Classical Mechanics - Lecture Notes Review

Vocabulary flashcards covering the core concepts of Newtonian mechanics recapitulation and the Lagrangian formulation as presented in the SRMIST Classical Mechanics lecture notes.

Inertial Frame

A reference frame in which space is homogeneous and isotropic and time is homogeneous, allowing Newton's second law to take the form $F = \frac{dp}{dt}$.

Newton's Second Law

In an inertial frame, the force is defined as the rate of change of linear momentum, expressed as $F = \frac{dp}{dt}$, where $p = mv$.

Strong form of the Third Law

The requirement that action and reaction forces are not only equal and opposite ($F_{ij} = -F_{ji}$) but also act along the line joining the two particles.

Noether's Theorem

A theorem stating that every continuous symmetry of the Lagrangian corresponds to a conserved quantity, known as a Noether charge.

Conservative Force

A force whose work done around any closed path vanishes (\text{\oint} F \cdot ds = 0), implying the force can be expressed as the negative gradient of a potential, $F = -\nabla V(r)$.

Reduced Mass ($\mu$)

An effective inertial mass used in the two-body problem, defined as $\mu = \frac{m_1 m_2}{m_1 + m_2}$.

Degrees of Freedom

The minimum number of independent quantities required to specify the configuration of a system.

Generalized Coordinates

Any set of independent quantities $\{q_1, ..., q_n\}$ that specify the configuration of a system, regardless of their dimensions or orthogonality.

Holonomic Constraint

An algebraic relation among coordinates and time of the form $f(q_1, ..., q_n, t) = 0$ that can be used to eliminate a coordinate.

Non-holonomic Constraint

A constraint that cannot be expressed as an algebraic relation among coordinates, typically represented as a non-integrable relation among velocities.

Scleronomic Constraint

A constraint that does not contain explicit time dependence, such as a bead on a fixed wire.

Rheonomic Constraint

A constraint that depends explicitly on time, such as a bead on a wire that is being spun or a driven pendulum support.

Mass Matrix ($a_{jk}$)

The configuration-dependent coefficients in the kinetic energy expression $T = \frac{1}{2} \sum_{j,k} a_{jk}(q, t) \dot{q}_j \dot{q}_k$.

Virtual Displacement ($\delta r_i$)

An infinitesimal change of configuration consistent with constraints at a 'frozen' instant of time, where $\delta t = 0$.

D'Alembert's Principle

The principle that the sum of applied forces minus the rate of change of momentum, dotted with virtual displacements, vanishes: $\sum_i (F_i^{(a)} - \dot{p}_i) \cdot \delta r_i = 0$.

Lagrangian ($L$)

A scalar function defined as the difference between kinetic energy and potential energy, $L = T - V$.

Euler-Lagrange Equations

The differential equations of motion for a system, given by $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_j}\right) - \frac{\partial L}{\partial q_j} = 0$.

Canonical Momentum ($p_j$)

The generalized or conjugate momentum belonging to coordinate $q_j$, defined as $p_j = \frac{\partial L}{\partial \dot{q}_j}$.

Cyclic (Ignorable) Coordinate

A generalized coordinate that does not appear explicitly in the Lagrangian ($\frac{\partial L}{\partial q_k} = 0$), leading to the conservation of its conjugate momentum.

Jacobi Integral ($h$)

The energy function defined as $h = \sum_j p_j \dot{q}_j - L$, which is conserved if the Lagrangian has no explicit time dependence ($\frac{\partial L}{\partial t} = 0$).

Natural System

A system where constraints are scleronomic and the potential is velocity-independent, resulting in the Jacobi integral equaling the total mechanical energy ($h = T + V = E$).

Action ($S$)

The time integral of the Lagrangian along a path, defined as $S = \int_{t_1}^{t_2} L(q, \dot{q}, t) dt$.

Hamilton's Principle of Least Action

The principle stating that the actual physical path taken by a system is the one for which the action $S$ is stationary ($\delta S = 0$).

Effective Potential ($V_{\text{eff}}$)

In the central force problem, the sum of the actual potential and the centrifugal barrier term, given by $V_{\text{eff}}(r) = V(r) + \frac{l^2}{2\mu r^2}$.

Bertrand's Theorem

The theorem stating that only two types of central force laws produce closed orbits for all bound motion: the inverse-square law ($F \propto r^{-2}$) and the linear spring law ($F \propto r$).