowan Hamilton. It uses a different set of variables and equations compared to Lagrangian mechanics. Instead of generalised coordinates (q_j) and generalised velocities (q̇_j), it uses generalised coordinates (q_j) and generalised momenta (p_j). The central function is the Hamiltonian, H.

The generalised momentum p_j conjugate to the coordinate q_j is defined as: p_j = ∂L/∂q̇_j

The Hamiltonian H is obtained from the Lagrangian L via a Legendre transformation: H(q_j, p_j, t) = Σ_j p_j q̇_j - L(q_j, q̇_j, t) Crucially, in the expression for H, the velocities q̇_j must be expressed in terms of coordinates q_j and momenta p_j. This is possible if the kinetic energy is a quadratic function of velocities and there are no explicit time dependencies in the kinetic energy.

If the forces are conservative and the coordinate transformation does not explicitly depend on time, the Hamiltonian H is equal to the total energy of the system (T + V).

Hamilton's equations of motion are a set of 2n first-order differential equations: q̇_j = ∂H/∂p_j ṗ_j = -∂H/∂q_j for j = 1, 2, ..., n.

These equations describe how the generalised coordinates and their conjugate momenta evolve over time. They form the foundation for much of advanced theoretical physics, including quantum mechanics and statistical mechanics.

Example: Simple Pendulum (Revisited)

From the previous example, L = 1/2 ml²θ̇² - mgl(1 - cosθ). The conjugate momentum to θ is: p_θ = ∂L/∂θ̇ = ml²θ̇ So, θ̇ = p_θ / (ml²).

The Hamiltonian is: H = p_θ θ̇ - L H = p_θ (p_θ / (ml²)) - [1/2 ml²(p_θ / (ml²))² - mgl(1 - cosθ)] H = p_θ² / (ml²) - [1/2 p_θ² / (ml²) - mgl(1 - cosθ)] H = 1/2 p_θ² / (ml²) + mgl(1 - cosθ) Note that H = T + V, as expected for a conservative system with time-independent kinetic energy.

Now, Hamilton's equations for θ and p_θ: θ̇ = ∂H/∂p_θ = 1/2 (2 p_θ / (ml²)) = p_θ / (ml²) This recovers the definition of p_θ. ṗ_θ = -∂H/∂θ = -∂/∂θ [1/2 p_θ² / (ml²) + mgl(1 - cosθ)] ṗ_θ = -mgl sinθ

Combining θ̇ = p_θ / (ml²) and ṗ_θ = -mgl sinθ, we can differentiate the first equation with respect to time: θ̈ = ṗ_θ / (ml²) Substitute ṗ_θ: θ̈ = (-mgl sinθ) / (ml²) = -(g/l) sinθ θ̈ + (g/l) sinθ = 0 This yields the same equation of motion as derived from Lagrange's equations.

Conservative and Non-Conservative Systems

In mechanics, systems are classified based on the nature of the forces acting upon them. This classification is important because it affects the applicability of certain energy conservation principles and the form of the equations of motion.

Conservative Systems

A system is called conservative if the work done by the net force acting on it is independent of the path taken and depends only on the initial and final positions. Equivalently, a system is conservative if the work done by the forces around any closed path is zero.

For a conservative system, the forces can be derived from a scalar potential energy function V. The force F_i acting on the i-th particle is related to the potential energy by: F_i = -∇_i V where ∇_i is the gradient operator with respect to the coordinates of the i-th particle.

In terms of generalised coordinates, if the potential energy V depends only on the generalised coordinates (and not explicitly on time), then the generalised forces Q_j related to these conservative forces are given by: Q_j = -∂V/∂q_j

A key consequence of a conservative system is the conservation of mechanical energy. The total mechanical energy E = T + V remains constant over time if only conservative forces are doing work.

In Lagrangian mechanics, if the potential energy V does not depend explicitly on time, the Lagrangian L = T - V is conserved if T is a quadratic function of velocities and V is independent of velocities. If L is conserved, then dL/dt = 0.

In Hamiltonian mechanics, if the Hamiltonian H = T + V does not explicitly depend on time, then H is conserved (dH/dt = 0). This means the total energy is conserved.

Non-Conservative Systems

A system is non-conservative if the work done by the net force depends on the path taken, or if the work done around a closed path is non-zero. Examples of non-conservative forces include friction, air resistance, and applied driving forces that are not derivable from a potential energy function.

For non-conservative forces, we cannot define a potential energy function V in the same way. The work done by these forces must be accounted for separately.

In Lagrangian mechanics, non-conservative forces contribute generalised forces Q_j^nc to the equations of motion: d/dt (∂L/∂q̇_j) - ∂L/∂q_j = Q_j^nc where Q_j^nc represents the generalised force due to non-conservative effects.

In Hamiltonian mechanics, the Hamiltonian H may not be conserved because the system's energy can change due to non-conservative forces. If there are dissipative forces like friction, energy is typically lost from the mechanical system (converted to heat, etc.).

Example: A block sliding down an inclined plane with friction. The gravitational force is conservative. However, the frictional force is non-conservative. The work done by friction depends on the distance slid, and it dissipates mechanical energy. The total mechanical energy of the block-Earth system will decrease.

Cyclic Variables

Cyclic variables, also known as ignorable coordinates, are a crucial concept in Lagrangian and Hamiltonian mechanics. A generalised coordinate q_j is called cyclic if the Lagrangian L does not depend explicitly on it.

Mathematically, a coordinate q_j is cyclic if: ∂L/∂q_j = 0

When a coordinate q_j is cyclic, Lagrange's equation of motion simplifies significantly. Recall Lagrange's equation: d/dt (∂L/∂q̇_j) - ∂L/∂q_j = 0 If ∂L/∂q_j = 0, the equation becomes: d/dt (∂L/∂q̇_j) = 0

This implies that the quantity ∂L/∂q̇_j is a constant of motion. This quantity is precisely the generalised momentum p_j conjugate to the coordinate q_j: p_j = ∂L/∂q̇_j = constant

Therefore, for a cyclic coordinate, its conjugate momentum is conserved. This is a powerful result, as it simplifies the analysis of systems where certain coordinates do not directly influence the dynamics.

Significance and Examples

1. Conservation Laws: The conservation of conjugate momenta associated with cyclic coordinates is a direct consequence of Noether's theorem, which links symmetries in a physical system to conserved quantities. The symmetry associated with a cyclic coordinate is translation invariance in that coordinate direction (or rotation invariance if it's an angle). 2. Simplification: Identifying cyclic variables can greatly simplify the problem. For instance, if a system has multiple degrees of freedom and several are cyclic, the equations of motion for those degrees of freedom reduce to simple statements of momentum conservation, reducing the number of differential equations that need to be solved.

Example 1: Simple Pendulum The Lagrangian was L = 1/2 ml²θ̇² - mgl(1 - cosθ). The coordinate θ is not cyclic because ∂L/∂θ = -mgl sinθ ≠ 0. If we consider a pendulum whose point of suspension moves horizontally with velocity v₀, the Lagrangian might involve both θ and the horizontal position x of the suspension. If the potential energy does not depend on x (e.g., no external forces in the x-direction other than constraints), then x would be a cyclic coordinate. Its conjugate momentum, p_x = ∂L/∂ẋ, would be conserved.

Example 2: Particle in a Central Potential Consider a particle moving in a central potential V(r). The Lagrangian in spherical coordinates (r, θ, φ) is: L = 1/2 m (ṙ² + r²θ̇² + r²sin²θ φ̇²) - V(r) Here, the coordinate φ (the azimuthal angle) does not appear in L. Thus, ∂L/∂φ = 0. φ is a cyclic coordinate. Its conjugate momentum is: p_φ = ∂L/∂φ̇ = mr²sin²θ φ̇ Since φ is cyclic, p_φ is conserved. This conserved quantity is the angular momentum component along the z-axis. This directly leads to the conservation of angular momentum in central force motion.

In Hamiltonian mechanics, a coordinate q_j is cyclic if H does not depend explicitly on q_j (∂H/∂q_j = 0). In this case, Hamilton's equation ṗ_j = -∂H/∂q_j implies ṗ_j = 0, meaning p_j is conserved.

Principle of Least Action

The Principle of Least Action is a fundamental variational principle in physics that provides an alternative and elegant way to formulate the laws of mechanics. It states that the actual path taken by a mechanical system between two points in time is the one for which the action is stationary (usually a minimum).

The action, denoted by S, is defined as the time integral of the Lagrangian L of the system between the initial time t₁ and the final time t₂: S = ∫_t1^t₂ L(q_j(t), q̇_j(t), t) dt

The principle states that for the true path of motion, the variation of the action δS is zero, provided the endpoints (positions at t₁ and t₂) are held fixed: δS = δ ∫_t1^t₂ L dt = 0

Applying the calculus of variations to this integral leads directly to Lagrange's equations of motion. Let's see how: Consider a variation δq_j(t) in the path q_j(t). The corresponding variation in the action is: δS = ∫_t1^t₂ δL dt δL = Σ_j (∂L/∂q_j δq_j + ∂L/∂q̇_j δq̇_j) So, δS = ∫_t1^t₂ Σ_j (∂L/∂q_j δq_j + ∂L/∂q̇_j δq̇_j) dt

We use integration by parts on the second term: ∫_t1^t₂ (∂L/∂q̇_j) δq̇_j dt = ∫_t1^t₂ (∂L/∂q̇_j) (d/dt δq_j) dt = [(∂L/∂q̇_j) δq_j]_t1^t₂ - ∫_t1^t₂ (d/dt ∂L/∂q̇_j) δq_j dt

Since the endpoints are fixed, δq_j(t₁) = δq_j(t₂) = 0. Thus, the boundary term vanishes. Substituting back into the expression for δS: δS = ∫_t1^t₂ Σ_j (∂L/∂q_j δq_j - ∂L/∂q̇_j δq_j) dt δS = ∫_t1^t₂ Σ_j (∂L/∂q_j - d/dt ∂L/∂q̇_j) δq_j dt

For δS to be zero for arbitrary variations δq_j, the term in the parenthesis must be zero for each j: ∂L/∂q_j - d/dt (∂L/∂q̇_j) = 0 d/dt (∂L/∂q̇_j) - ∂L/∂q_j = 0 This is precisely Lagrange's equation of motion.

Significance

1. **Universality:** The Principle of Least Action is remarkably universal. It applies not only to classical mechanics but also to electromagnetism, relativity, and quantum field theory. It provides a unifying framework for diverse physical theories. 2. **Symmetries and Conservation Laws:** As mentioned with cyclic variables, the principle is intrinsically linked to symmetries and conservation laws via Noether's theorem. 3. **Foundation for Advanced Theories:** It forms the bedrock for developing more advanced theoretical frameworks.

While often called the "Principle of Least Action," it's more accurately the "Principle of Stationary Action" because the action can sometimes be a maximum or saddle point, though minimum is most common in basic mechanics.