Question

Prove that a particle constrained to stay on a surface \(f(x, y, z)=0,\) but subject to no other forces, moves along a geodesic of the surface. Hint: The potential energy \(V\) is constant, since constraint forces are normal to the surface and so do no work on the particle. Use Hamilton's principle and show that the problem of finding a geodesic and the problem of finding the path of the particle are identical mathematics problems.

Short answer

The particle constrained to the surface moves along a geodesic as proven by the equivalence in minimization conditions from Hamilton's principle.

Step-by-step solution

- Understanding the Problem

A particle is constrained to stay on a surface defined by the equation \( f(x, y, z) = 0 \). There are no other forces acting on the particle. The goal is to show that its path is a geodesic of the surface.

- Using Hamilton's Principle

Hamilton's principle states that the path of a particle is such that the action integral \( S \) is minimized, where \[ S = \int L \, dt \]with \( L = T - V \) being the Lagrangian, \( T \) the kinetic energy, and \( V \) the potential energy. Here, given that the constraint forces do no work, \( V \) is constant.

- Expressing Kinetic Energy

The kinetic energy \( T \) of the particle can be written as: \[ T = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2 + \dot{z}^2) \] where \( m \) is the mass and \( \dot{x}, \dot{y}, \dot{z} \) are the time derivatives of the coordinates.

- Lagrangian Formulation

Since \( V \) is constant and can be ignored in the minimization of \( S \), the Lagrangian \( L \) simply equals the kinetic energy: \[ L = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2 + \dot{z}^2) \]

- Impact of the Constraint

The constraint \( f(x, y, z) = 0 \) restricts the motion of the particle to the surface. This constraint must be considered to eliminate one degree of freedom.

- Reducing Dimensions via Constraint

Use the constraint to reduce the dimensionality of the problem by expressing one coordinate in terms of the others (e.g., solve \( f(x, y, z) = 0 \) for \( z \) in terms of \( x \) and \( y \)). The kinetic energy and the Lagrangian now involve only the independent coordinates.

- Geodesic Problem Formulation

The problem is now to find the path (\( x(t), y(t) \)) which minimizes the action integral:\[ S = \int \frac{1}{2} m (\dot{x}^2 + \dot{y}^2 + (\dot{z})^2) \, dt \]where \( \dot{z} \) is derived from the constraint equation.

- Geodesics in Classical Mechanics

In classical mechanics, the minimization of the action integral for this specific scenario leads to the same equations as those derived from the geodesic equation on the surface defined by \( f(x, y, z) = 0 \). This equivalence shows that the particle must move along a geodesic of the surface.

Conclusion

Both the geodesic problem and the particle constraint problem reduce to finding the same set of minimization conditions, proving they are mathematically identical. Therefore, the particle moves along a geodesic of the surface.

Key concepts

Hamilton's Principle

Hamilton's principle is a cornerstone of classical mechanics that states a particle's path minimizes the action integral. The action integral, denoted as \( S \), is the integral of the Lagrangian \( L \) over time. Mathematically, it is expressed as: \[ S = \int L \ dt \] \ The Lagrangian \( L \) is a function that combines kinetic and potential energy. For this problem, since the forces are constrained and the potential energy is constant, it simplifies to only the kinetic energy term. With this setup, Hamilton's principle helps us determine the trajectory of the particle.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. It is calculated based on the mass \( m \) and velocity of the object. In this context, the kinetic energy \( T \) of the particle can be described as: \[ T = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2 + \dot{z}^2) \] \ This formula sums up the energy contributions from the particle's velocities in the x, y, and z directions. In problems involving constraints, the kinetic energy often plays a crucial role, as it helps form the Lagrangian, crucial for deriving the particle's equations of motion.

Lagrangian Mechanics

Lagrangian mechanics is another fundamental framework in classical mechanics. It uses the Lagrangian \( L \), defined as \( L = T - V \) (kinetic energy \( T \) minus potential energy \( V \)). Here, the potential energy \( V \) is constant due to the constraints, which results in: \[ L = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2 + \dot{z}^2) \] \ This Lagrangian simplifies our calculations, as we only focus on minimizing kinetic energy. In constrained motion, we often use the Lagrangian to find the equations that describe the path or trajectory of the particle.

Constraint Forces

Constraint forces are forces that restrict a particle's motion to a specific path or surface. In this problem, the particle is constrained to move on the surface \( f(x, y, z) = 0 \). These forces are always perpendicular to the surface, meaning they do no work on the particle—this ensures the potential energy remains constant. Constraint forces simplify the problem by reducing the degrees of freedom, making it easier to describe the motion using fewer variables. By applying these constraints, we reduce the system's complexity while still accurately describing the particle's path.

Classical Mechanics

Classical mechanics is the branch of physics that describes the motion of particles using principles like Newton's laws, the Lagrangian, and Hamiltonian mechanics. In this context, we're interested in how these principles apply to a particle constrained by a surface. By minimizing the action integral using Hamilton's principle and considering only the kinetic energy via the Lagrangian, we see that the equations derived are mathematically identical to the geodesic equations. This confirms that the particle follows a geodesic path on the surface. Such problems showcase the power and versatility of classical mechanics in addressing a wide range of physical phenomena.