• Images
• Videos
• Documents
• Books
• Articles

Notations

Let M {\displaystyle M} be the configuration space of the mechanical system, t 0 , t 1 ∈ R {\displaystyle t_{0},t_{1}\in \mathbb {R} } be time instants, q 0 , q 1 ∈ M , {\displaystyle q_{0},q_{1}\in M,} C ∞ [ t 0 , t 1 ] {\displaystyle C^{\infty }[t_{0},t_{1}]} consists of smooth functions on [ t 0 , t 1 ] {\displaystyle [t_{0},t_{1}]} , and

P ( M ) = { γ ∈ C ∞ ( [ t 0 , t 1 ] , M ) ∣ γ ( t 0 ) = q 0 ,   γ ( t 1 ) = q 1 } . {\displaystyle P(M)=\{\gamma \in C^{\infty }([t_{0},t_{1}],M)\mid \gamma (t_{0})=q_{0},\ \gamma (t_{1})=q_{1}\}.}

The constraints γ ( t 0 ) = q 0 , {\displaystyle \gamma (t_{0})=q_{0},} γ ( t 1 ) = q 1 {\displaystyle \gamma (t_{1})=q_{1}} are here for illustration only. In practice, for each individual system, an individual set of constraints is required.

Definition

For each path γ ∈ P ( M ) {\displaystyle \gamma \in P(M)} and ϵ 0 > 0 , {\displaystyle \epsilon _{0}>0,} a variation of γ {\displaystyle \gamma } is a function Γ : [ t 0 , t 1 ] × [ − ϵ 0 , ϵ 0 ] → M {\displaystyle \Gamma :[t_{0},t_{1}]\times [-\epsilon _{0},\epsilon _{0}]\to M} such that, for every ϵ ∈ [ − ϵ 0 , ϵ 0 ] , {\displaystyle \epsilon \in [-\epsilon _{0},\epsilon _{0}],} Γ ( ⋅ , ϵ ) ∈ P ( M ) {\displaystyle \Gamma (\cdot ,\epsilon )\in P(M)} and Γ ( t , 0 ) = γ ( t ) . {\displaystyle \Gamma (t,0)=\gamma (t).} The virtual displacement δ γ : [ t 0 , t 1 ] → T M {\displaystyle \delta \gamma :[t_{0},t_{1}]\to TM} ( T M {\displaystyle (TM} being the tangent bundle of M ) {\displaystyle M)} corresponding to the variation Γ {\displaystyle \Gamma } assigns⁴ to every t ∈ [ t 0 , t 1 ] {\displaystyle t\in [t_{0},t_{1}]} the tangent vector

δ γ ( t ) = d Γ ( t , ϵ ) d ϵ | ϵ = 0 ∈ T γ ( t ) M . {\displaystyle \delta \gamma (t)=\left.{\frac {d\Gamma (t,\epsilon )}{d\epsilon }}\right|_{\epsilon =0}\in T_{\gamma (t)}M.}

In terms of the tangent map,

δ γ ( t ) = Γ ∗ t ( d d ϵ | ϵ = 0 ) . {\displaystyle \delta \gamma (t)=\Gamma _{*}^{t}\left(\left.{\frac {d}{d\epsilon }}\right|_{\epsilon =0}\right).}

Here Γ ∗ t : T 0 [ − ϵ , ϵ ] → T Γ ( t , 0 ) M = T γ ( t ) M {\displaystyle \Gamma _{*}^{t}:T_{0}[-\epsilon ,\epsilon ]\to T_{\Gamma (t,0)}M=T_{\gamma (t)}M} is the tangent map of Γ t : [ − ϵ , ϵ ] → M , {\displaystyle \Gamma ^{t}:[-\epsilon ,\epsilon ]\to M,} where Γ t ( ϵ ) = Γ ( t , ϵ ) , {\displaystyle \Gamma ^{t}(\epsilon )=\Gamma (t,\epsilon ),} and d d ϵ | ϵ = 0 ∈ T 0 [ − ϵ , ϵ ] . {\displaystyle \textstyle {\frac {d}{d\epsilon }}{\Bigl |}_{\epsilon =0}\in T_{0}[-\epsilon ,\epsilon ].}

Properties

• Coordinate representation. If { q i } i = 1 n {\displaystyle \{q_{i}\}_{i=1}^{n}} are the coordinates in an arbitrary chart on M {\displaystyle M} and n = dim ⁡ M , {\displaystyle n=\dim M,} then δ γ ( t ) = ∑ i = 1 n d [ q i ( Γ ( t , ϵ ) ) ] d ϵ | ϵ = 0 ⋅ d d q i | γ ( t ) . {\displaystyle \delta \gamma (t)=\sum _{i=1}^{n}{\frac {d[q_{i}(\Gamma (t,\epsilon ))]}{d\epsilon }}{\Biggl |}_{\epsilon =0}\cdot {\frac {d}{dq_{i}}}{\Biggl |}_{\gamma (t)}.}
• If, for some time instant τ {\displaystyle \tau } and every γ ∈ P ( M ) , {\displaystyle \gamma \in P(M),} γ ( τ ) = const , {\displaystyle \gamma (\tau )={\text{const}},} then, for every γ ∈ P ( M ) , {\displaystyle \gamma \in P(M),} δ γ ( τ ) = 0. {\displaystyle \delta \gamma (\tau )=0.}
• If γ , d γ d t ∈ P ( M ) , {\displaystyle \textstyle \gamma ,{\frac {d\gamma }{dt}}\in P(M),} then δ d γ d t = d d t δ γ . {\displaystyle \delta {\frac {d\gamma }{dt}}={\frac {d}{dt}}\delta \gamma .}

Examples

Free particle in R3

A single particle freely moving in R 3 {\displaystyle \mathbb {R} ^{3}} has 3 degrees of freedom. The configuration space is M = R 3 , {\displaystyle M=\mathbb {R} ^{3},} and P ( M ) = C ∞ ( [ t 0 , t 1 ] , M ) . {\displaystyle P(M)=C^{\infty }([t_{0},t_{1}],M).} For every path γ ∈ P ( M ) {\displaystyle \gamma \in P(M)} and a variation Γ ( t , ϵ ) {\displaystyle \Gamma (t,\epsilon )} of γ , {\displaystyle \gamma ,} there exists a unique σ ∈ T 0 R 3 {\displaystyle \sigma \in T_{0}\mathbb {R} ^{3}} such that Γ ( t , ϵ ) = γ ( t ) + σ ( t ) ϵ + o ( ϵ ) , {\displaystyle \Gamma (t,\epsilon )=\gamma (t)+\sigma (t)\epsilon +o(\epsilon ),} as ϵ → 0. {\displaystyle \epsilon \to 0.} By the definition,

δ γ ( t ) = ( d d ϵ ( γ ( t ) + σ ( t ) ϵ + o ( ϵ ) ) ) | ϵ = 0 {\displaystyle \delta \gamma (t)=\left.\left({\frac {d}{d\epsilon }}{\Bigl (}\gamma (t)+\sigma (t)\epsilon +o(\epsilon ){\Bigr )}\right)\right|_{\epsilon =0}}

which leads to

δ γ ( t ) = σ ( t ) ∈ T γ ( t ) R 3 . {\displaystyle \delta \gamma (t)=\sigma (t)\in T_{\gamma (t)}\mathbb {R} ^{3}.}

Free particles on a surface

N {\displaystyle N} particles moving freely on a two-dimensional surface S ⊂ R 3 {\displaystyle S\subset \mathbb {R} ^{3}} have 2 N {\displaystyle 2N} degree of freedom. The configuration space here is

M = { ( r 1 , … , r N ) ∈ R 3 N ∣ r i ∈ R 3 ;   r i ≠ r j   if   i ≠ j } , {\displaystyle M=\{(\mathbf {r} _{1},\ldots ,\mathbf {r} _{N})\in \mathbb {R} ^{3\,N}\mid \mathbf {r} _{i}\in \mathbb {R} ^{3};\ \mathbf {r} _{i}\neq \mathbf {r} _{j}\ {\text{if}}\ i\neq j\},}

where r i ∈ R 3 {\displaystyle \mathbf {r} _{i}\in \mathbb {R} ^{3}} is the radius vector of the i th {\displaystyle i^{\text{th}}} particle. It follows that

T ( r 1 , … , r N ) M = T r 1 S ⊕ … ⊕ T r N S , {\displaystyle T_{(\mathbf {r} _{1},\ldots ,\mathbf {r} _{N})}M=T_{\mathbf {r} _{1}}S\oplus \ldots \oplus T_{\mathbf {r} _{N}}S,}

and every path γ ∈ P ( M ) {\displaystyle \gamma \in P(M)} may be described using the radius vectors r i {\displaystyle \mathbf {r} _{i}} of each individual particle, i.e.

γ ( t ) = ( r 1 ( t ) , … , r N ( t ) ) . {\displaystyle \gamma (t)=(\mathbf {r} _{1}(t),\ldots ,\mathbf {r} _{N}(t)).}

This implies that, for every δ γ ( t ) ∈ T ( r 1 ( t ) , … , r N ( t ) ) M , {\displaystyle \delta \gamma (t)\in T_{(\mathbf {r} _{1}(t),\ldots ,\mathbf {r} _{N}(t))}M,}

δ γ ( t ) = δ r 1 ( t ) ⊕ … ⊕ δ r N ( t ) , {\displaystyle \delta \gamma (t)=\delta \mathbf {r} _{1}(t)\oplus \ldots \oplus \delta \mathbf {r} _{N}(t),}

where δ r i ( t ) ∈ T r i ( t ) S . {\displaystyle \delta \mathbf {r} _{i}(t)\in T_{\mathbf {r} _{i}(t)}S.} Some authors express this as

δ γ = ( δ r 1 , … , δ r N ) . {\displaystyle \delta \gamma =(\delta \mathbf {r} _{1},\ldots ,\delta \mathbf {r} _{N}).}

Rigid body rotating around fixed point

A rigid body rotating around a fixed point with no additional constraints has 3 degrees of freedom. The configuration space here is M = S O ( 3 ) , {\displaystyle M=SO(3),} the special orthogonal group of dimension 3 (otherwise known as 3D rotation group), and P ( M ) = C ∞ ( [ t 0 , t 1 ] , M ) . {\displaystyle P(M)=C^{\infty }([t_{0},t_{1}],M).} We use the standard notation s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices. The exponential map exp : s o ( 3 ) → S O ( 3 ) {\displaystyle \exp :{\mathfrak {so}}(3)\to SO(3)} guarantees the existence of ϵ 0 > 0 {\displaystyle \epsilon _{0}>0} such that, for every path γ ∈ P ( M ) , {\displaystyle \gamma \in P(M),} its variation Γ ( t , ϵ ) , {\displaystyle \Gamma (t,\epsilon ),} and t ∈ [ t 0 , t 1 ] , {\displaystyle t\in [t_{0},t_{1}],} there is a unique path Θ t ∈ C ∞ ( [ − ϵ 0 , ϵ 0 ] , s o ( 3 ) ) {\displaystyle \Theta ^{t}\in C^{\infty }([-\epsilon _{0},\epsilon _{0}],{\mathfrak {so}}(3))} such that Θ t ( 0 ) = 0 {\displaystyle \Theta ^{t}(0)=0} and, for every ϵ ∈ [ − ϵ 0 , ϵ 0 ] , {\displaystyle \epsilon \in [-\epsilon _{0},\epsilon _{0}],} Γ ( t , ϵ ) = γ ( t ) exp ⁡ ( Θ t ( ϵ ) ) . {\displaystyle \Gamma (t,\epsilon )=\gamma (t)\exp(\Theta ^{t}(\epsilon )).} By the definition,

δ γ ( t ) = ( d d ϵ ( γ ( t ) exp ⁡ ( Θ t ( ϵ ) ) ) ) | ϵ = 0 = γ ( t ) d Θ t ( ϵ ) d ϵ | ϵ = 0 . {\displaystyle \delta \gamma (t)=\left.\left({\frac {d}{d\epsilon }}{\Bigl (}\gamma (t)\exp(\Theta ^{t}(\epsilon )){\Bigr )}\right)\right|_{\epsilon =0}=\gamma (t)\left.{\frac {d\Theta ^{t}(\epsilon )}{d\epsilon }}\right|_{\epsilon =0}.}

Since, for some function σ : [ t 0 , t 1 ] → s o ( 3 ) , {\displaystyle \sigma :[t_{0},t_{1}]\to {\mathfrak {so}}(3),} Θ t ( ϵ ) = ϵ σ ( t ) + o ( ϵ ) {\displaystyle \Theta ^{t}(\epsilon )=\epsilon \sigma (t)+o(\epsilon )} , as ϵ → 0 {\displaystyle \epsilon \to 0} ,

δ γ ( t ) = γ ( t ) σ ( t ) ∈ T γ ( t ) S O ( 3 ) . {\displaystyle \delta \gamma (t)=\gamma (t)\sigma (t)\in T_{\gamma (t)}\mathrm {SO} (3).}

See also

• D'Alembert principle
• Virtual work

References

1. Takhtajan, Leon A. (2017). "Part 1. Classical Mechanics". Classical Field Theory (PDF). Department of Mathematics, Stony Brook University, Stony Brook, NY. /wiki/Leon_Takhtajan ↩

2. Goldstein, H.; Poole, C. P.; Safko, J. L. (2001). Classical Mechanics (3rd ed.). Addison-Wesley. p. 16. ISBN 978-0-201-65702-9. 978-0-201-65702-9 ↩

3. Torby, Bruce (1984). "Energy Methods". Advanced Dynamics for Engineers. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. ISBN 0-03-063366-4. 0-03-063366-4 ↩

4. Takhtajan, Leon A. (2017). "Part 1. Classical Mechanics". Classical Field Theory (PDF). Department of Mathematics, Stony Brook University, Stony Brook, NY. /wiki/Leon_Takhtajan ↩