Classical Mechanics - Lagrangian & Hamiltonian Formalisms Flashcards

Question 1

Generalized Coordinates

Accepted Answer

A set of $n$ independent parameters $q_1, q_2, \dots, q_n$ that completely specify the state or spatial configuration of a system with $n$ degrees of freedom without requiring constraint forces.

Question 2

Holonomic vs. Non-Holonomic Constraints

Accepted Answer

**Holonomic constraints** can be expressed as algebraic equations involving coordinates and time: $f(q_1, \dots, q_n, t) = 0$.

**Non-holonomic constraints** cannot be written in algebraic coordinate form, often involving inequalities or non-integrable differential velocity constraints (e.g., rolling without slipping).

Question 3

Principle of Virtual Work

Accepted Answer

States that a mechanical system is in static equilibrium if and only if the total virtual work done by applied forces $\mathbf{F}_i^{(a)}$ during any arbitrary virtual displacement $\delta \mathbf{r}_i$ consistent with constraints is zero: $$\sum_i \mathbf{F}_i^{(a)} \cdot \delta \mathbf{r}_i = 0$$

Question 4

D'Alembert's Principle

Accepted Answer

Extends virtual work to dynamic systems by including inertial forces $-m_i \ddot{\mathbf{r}}_i$. The dynamic equilibrium condition is: $$\sum_i \left(\mathbf{F}_i^{(a)} - m_i \ddot{\mathbf{r}}_i\right) \cdot \delta \mathbf{r}_i = 0$$

Question 5

Hamilton's Principle (Principle of Least Action)

Accepted Answer

States that the actual trajectory of a system between times $t_1$ and $t_2$ renders the action integral $S$ stationary ($
\delta S = 0$): $$S = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt$$

Question 6

The Lagrangian ($L$)

Accepted Answer

A scalar function characterising a dynamic system, defined as kinetic energy minus potential energy: $$L(q, \dot{q}, t) = T - V$$ formulated in terms of generalized coordinates $q_i$ and generalized velocities $\dot{q}_i$.

Question 7

Euler-Lagrange Equations

Accepted Answer

The fundamental second-order differential equations of motion derived from Hamilton's Principle: $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0$$

Question 8

Generalized (Conjugate) Momentum

Accepted Answer

The canonical momentum $p_i$ conjugate to the generalized coordinate $q_i$, defined as: $$p_i = \frac{\partial L}{\partial \dot{q}_i}$$

Question 9

Cyclic (Ignorable) Coordinates

Accepted Answer

A coordinate $q_k$ that does not explicitly appear in the Lagrangian ($L$). By the Euler-Lagrange equations, its conjugate momentum $p_k$ is a conserved quantity (constant of motion): $$\frac{dp_k}{dt} = \frac{\partial L}{\partial q_k} = 0$$

Question 10

Noether's Theorem

Accepted Answer

States that every continuous differentiable symmetry of the action of a system corresponds to a conserved quantity. For example:
- Time translation symmetry $\implies$ Energy conservation
- Spatial translation symmetry $\implies$ Linear momentum conservation
- Rotational symmetry $\implies$ Angular momentum conservation

Question 11

Configuration Space vs. Phase Space

Accepted Answer

**Configuration Space**: $n$-dimensional space spanned by $n$ generalized coordinates $(q_1, \dots, q_n)$.

**Phase Space**: $2n$-dimensional space spanned by both $n$ coordinates and $n$ conjugate momenta $(q_1, \dots, q_n, p_1, \dots, p_n)$.

Question 12

Legendre Transformation

Accepted Answer

A mathematical transformation that switches independent variables from velocities $\dot{q}$ to conjugate momenta $p$. It converts the Lagrangian $L(q, \dot{q}, t)$ into the Hamiltonian $H(q, p, t)$.

Question 13

The Hamiltonian ($H$)

Accepted Answer

The state function defined via the Legendre transformation of $L$: $$H(q, p, t) = \sum_i p_i \dot{q}_i - L(q, \dot{q}, t)$$ If constraints are scleronomic (time-independent) and $V$ is velocity-independent, $H$ equals total mechanical energy $E = T + V$.

Question 14

Hamilton's Canonical Equations

Accepted Answer

A set of $2n$ first-order differential equations governing system trajectory in phase space: $$\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}$$

Question 15

Liouville's Theorem

Accepted Answer

States that the volume element $d\Omega$ of a region in phase space is conserved along the system trajectory under Hamiltonian dynamics ($
\frac{d\rho}{dt} = 0$).

Question 16

Poisson Bracket

Accepted Answer

An operation on phase-space functions $f(q, p)$ and $g(q, p)$ defined as: $$\{f, g\} = \sum_i \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right)$$ Serves as the classical precursor to the quantum commutator.

Question 17

Time Evolution via Poisson Brackets

Accepted Answer

The total rate of change of any dynamic variable $A(q, p, t)$ is given by: $$\frac{dA}{dt} = \{A, H\} + \frac{\partial A}{\partial t}$$ $A$ is a constant of motion if $\partial A / \partial t = 0$ and $\{A, H\} = 0$.

Question 18

Canonical Transformations

Accepted Answer

Phase space coordinate transformations $(q, p) 	o (Q, P)$ that preserve the canonical form of Hamilton's equations and fundamental Poisson brackets: $$\{Q_i, Q_j\} = 0, \quad \{P_i, P_j\} = 0, \quad \{Q_i, P_j\} = \delta_{ij}$$

Question 19

Scleronomic vs. Rheonomic Constraints

Accepted Answer

**Scleronomic**: Constraints do not explicitly depend on time ($f(q) = 0$).

**Rheonomic**: Constraints depend explicitly on time ($f(q, t) = 0$).

Question 20

Simple Harmonic Oscillator (Lagrangian Approach)

Accepted Answer

Given $L = T - V = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2$, the Euler-Lagrange equation yields: $$\frac{d}{dt}(m\dot{x}) - (-kx) = 0 \implies m\ddot{x} + kx = 0$$

Question 21

Simple Harmonic Oscillator (Hamiltonian Approach)

Accepted Answer

With $p = m\dot{x}$, $H = \frac{p^2}{2m} + \frac{1}{2}kx^2$. Hamilton's canonical equations give: $$\dot{x} = \frac{\partial H}{\partial p} = \frac{p}{m}, \quad \dot{p} = -\frac{\partial H}{\partial x} = -kx$$ Combining yields $m\ddot{x} + kx = 0$.

Classical Mechanics - Lagrangian & Hamiltonian Formalisms

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